http://www.countbayesie.com/blog daily 1.0 2026-02-14 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1426017266650-Z8NZM2ME1GF7IZWSL9OI/Capture.PNG Explore Probability with Count Bayesie http://www.countbayesie.com/blog/2026/2/14/automating-tiktok-video-generation-with-open-models monthly 0.5 2026-02-16 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/08e452d9-1148-417f-b6b1-3e5850edf84d/last_frame.png Explore Probability with Count Bayesie - Building an AI Director for Continuous Video Generation - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0a11a63b-190a-4311-a6ab-c6a2f5219894/architecture.png Explore Probability with Count Bayesie - Building an AI Director for Continuous Video Generation - Make it stand out Thankfully AI built nearly all of this for me! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/8224f1a8-316a-452c-8b9f-bf98f26d2ece/kimi_code.png Explore Probability with Count Bayesie - Building an AI Director for Continuous Video Generation - Make it stand out The Kimi Code CLI is very intuitive for anyone coming from Claude Code https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/76ed666e-8e10-45da-95e9-0c0d7d5d3cb5/context.png Explore Probability with Count Bayesie - Building an AI Director for Continuous Video Generation - Make it stand out Consumed a ton of documentation and still have a ways to go before worrying about context! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/85a713b4-09c0-4072-aa85-6963b4868eb1/full_workflow.png Explore Probability with Count Bayesie - Building an AI Director for Continuous Video Generation - Make it stand out ComfyUI workflows are incredibly useful and can quickly grow quite complex. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/f1b23377-e7f0-453b-84c4-7bef984964ad/damn_fine_stable_diffusion.png Explore Probability with Count Bayesie - Building an AI Director for Continuous Video Generation - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2023/4/21/linear-diffusion monthly 0.5 2024-11-29 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/69239cd8-a90c-44fa-95e4-c821bd9f67b4/linear-diffusion-arch.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Visualization of the modifications to the standard diffusion model used by Linear Diffusion. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9f34eaf0-1513-46c0-a136-9363a73fe50f/linear_diffusion.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Digits generated from Linear Diffusion https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/f126a38f-cf09-4670-852c-a83771e1b96b/sandwich.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out How in the world could a model figure this out? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9b2ecd4f-f11e-4ad7-b84e-884556348174/five.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out A trained Linear Diffusion model was fed only the string “5” and output this image! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/29a7b777-dbca-4edb-a9b1-f95475f2abad/Linear+Diffusion+Images.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out The basic journey through a diffusion model https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/f5691dbd-dc7b-407a-8323-07b4e9a59912/Linear+Diffusion+Images%281%29.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out The major difference here is that we start with noise rather than a latent image, then “denoise the noise”. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/74e70b39-3e93-49a6-ae7f-a20da5dc6ab4/mnist_sample.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out The very familiar, MNIST digits. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/42c22103-8083-46d0-aac4-c23295970f74/mnist_latents.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Here we have created a lower dimensional (12x12 vs 28x28) embedding for our images. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/4a37c40a-701c-4b41-af10-e31d22434c14/mnist_recon.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out From 12x12 back to 28x28 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/ad32719f-3e67-41a0-a91b-66ba4a6c4d62/mnist_noisy_recon_1std.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Adding noise in the latent space makes these images blurry rather than “snowy” https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9c57ed5e-6827-4c34-b59a-f34e2191d7d0/mnist_noisy_recon_5std.png.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Our images are now hard to make out, but still look vaguely like digits. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0c0c8a93-b5a1-4a0b-a1bf-9a083b2cda67/mnist_just_noise.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out This is just pure noise, still looks surprisingly close to digits. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/89453d6b-5137-4906-84f7-abd896d97702/Linear+Diffusion+Images%283%29.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out A neural network would just learn these interactions if necessary, but we need to help our linear model out just a bit. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/2e14f7c3-8da5-4398-8ee9-e1ffb530d5a9/linear_diffusion_results.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Not too bad! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/44217036-394b-448e-913a-00225b1cb750/Kurt-MEAP-HI.png Explore Probability with Count Bayesie - Linear Diffusion: Building a Diffusion Model from linear Components - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2023/3/23/replacing-an-ab-test-with-gpt monthly 0.5 2023-03-27 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/4d7bd400-a30a-4f4b-b776-121d79ab0066/Selection_011.png Explore Probability with Count Bayesie - Replacing an A/B Test with GPT - Make it stand out A single experiment involves the comparison of multiple variants. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/c7607e78-5618-4459-9685-e900e552357b/Selection_013.png Explore Probability with Count Bayesie - Replacing an A/B Test with GPT - Make it stand out If we represent our labels honestly, they are probabilistic labels, which makes our problem a bit different than usual. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/488543b0-390d-495d-845f-ac43b8e2c3ca/A_B+testing+modeling+notes.jpg Explore Probability with Count Bayesie - Replacing an A/B Test with GPT - Make it stand out The basic flow of our model, the key insight is computing the difference of the vector representations. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/348a6793-6132-4b6b-9621-1fe4c1dbb800/Selection_015.png Explore Probability with Count Bayesie - Replacing an A/B Test with GPT - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9bb7b743-3966-4b8d-aa99-de7e3226adf6/prob_calibrations.png Explore Probability with Count Bayesie - Replacing an A/B Test with GPT - Make it stand out We want to see a “V” shape in our results because a 0.1 probability in winning reflects the same confidence as 0.9 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/8f7d75d5-5ab6-462d-a176-0b6fabdec8f1/patreon.png Explore Probability with Count Bayesie - Replacing an A/B Test with GPT - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2022/11/30/understanding-convolutions-in-probability-a-mad-science-perspective monthly 0.5 2022-12-01 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/dc75beb0-a913-4ab7-9603-c85cd5ce4eae/mad_science2.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out A bit too much data science ultimately leads to other career options. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/678e1da3-e696-485e-995d-79f92187c805/monster2.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out A good example of a Crab-Man monster https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/01478d65-da7f-442f-b4cf-14a2afa29822/crab_fight_2.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out Crab fights (and chainsaw fights) typically end up in the loss of limbs. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/df189a50-2727-4161-8837-58c0a8b44bc4/fig1.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out The distribution of human limbs remaining after our brutal chainsaw tournament. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/37184ce2-5191-4b2f-bdfe-46cd141022b6/fig2.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out I actually didn’t know crabs had 10 limbs, I thought only 6 legs and 2 claws, but it is in fact 10! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/8940efe9-b34b-4341-a535-049cd5f32370/fig3.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out Here we view the final output as the result of summing up all the individual distributions we’ve computed. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/a405fbd2-75d2-4c25-9523-1e5f853ff678/fig4.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out Starting at the output helps us get to the final formula for convolution. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/335f232c-42b4-410d-b337-3bb6da2d8219/video_screen_shot.png Explore Probability with Count Bayesie - Understanding Convolutions in Probability: A Mad-Science Perspective - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2022/10/19/what-is-the-probability-elon-musk-will-buy-twitter-using-the-volatility-smile-to-infer-market-probability-distributions monthly 0.5 2022-10-20 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/8c4f4b2d-45b1-4ddb-97af-65b971061027/fig_1_musk.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out If Musk buys TWTR the future price will be certain and not stochastic, this impacts the current price https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/a1ea1e28-51d0-48e9-adff-0d5d2bbc526c/iv_screenshot_2.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out Showing the implied volatility of a given option. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/af420d74-1ff3-4042-ac0b-554178a49274/fig_2_musk.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out The ‘volatility smile’ shows that real pricing disagrees with the BSM model. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/f8401c41-49a7-4e03-b716-5bd2a424d78f/fig_3_musk.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out Notice the difference at the lower strikes between the theoretical and the empirical. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/376579cd-8582-476f-9fcc-1b06603c74aa/fig_4_musk.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out Volatility Smile for Twitter options expiring Nov 4th 2022. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/8f3511e0-b117-4fcd-9f36-91fba8543471/fig_5_musk.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out Compared to the AMZN CDF this one is much stranger looking! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/79ed4b72-5b43-439a-8f5d-6bc2a445137c/Selection_007.png Explore Probability with Count Bayesie - What is the Probability Elon Musk will buy Twitter? Using the Volatility Smile to Infer Market Probability Distributions - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2022/10/9/using-censored-data-to-estimating-a-normal-distribution monthly 0.5 2022-10-10 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/bdb1d893-8428-4fc1-b1ac-47da4e021ba4/scientists_guessing.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Surreal scientists contemplating the weather https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0c113bf5-c5f1-4709-9e2a-a221e4f1bec5/answers_with_true_dist.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Visually we start to get a clue how we might use these direction answers to estimate our parameters. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/e0b1ef68-bed1-438f-b939-3fd53978568a/fig_2_twitter.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Our answers map to our initial estimates show us that we should move our estimated mean. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0564ffa3-d515-4817-aefa-660741c084f4/equation.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/38d306c7-ab1e-4835-8050-1dd0121978a4/fig_sup_twitter.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Comparing the observation that “the temperature will greater that 78.6” given two different parameterizations for our model. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/ed99f240-624a-42c1-95d7-a96e94352236/fig_sup2_twitter.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Here we use the regular CDF which is the equivalent of integrating the PDF to the specified value. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/c36ee16b-0306-4d8c-9853-4c18bf7dc696/fig_3_twitter.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out We got the mean perfect, but our variances seems way off (that’s a problem with our simulation, not the model itself) https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/fd113833-09a5-4753-b27d-98bcbeae9165/fig_4_twitter.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out As we can see, we know have variance in beliefs, and not all of them point towards the true mean. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/d7d61b20-34f7-452c-87ee-6d200ffa392d/fig_5_twitter.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Correctly simulating our problem gives us a much better model! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/3c40ed9e-943f-4180-b8b3-8e6413f69781/patreon_preview.png Explore Probability with Count Bayesie - Using Censored Data to Estimate a Normal Distribution - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2022/8/7/modern-portfolio-theory-and-optimization-with-jax monthly 0.5 2022-08-08 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0a42d48f-2803-4431-a2c2-9aa3cc10ce87/ticker_data.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out What our stock price time series data looks like. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/04c7803a-f048-42e2-a826-a90f15e9a36d/daily_prices.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Typically when we look at stocks we consider the daily prices over a period of time. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0b9a9f48-78e5-4353-8572-6635a56d2cac/log_return_stock_prices.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/a96d60e9-a098-4690-b23b-586f8996b8f6/aal_log_returns.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/136eccee-3e96-4781-861b-ac72aba2699b/modeling_returns.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Our normal model for our data fits reasonably well. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/adbd98df-95c5-4c13-801f-838c26e73dea/log_returns_sample.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Looking at a simulated sampling of our AAL log returns. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/02628f88-6350-4168-b521-af87bcb199e2/cumulative_log_returns.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out By looking at the cumulative log returns, or simulation looks a bit more like a real ticker chart. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/fe9dc607-6591-44b5-9956-f649d5141b8a/simulated_returns.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Not too much different in shape, but now we are looking at the actual, not log returns https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9a4562cc-8dd2-463d-92a8-6eee9b9b2f0c/simulated_stock_price.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Here we can see that we are ultimately simulating the actual price movements of the stock. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/a93f2952-3a1f-4bd1-983d-6ccf39d66eb1/multiple_simulations_of_AAL.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Here we can imagine all the possible paths AAL could have taken according to our model. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/a1d08be2-9779-44bb-9f70-375474259fd6/log_return_correlation_matrix.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Correlation matrix of our log returns. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/e95aeb38-f64c-47fd-b801-bd3a87d906f5/amzn_to_aapl_comparison.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out We can see visually the prices of these two stocks are very correlated. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/faf301b2-4be6-4973-bf65-f4d169c86bf9/two_univaritate_sample_path.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Something isn’t quite right with our model as the correlation of the real stocks is not reflected here. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/69ac92f8-fd47-4302-9e79-d52f03b30f52/multivariate_normal_sample_path.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Modeling with a multivariate normal does a much better job of capturing the correlation in price movements. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/85241f34-9dc0-4027-8ed2-9061b8916761/portfolio_returns_accutal.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Had we had this portfolio this is what our cumulative log returns would have look like. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/b43e3d5c-c52a-4bd9-9d13-21ef7b05439d/correlation_impacting_portfolio_returns.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out While the expectations of all of these are the same, there is a major difference in variance of the final outcome https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/b5ec8e8d-dacd-4ee4-93e8-eb4b8bb54401/random_weights.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/5c10a165-17fe-4b72-b8f9-3ae4d10b6844/initial_portfolio_weights.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Our initial weights give us a Sharpe ratio of 0.173 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/de22ee65-184b-46c9-b935-b2613bcd80b3/optimized_portfolio_weights.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Now we see a much better Sharpe ratio and very different portfolio. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/02587b34-b804-482c-9687-503bc61c2ba5/patreon_video.png Explore Probability with Count Bayesie - Modern Portfolio Theory and Optimization with JAX - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2022/2/19/how-to-read-the-news-like-a-bayesian monthly 0.5 2022-02-20 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/5c953a54-7193-404a-a468-581eff291138/article_image.png Explore Probability with Count Bayesie - How to Read the News like a Bayesian - Make it stand out You can learn a lot more from the news if you read it like a Bayesian! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/f09291b9-4149-401e-aac5-8c9e44e1d831/evidence_1.png Explore Probability with Count Bayesie - How to Read the News like a Bayesian - Make it stand out The actual evidence supporting the hypothesis of the article is fairly sparse. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/96a7ee3d-7e8b-4db1-87e8-89c0de1e4385/Screen+Shot+2022-02-19+at+2.02.19+PM.png Explore Probability with Count Bayesie - How to Read the News like a Bayesian - Make it stand out I’m not even sure this fact supports the hypothesis! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/c4839161-56bd-4530-a7a4-c64d317dd5ad/assuming_hypothesis.png Explore Probability with Count Bayesie - How to Read the News like a Bayesian - Make it stand out This next section is only interesting is you assume the hypothesis is correct. http://www.countbayesie.com/blog/2022/1/9/is-december-getting-warmer-modeling-weather-data-in-nj monthly 0.5 2022-01-09 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/3ab55fc4-4c07-44b5-bcc5-f48ade9d0f5a/weather_data_newark.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out Example of the data we’ll be using to model NJ weather patterns. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/abe94cfd-1e86-4122-bfb1-36353cfd5197/newark_raw_data.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out It’s not obvious to me that, visually, anything is happening here. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/150b0544-774f-4cf7-a778-ceb635eab4da/newark2.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out Looking at the end of the year it does look like December is a bit warmer. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/7009b5cf-2ebb-4025-88b6-6cd27d5fd484/year_model.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out Results from our simple linear model assuming a constant increase in temperature each year. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/8d564c3e-06ad-40fa-8103-faf89ba45284/year_model_with_obs.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out With our simple regression we can see that it is getting warmer each year. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/b5f04d63-ac90-4fcb-aec5-8fe09c8b946a/Screen+Shot+2022-01-09+at+12.59.08+PM.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out Now we have an average temperature for each month and the average yearly change across all months. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/49edd181-839b-469f-a5ef-b62480ce1baf/month_year_model_obs.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out We can see that this model is a much better fit of the data. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/45457d18-cad9-4a47-9f8b-15d61df7e608/Screen+Shot+2022-01-09+at+1.04.05+PM.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out Month, year and month*year interactions are able to capture how much each month is changing over time. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/0a7c6054-49c8-4a67-aa2c-73a710dd9d63/newark5.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out As we can see, December is warming much faster than the other months in Newark, NJ. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9163c76c-8aab-4e2c-bcea-396ad4da6f29/video_image.png Explore Probability with Count Bayesie - Is December getting warmer? Modeling weather data in NJ - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2021/9/30/the-logit-normal-a-ubitiqutious-but-strange-distribution monthly 0.5 2021-10-01 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1633053011983-WOUB1GWHNQ9VUCQP3GHP/logitnormal_fig_1.png Explore Probability with Count Bayesie - The Logit-Normal: A ubiquitous but strange distribution! - Make it stand out Samples from a standard normal and those samples transformed into a logit normal. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1633053262373-8H506C7VARUL223SGAS3/comparing_means_logit_normal.png Explore Probability with Count Bayesie - The Logit-Normal: A ubiquitous but strange distribution! - Make it stand out The logistic of the mean of the normal distribution is not necessarily the mean of the logit-normal corresponding to it. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1633054418560-V43NOOAKUISH5AJ1YJ2B/statsmodels_results.png Explore Probability with Count Bayesie - The Logit-Normal: A ubiquitous but strange distribution! - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1633099183709-IJDPZAB0A773F2TMEBV9/Screen+Shot+2021-10-01+at+10.37.00+AM.png Explore Probability with Count Bayesie - The Logit-Normal: A ubiquitous but strange distribution! - Make it stand out Whatever it is, the way you tell your story online can make all the difference. http://www.countbayesie.com/blog/2021/4/27/technically-wrong-when-bayesian-and-frequentist-methods-differ monthly 0.5 2021-04-28 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1619576936801-3QGIY23D56704IKZSQA0/slug_gun.png Explore Probability with Count Bayesie - Technically Wrong: When Bayesian and Frequentist methods differ Inter-dimensional travel presents some fun probability problems! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1619578339092-VQJ5FF5KYYZ58WRZ6NQ0/standard_error_of_proportion.png Explore Probability with Count Bayesie - Technically Wrong: When Bayesian and Frequentist methods differ When we plot out these beliefs we find they are impossible https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1619579035594-707IROAE95M4416SCNXO/beta_likelihood.png Explore Probability with Count Bayesie - Technically Wrong: When Bayesian and Frequentist methods differ There are no impossible values now that we are representing this correctly https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1619579235143-DV6PRRVPKZ4D1MLEJBXB/beta_with_prior.png Explore Probability with Count Bayesie - Technically Wrong: When Bayesian and Frequentist methods differ The correct representation of our beliefs given the information we have. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1619580296616-VMV6RJ921MJU6TNDPKUJ/diminishing_difference.png Explore Probability with Count Bayesie - Technically Wrong: When Bayesian and Frequentist methods differ Very quickly the difference between “technically wrong” and not disappears. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1619625680166-KKBQS43Z8PR16U18VK1T/will_kurt_patreon.png Explore Probability with Count Bayesie - Technically Wrong: When Bayesian and Frequentist methods differ http://www.countbayesie.com/blog/2021/1/4/inference-and-prediction-part-2-statistics monthly 0.5 2021-01-05 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609861988401-RNPX4UF2SJO5OIIN4S8J/perceptron_as_logistic_regression.jpg Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics Even though the perceptron can feel quite different than logistic regression, our implementation is identical https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609796600854-NZM3UW6VTCYX9D98SETG/coefficient_values.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics These values are in log-odds which can be tricky to interpret at first https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609798083683-KARSAXW06MHF4XVHHF5U/adding_our_standard_error.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics That was a bit of work, but now all the rest of statistics falls out of this almost trivially! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609798592113-J7PNOEY6OY6TMGXWAPLJ/model_statistics.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics Here we have essentially replicated the results we would have gotten from a tool like statsmodels. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609798947616-ZM5XFK9U2BQHF7HQK2F7/visualizing_parameter_estimation.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics We can visualize everything we have learned about our weights https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609800544117-QFF4R7RQVN64FY72FCDU/experiment_results_no_causality.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics Looking at our test results it doesn’t look too good for our new rewriting tool https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609801097020-WJ4L6QIZG5561IV1JJQ7/test-results-with-causality.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics Now that we are correctly modeling our problem we see the results are significant! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1609866211875-5JAHHJOOYYRPJN37PQXI/support_on_patreon.png Explore Probability with Count Bayesie - Inference and Prediction Part 2: Statistics - Support on Patreon Support my writing on Patreon and gain access to the source code and video commentary for this article as well as access to much more of my writing! http://www.countbayesie.com/blog/2020/12/15/inference-and-prediction-part-1-machine-learning monthly 0.5 2020-12-24 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1608055807378-GH50699R0VJAUOPIFATB/prediction_vs_inference.jpg Explore Probability with Count Bayesie - Inference and Prediction Part 1: Machine Learning In the series we’ll ultimately see who these two are more connected than they are typically treated https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1608056870468-9SJI1F24HIJFTBDP7X01/looking_at_data.png Explore Probability with Count Bayesie - Inference and Prediction Part 1: Machine Learning https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1608057438750-M3CXFF927UBMZL76YJWV/perceptron.png Explore Probability with Count Bayesie - Inference and Prediction Part 1: Machine Learning The likely familiar image of the simplest neural network. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1608064623139-Q8L096KBN9LWOU1E8Q7S/support_on_patreon.jpg Explore Probability with Count Bayesie - Inference and Prediction Part 1: Machine Learning http://www.countbayesie.com/blog/2020/11/5/survivorship-bias-in-house-hunting-a-practical-modeling-example-using-jax monthly 0.5 2020-11-05 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1604603124648-J76FN65DANGZGV2VJ5ZQ/survivorship_bias_plane.png Explore Probability with Count Bayesie - Survivorship bias in House Hunting: A practical modeling example using JAX It is legally required to show this image of a plane whenever you write about survivorship bias https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1604603590535-UBTOMATCRMUFQ8MR76YK/visualizing_our_mental_model.png Explore Probability with Count Bayesie - Survivorship bias in House Hunting: A practical modeling example using JAX The first step in the modeling process is to understand you mental model of how the process works. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1604604056858-87UX7IX02R6JTH9O67QN/exponential_distribution_in_practice.png Explore Probability with Count Bayesie - Survivorship bias in House Hunting: A practical modeling example using JAX Individual listings can be modeled as observations of an exponentially distributed random variable. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1604608421036-R34S21INDALHT7E5WKXB/final_equation.png Explore Probability with Count Bayesie - Survivorship bias in House Hunting: A practical modeling example using JAX https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1604604561329-145T52VD1W9Z3YT45IXU/our_mathematical_model_of_mls.png Explore Probability with Count Bayesie - Survivorship bias in House Hunting: A practical modeling example using JAX Transforming our mental model into a mathematical one allows us to quantify our beliefs. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1604609655462-BZWRWQ3RQQJ00GE0MXO6/support_on_patreon.png Explore Probability with Count Bayesie - Survivorship bias in House Hunting: A practical modeling example using JAX http://www.countbayesie.com/blog/2020/9/26/learn-thompson-sampling-by-building-an-ad-auction monthly 0.5 2020-09-28 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601155197091-OUHIGTAWWVZLYWVSW55T/thompson_sampling_animated.gif Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! By the end of this post you’ll be able to implement the optimization seen here! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601155539134-5B3ABS5AKRCEIKXT66Q8/aution_bids.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601155737028-QPLA83TW8V8HWJAKLX5A/ad_auction_with_ctr.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601155861361-EYCM42XFDTSN1TLSBKI5/ad_auction_with_unknown_values.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601158640795-NJNWK9FM17J6KQ7091PF/beta_distributed_likelihood_estimate.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! Here we can see how strongly we believe in various CTRs for BuyMyStuff (without prior information). https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601158752215-UQAU01ORH5N8NDV6E4DV/estimating_expected_value.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! Our distribution of expected values is just our previous Beta distribution scaled by the bid price. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601159193691-829V0VJHX3CX2Z0ZGXXM/historical_ad_ctr.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! Having a history of previous ad performance allows us to estimate what future performance might look like. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601159623724-H27HIO3CME5ZBN772Y4O/beta_prior_learned_from_data.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! Now we have a distribution that we can expect future CTRs that describes our belief in how likely a given CTR is. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601172699845-KCYIZLXUT9ADH0VAOC4N/distributions_of_expected_value.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! At this stage two of the three ads rely exclusively on our prior distribution. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601173241631-MHPFZMKFQG2QZGN0UIF3/comparing_random_variables.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! Subtracting samples of one random variable from another lets us see what the distribution of their difference looks like. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601173670203-C73JYDZUJ4IN2M9F2ZMB/table_probability_of_improvement.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601174028729-O3UA4XILF8MA8D8TUYGJ/updated_probability_table.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601174143006-JLK1K830S61X5ZKJP2TY/probabilities_and_weights.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601174366296-11ITVK8R0XBBKDU8BAPM/adding_data.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601174506745-ICF33PPN3DT2DBJL5ZC6/updated_beliefs_after_500_more_impressions.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! Even just 500 more impressions changed our belief in how good the CTR for CommodityFetishismLCC is! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601174675997-BRBJM8XXK8TYTMQOKJFN/updated_weights.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601174911221-M978ITJETDZ702JOB1VC/experiment_over_time.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! The sampler quickly learns that the only two ads are worth showing, and since they are pretty similar the end up being shown about 50/50 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601309229503-VBYG8WEUBLA5RFXLMPBN/patreon_preview.png Explore Probability with Count Bayesie - Learn Thompson Sampling by Building an Ad Auction! http://www.countbayesie.com/blog/2020/8/16/why-bayesian-stats-need-monte-carlo-methods monthly 0.5 2020-09-27 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1597612396814-7EUDWRI731QEHCCQ3GFE/approximating_difference.jpg Explore Probability with Count Bayesie - Why Bayesian Stats Needs Monte-Carlo Methods If we look at the distribution of the difference of our samples we can see that it is approximating the PDF of the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1597612495029-6MK8C4MQ4B4WNGMWPQAU/approximating_the_integral.png Explore Probability with Count Bayesie - Why Bayesian Stats Needs Monte-Carlo Methods Our code hides much of the complexity, but we’re really approximating this integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1597612806633-HI3P5IRT4HQ3AEL2GWLN/normal_approximates_beta.png Explore Probability with Count Bayesie - Why Bayesian Stats Needs Monte-Carlo Methods In many cases the Normal distribution provides a very nice approximation of the Beta distribution. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1597613238943-CKVXBB32SIT50TQNJZ05/comparing_our_normal_approximation.jpg Explore Probability with Count Bayesie - Why Bayesian Stats Needs Monte-Carlo Methods As we can see, with a low value for our total observations the Normal approximation is not that great https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1597613339688-IWHC6TVO5R5LQBQRWK1B/difference_approximation.jpg Explore Probability with Count Bayesie - Why Bayesian Stats Needs Monte-Carlo Methods Our approximation for the PDF of the difference is pretty bad, but we only really care about P(B>A) for this example. http://www.countbayesie.com/blog/2020/6/5/on-audre-lordes-the-masters-tools monthly 0.5 2020-06-05 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1591389793415-EG8E59KPNMAOPQ7VNJ2S/audre_lorde Explore Probability with Count Bayesie - On Audre Lorde's "The Master's Tools..." “The Master’s Tools Will Never Dismantle the Master’s House” http://www.countbayesie.com/blog/2019/12/1/probability-and-statistics-in-90-minutes monthly 0.5 2019-12-02 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1575251518097-ZSV5EB2DUH1NTH1B349Y/mind_1.jpg Explore Probability with Count Bayesie - Probability and Statistics in 90 minutes Very excited I got to make a few points about the mind projection fallacy! http://www.countbayesie.com/blog/2019/8/14/prior-probability-in-logistic-regression monthly 0.5 2019-08-20 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1566309281986-RC9VJ4PQI9ATVY684ZDN/Odds-version-logistic.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1565830258604-P1O4OIJUGNTU7ZKK475D/symptoms_table.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1565912754976-0GU5BZ5TUYTGIXDQSJJG/logistic-mode-original-prior-predictions.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression The distribution of the prediction for our logistic model given the prior in the training data. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1565917160618-HTIRGTB02E4O5UK7JBUI/observed-vs-empirical-original-prior.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression Because it is trained on the wrong prior, the model drastically overestimates the empirical rate of Bayes’ Pox. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1565917693943-6CULFOCH3NFUCVR85NW2/distribution-corrected-prior.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression Distribution of predictions after updating our prior to the correct one. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1565917803262-EZ0C39JKTZMWWLP2VKTJ/updated-prior-vs-empirical-rates.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression With our updated prior, our model’s beliefs are much closer to the empirical ratio of Bayes’ Box to Frequentist Fever. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Prior Probability in Logistic Regression Bayesian Statistics the Fun Way is out now! http://www.countbayesie.com/blog/2019/6/12/logistic-regression-from-bayes-theorem monthly 0.5 2019-08-20 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1560340419704-W6NNEZH31GR8RSW2P6BM/logistic-regression-s-shape.png Explore Probability with Count Bayesie - Logistic Regression from Bayes' Theorem Logistic regression is often described as an s-shaped function that squishes values to 0 or 1 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1560340658262-GM89RPCDGJW1RILA6VVL/pour-over-coffee-data.jpg Explore Probability with Count Bayesie - Logistic Regression from Bayes' Theorem Making a good cup of coffee requires a surprising amount of data! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1560344484316-BDGIFL58XTFIHVPT52EL/logistic-regression-diagram.png Explore Probability with Count Bayesie - Logistic Regression from Bayes' Theorem This diagram show how we transformed Bayes’ Theorem into Logistic Regression https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1560425703102-KTU28U4RVSEV2Q67KB5R/bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Logistic Regression from Bayes' Theorem http://www.countbayesie.com/blog/2019/1/30/a-deeper-look-at-mean-squared-error monthly 0.5 2021-01-08 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1548902692177-ZN0OGBWVNTT9KSTSZ77N/simulating_data.png Explore Probability with Count Bayesie - A Deeper look at Mean Squared Error https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1548902853419-L4VPXTAFMOL7NTO7DRAT/high_bias_model.png Explore Probability with Count Bayesie - A Deeper look at Mean Squared Error Our high bias model is consistently off in its estimate. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1548944682033-93TTNQVTYPAEYEWX494V/high_variance_no_bias.png Explore Probability with Count Bayesie - A Deeper look at Mean Squared Error When we use the mean to predict the data, nearly all of our error is cause by the variance in our model. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1548944865089-MYEIBM2VX7WTAZ88LWI9/linear_model_no_variance_no_bias.png Explore Probability with Count Bayesie - A Deeper look at Mean Squared Error When we use the actual model that generates the data we get 0 variance and 0 bias, but still have error due to uncertainty. http://www.countbayesie.com/blog/2017/5/9/kullback-leibler-divergence-explained monthly 0.5 2020-09-27 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494401025139-ODE7CP2043TS1CO9MQSN/biting-worms.jpg Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained Space worms and KL divergence!!! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494396254926-SWY85XI22T1M4Q1ZEPCO/empirical-distribution-of-data.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained The empirical probability distribution of the data collected https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494397124633-IC3E9LB2IML2JXHJQVFQ/uniform-approximation.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained Our uniform approximation wipes out any nuance in our data https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494397201106-RKMWRQ4GUNY1ZTKCM1S0/binomial-approximation.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained Our binomial approximation has more subtlety, but doesn't perfectly model our data either https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494397358518-09MZORGNU1VQK4EBQ1ZL/all-approximations.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained Compared with the original data, it's clear that both approximations are limited. How can we choose which one to use? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494397981106-SRB7017RCPXJGTZY4WPY/finding-the-optimal-parameter.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained It turns out we did choose the correct approach for finding the best Binomial distribution to model our data https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494398217552-EC6J4VA3MU2PZKI4UKKM/optimizing-our-ad-hoc-function.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained By finding the minimum for KL Divergence as we change our parameter we can find the optimal value for p. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1494398337695-19C7OTCF6UBAPNQW4CLU/optimal-value-for-ad-hoc.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained Our ad hoc model ends up being optimal very close to the uniform distribution https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563718580421-2MC2ICMWIQU76F8M0NN5/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Kullback-Leibler Divergence Explained Bayesian Statistics the Fun Way is out now! http://www.countbayesie.com/blog/2016/5/1/a-guide-to-bayesian-statistics monthly 0.5 2019-07-21 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563717711212-B6LDKYI2Z1LIAY7MUA6I/BayesianStats.png Explore Probability with Count Bayesie - A Guide to Bayesian Statistics https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462139479914-7A8QSM5ERNHOKV3BN72H/image-asset.jpeg Explore Probability with Count Bayesie - A Guide to Bayesian Statistics This is a great book to get you excited about Bayesian Stats! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462139647115-HYZ6ND4I21I4P2SSINHW/bayes-lego.jpg Explore Probability with Count Bayesie - A Guide to Bayesian Statistics Bayes' Theorem is much easier to understand visually https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462139849926-YO8U8INKDSR08OHNLX20/han-solo-priors Explore Probability with Count Bayesie - A Guide to Bayesian Statistics "Never tell me the Odds!" https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462139983227-5YS1Y4IAI1MIIAG226PD/applying-bayes-theorem Explore Probability with Count Bayesie - A Guide to Bayesian Statistics Let's put Bayes' theorem to use! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462140180087-JFRA9FV5QLRARJ0XN9YQ/bayes-factor-twilight-zone Explore Probability with Count Bayesie - A Guide to Bayesian Statistics That glimmer in his eye is from Bayesian Statistics! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462140319882-5EKMU2K8FRYSHW2TIS5H/no-p-value Explore Probability with Count Bayesie - A Guide to Bayesian Statistics Just say no to p-values https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462140459935-6BLUB6CS5YX8EDBDW8DP/probability-theory-the-logic-of-science Explore Probability with Count Bayesie - A Guide to Bayesian Statistics You could easily spend a lifetime pondering this Jaynes' masterpiece. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462141258162-NYYB3J8TUNGAQEL95K66/parameter-estimation Explore Probability with Count Bayesie - A Guide to Bayesian Statistics Understanding CDF plots is very important https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462141354662-2IKTYT1HZRZVTEVCORRD/beta-priors Explore Probability with Count Bayesie - A Guide to Bayesian Statistics Three priors from the same data https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462141430810-8XP0ZSOFDGFOZOOVO6PG/rejection-sampling Explore Probability with Count Bayesie - A Guide to Bayesian Statistics Rejection sampling looks pretty fun! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462141540066-7MVCPYJP42MONDN4QG1A/bayesian-ab-test Explore Probability with Count Bayesie - A Guide to Bayesian Statistics The beauty of two competing posteriors! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462141756528-95M2HWY9BZUOFQR1YIKN/doing-bayesian-data-analysis Explore Probability with Count Bayesie - A Guide to Bayesian Statistics The next logic step after the posts here https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462141840752-19BP893UN190RWIOU626/bayesian-data-analysis-gelman Explore Probability with Count Bayesie - A Guide to Bayesian Statistics This book is full of wonderful, practical examples http://www.countbayesie.com/blog/2016/3/16/bayesian-reasoning-in-the-twilight-zone monthly 0.5 2016-05-08 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1458155681697-B585KYI798X56TES5A3W/gathering-evidence.png Explore Probability with Count Bayesie - Bayesian Reasoning in The Twilight Zone! Good thing we have lots of pennies left from all those coin tossing experiments! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1458155731519-NCYC651RONI95V1YF1NQ/bayesian-reasoning-devil.png Explore Probability with Count Bayesie - Bayesian Reasoning in The Twilight Zone! Behold! The mystic art of Bayesian Reasoning! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1458156064406-SNN7OSXXHJ5XGWHL0Y8E/bayes-factor-interpretations.png Explore Probability with Count Bayesie - Bayesian Reasoning in The Twilight Zone! Common interpretations for Bayes' Factor https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1458155795887-P7SCBWSE69OINYTEL75Z/prior-card.png Explore Probability with Count Bayesie - Bayesian Reasoning in The Twilight Zone! What is your prior belief in the Mystic Seer? http://www.countbayesie.com/blog/2016/1/22/why-you-should-believe-your-friends-claims-about-food-allergies monthly 0.5 2016-05-08 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1453491431585-FNRNTY2BECRMEA3XE999/image-asset.jpeg Explore Probability with Count Bayesie - Use Bayes' Theorem to Investigate Food Allergies ...actually tastes really good! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1453490707056-2PF9LAZEDO3GBJYWYU00/image-asset.jpeg Explore Probability with Count Bayesie - Use Bayes' Theorem to Investigate Food Allergies What all the terms in Bayes' Theorem mean for our example http://www.countbayesie.com/blog/2015/11/21/the-black-friday-puzzle-understanding-markov-chains monthly 0.5 2020-09-06 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1448234215635-SCG7EGM9M9Y0UMUMY011/probabilistic+black+friday.png Explore Probability with Count Bayesie - Understanding Markov Chains with the Black Friday Puzzle Can probability save us from our madness? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1448137053064-YBIVIR23ZM76W85L77DW/books+simple+transition+graph.png Explore Probability with Count Bayesie - Understanding Markov Chains with the Black Friday Puzzle Given that you are in the Books section, where will you be next? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1448137070997-ZQU06LAMPY1RJ9HYR30Y/markov+chain+graph+all.png Explore Probability with Count Bayesie - Understanding Markov Chains with the Black Friday Puzzle The full Markov Chain representation of our problem https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1448137205595-I4CLYNYRBM6FEJ8634YO/markov+chain+transition+table.png Explore Probability with Count Bayesie - Understanding Markov Chains with the Black Friday Puzzle Transition graph in table form http://www.countbayesie.com/blog/2015/10/13/the-toy-collectors-puzzle monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1444714281286-XYNOKNB5JEZ4V253LEMP/image-asset.jpeg Explore Probability with Count Bayesie - The Toy Collector's Puzzle https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1444714533350-W3KQSKPFXJJ3NFNDSQG0/image-asset.jpeg Explore Probability with Count Bayesie - The Toy Collector's Puzzle https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1444714973769-VCPC1KPX6CQC6DIWIF2Z/image-asset.png Explore Probability with Count Bayesie - The Toy Collector's Puzzle https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1444714713607-64LAY3KPL26UI458L4KU/image-asset.jpeg Explore Probability with Count Bayesie - The Toy Collector's Puzzle http://www.countbayesie.com/blog/2015/9/29/curious-tea-parties-and-the-lebesgue-integral-part-2 monthly 0.5 2016-04-28 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1443509736007-N0IT7AD6WIM6M9LEUYYM/dist-needing-lebesgue.png Explore Probability with Count Bayesie - The Lebesgue Integral: a Curious Tea Party (Part 2) A distribution that is both discrete and continous https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1443510088340-GMKVJUSNU9FZ6S14R0FV/riemann-towers.png Explore Probability with Count Bayesie - The Lebesgue Integral: a Curious Tea Party (Part 2) The Riemann Integral can be visualized as building towers under the curve https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1443509937783-C840L6ILY2SBBCE97XNO/image-asset.png Explore Probability with Count Bayesie - The Lebesgue Integral: a Curious Tea Party (Part 2) https://en.wikipedia.org/wiki/Probability_distribution#/media/File:Dice_Distribution_(bar).svg https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1443510031066-23YA075SLXKJP0LTWTCE/image-asset.jpeg Explore Probability with Count Bayesie - The Lebesgue Integral: a Curious Tea Party (Part 2) Sometimes pretending discrete distributions are continuous works.... https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1443510234644-QH7D35A10OB8R4VUBMXX/image-asset.png Explore Probability with Count Bayesie - The Lebesgue Integral: a Curious Tea Party (Part 2) "Ever get the feeling you've been cheated?" http://www.countbayesie.com/blog/2015/9/15/curious-tea-parties-and-the-lebesgue-integral-part-1 monthly 0.5 2020-11-16 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1442294135723-EZ8W4TAG5OPJGY8APWZE/image-asset.jpeg Explore Probability with Count Bayesie - The Lebesgue Integral: A Curious Tea Party (Part 1) Sometimes Random Variables can be both discrete and continuous. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1442297865864-DL95SHCU72ZQFJUD2K13/image-asset.jpeg Explore Probability with Count Bayesie - The Lebesgue Integral: A Curious Tea Party (Part 1) Like our data, our distribution has both discrete and continuous components http://www.countbayesie.com/blog/2015/8/30/picture-guide-to-probability-spaces monthly 0.5 2016-04-28 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1440988688655-9ZKJQPS25PWSQ73GBQZG/probability-triplet.png Explore Probability with Count Bayesie - Probability Spaces: An Illustrated Introduction A simple triplet describing a probability space. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1440989105315-GP71CCGUXF7OLHFUODGD/omega-image.png Explore Probability with Count Bayesie - Probability Spaces: An Illustrated Introduction Omega in our Probability Space represents all possibilities you can imagine. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1440989855627-A8NVLC81NY3XGZI4KDJU/sigma-algebra.png Explore Probability with Count Bayesie - Probability Spaces: An Illustrated Introduction The Sigma-algebra can be understood as all the valid questions you can ask about Omega. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1440989905977-FYFGV0C8B5F9NNU4V33Y/sensible-questions.png Explore Probability with Count Bayesie - Probability Spaces: An Illustrated Introduction Not every question is sensible, our Sigma-algebra ensures that we only ask sensilbe questions. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1440990011128-W09M3AQWJX600ZNVGPO5/image-asset.jpeg Explore Probability with Count Bayesie - Probability Spaces: An Illustrated Introduction P is the set of answers to all the questions in our sigma-algebra. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1440990271572-VT4I06BDX3SHJVVDY1XL/probability-triplet-illustration.png Explore Probability with Count Bayesie - Probability Spaces: An Illustrated Introduction A much better way to visualize our Probability triplet! http://www.countbayesie.com/blog/2015/8/17/a-very-brief-and-non-mathematical-introduction-to-measure-theory-for-probability monthly 0.5 2016-05-07 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1439872564554-67292JSJIIX2LLMOKMD1/image-asset.jpeg Explore Probability with Count Bayesie - Measure Theory for Probability: A Very Brief Introduction Though physically absurd this is one way a non-measurable object could behave. http://www.countbayesie.com/blog/2015/8/3/the-riemann-integral monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638045818-3S5QFHPMQWNCVMA91ZJA/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638076293-YC4CUZ4PW1PAL6WJFBGR/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638115053-R22VAA7SN7T9JF3FKYWT/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638143481-NV2TVHLL9XHUMY2F5H2O/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638170186-F4GOSWZN6EEG5AKN8BSU/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638200710-6EOSIUKBLPADM7UZS514/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638231225-CNBHHK50ETET5QC9OLP1/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1438638269397-0336WJTEQOC3Q7UA1OYX/image-asset.png Explore Probability with Count Bayesie - The Riemann Integral http://www.countbayesie.com/blog/2015/7/19/fundamental-theorem-of-calculus monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1437344543453-5TS0L4ZQUAPIHT1HYLX7/image-asset.png Explore Probability with Count Bayesie - Programming the Fundamental Theorem of Calculus https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1437344616761-YXXOG09VGQPMYW1MG8NC/image-asset.png Explore Probability with Count Bayesie - Programming the Fundamental Theorem of Calculus https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1437344768371-H1LO9TZFBG5GHX1GJUIW/image-asset.png Explore Probability with Count Bayesie - Programming the Fundamental Theorem of Calculus https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1437345734838-PTAPL08MYG2TMQ11VIMU/image-asset.png Explore Probability with Count Bayesie - Programming the Fundamental Theorem of Calculus https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1437345796724-6KX54Z43B68JLCRT5KPS/image-asset.png Explore Probability with Count Bayesie - Programming the Fundamental Theorem of Calculus https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1437344921821-HLM34HQ2XSC1A1HEB38G/image-asset.png Explore Probability with Count Bayesie - Programming the Fundamental Theorem of Calculus http://www.countbayesie.com/blog/2015/7/6/how-big-is-the-difference-between-being-90-and-99-certain monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1436169541991-Y3C7VXZ81IY4UFSL0AE9/image-asset.png Explore Probability with Count Bayesie - How Big is the Difference Between Being 90% and 99% Certain? A poorly drawn Homunculus https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1436169609758-RSZC8SQMS16XEB0OGK5X/image-asset.jpeg Explore Probability with Count Bayesie - How Big is the Difference Between Being 90% and 99% Certain? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1436169646160-XJ748AM3N6Q1Q3BTZ9LU/image-asset.jpeg Explore Probability with Count Bayesie - How Big is the Difference Between Being 90% and 99% Certain? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1436170103021-ZI8E26UBYGXKQDK42FRB/image-asset.jpeg Explore Probability with Count Bayesie - How Big is the Difference Between Being 90% and 99% Certain? http://www.countbayesie.com/blog/2015/6/20/tricky-priors-and-rejection-sampling monthly 0.5 2016-05-08 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1434842095719-LHLPTF53456K2KQDX9P6/image-asset.png Explore Probability with Count Bayesie - Rejection Sampling and Tricky Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1434842128149-9UWUA2922WPX9X3SZKVW/image-asset.png Explore Probability with Count Bayesie - Rejection Sampling and Tricky Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1434842242272-JGV332VTZILQUX7J1MHC/image-asset.png Explore Probability with Count Bayesie - Rejection Sampling and Tricky Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1434842414819-KE2RLLSVKRG8PLKWTZFJ/image-asset.png Explore Probability with Count Bayesie - Rejection Sampling and Tricky Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1434842749997-SFRR5L4PM7XGQ02BGIOD/image-asset.png Explore Probability with Count Bayesie - Rejection Sampling and Tricky Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1434842837150-LOV123WRSNH723DK8DB4/image-asset.png Explore Probability with Count Bayesie - Rejection Sampling and Tricky Priors http://www.countbayesie.com/blog/2015/4/23/why-so-square-jensens-inequality-and-moments-of-a-random-variable monthly 0.5 2020-06-11 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429766476106-PHPSBGHX2G6FRPICGU2Y/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429766625021-28YAVJ0ZNTZM7PKIC3JF/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429766670041-FCL7WWL9YGOFYPU56PV9/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429766737241-BDJUMWE7ODINOFHW30VE/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429766802371-GLTDP0LILEG4AX89R2IQ/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1433869029184-CVKMWO9XDXSWMRD8ITWK/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429766898031-GXTCDI3FMVIY9UVD8BFA/image-asset.png Explore Probability with Count Bayesie - Moments of a Random Variable Explained http://www.countbayesie.com/blog/2015/5/24/writing-finnegans-wake-with-a-recurrent-neural-net monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1432513160022-BJINTN4T4P0LXHSVUTGB/image-asset.jpeg Explore Probability with Count Bayesie - Writing Finnegans Wake with a Recurrent Neural Net http://www.countbayesie.com/blog/2015/5/20/small-list-of-gigantic-books-for-summer-reading monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1432154421016-DAC2TLTE0MN644T5G7T9/image-asset.png Explore Probability with Count Bayesie - Small List of Gigantic Books for Summer Reading! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1432154222343-7RFO4S1AE4KQH3TFIH5I/image-asset.jpeg Explore Probability with Count Bayesie - Small List of Gigantic Books for Summer Reading! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1432154264399-FLOSYCD87YHP68SVHKMZ/image-asset.png Explore Probability with Count Bayesie - Small List of Gigantic Books for Summer Reading! http://www.countbayesie.com/blog/2015/4/25/bayesian-ab-testing monthly 0.5 2019-08-21 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429945207684-65WQF89EU25SL6B00G0K/weak-prior-probability.jpg Explore Probability with Count Bayesie - Bayesian A/B Testing: A Hypothesis Test that Makes Sense Different Beta distributions can represent varying strengths in belief in known priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429945600399-SR4VMWE2RBJMWDD49H6Q/image-asset.png Explore Probability with Count Bayesie - Bayesian A/B Testing: A Hypothesis Test that Makes Sense Our observed evidence https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429945260198-XPTE4P1I727SWNMJV4YF/comparing-beta-distributions.jpg Explore Probability with Count Bayesie - Bayesian A/B Testing: A Hypothesis Test that Makes Sense The overlap between the distributions is what we care about. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429945301167-I7CW7RJQ79RGZ2DBNU8H/analyzing-simulation-results.jpg Explore Probability with Count Bayesie - Bayesian A/B Testing: A Hypothesis Test that Makes Sense This histogram describes all the possible differences between A and B https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1429945373161-RY3K9JZ0HGAEOI2F32NS/image-asset.png Explore Probability with Count Bayesie - Bayesian A/B Testing: A Hypothesis Test that Makes Sense The line here represents the median improvement seen in the simulation https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Bayesian A/B Testing: A Hypothesis Test that Makes Sense http://www.countbayesie.com/blog/2015/2/27/building-a-bayesian-voight-kampff-test monthly 0.5 2016-05-08 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425082959993-ODQ6S8YL5DMY9WKAQI0L/observing-evidence.png Explore Probability with Count Bayesie - Using Bayes' Factor to Build a Voight-Kampff Test! Observing Evidence is an important part of Bayesian Reasoning. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425083707452-3K4EC8WNG9YY4SK2MJ3T/question-p-values.png Explore Probability with Count Bayesie - Using Bayes' Factor to Build a Voight-Kampff Test! You aren't going to use a p-value for that are you? https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1431216330959-CRWF4YSP1ZZWY0P3X3MJ/image-asset.png Explore Probability with Count Bayesie - Using Bayes' Factor to Build a Voight-Kampff Test! Evidence for five questions. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1431216418879-VINZXCDIHYEPD55UTCVD/image-asset.png Explore Probability with Count Bayesie - Using Bayes' Factor to Build a Voight-Kampff Test! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1431216463541-M19PH7SCSRCXT1KN1O97/image-asset.png Explore Probability with Count Bayesie - Using Bayes' Factor to Build a Voight-Kampff Test! http://www.countbayesie.com/blog/2015/4/4/parameter-estimation-adding-bayesian-priors monthly 0.5 2020-09-24 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428170171080-X65QTCDJ61XZ9FV1VAMK/beta-representing-conversion Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation With a small sample, our Beta distribution covers a wide range of possibilities https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428170217494-ZTP32QGS53WRNMSQRDU7/beta-cdf Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation Using the CDF allows for easier estimation of interesting quantiles and confidence intervals https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428176407757-XWV9D801RCJMHMFJOU59/possible-data-backed-priors Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation Different Beta distributions can be used to represent different strengths of belief in the same data. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428170294697-Q3ZE2PPYHH5NOMLKM1IN/adding-prior-to-parameter-estimate Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation Adding a Prior Probability can drastically improve estimate from small sample sizes. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428170408000-3Q5A0IPBZ3WK7Z9QEVCK/comparing-estimates-with-more-data Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation The more data we collect the less our prior influences of final beliefs https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428170476813-CZ29X1TBMTJH59T33QGR/converging-estimates Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation As we can see, more data means the eventual convergence of different priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Bayesian Priors for Parameter Estimation http://www.countbayesie.com/blog/2015/4/4/parameter-estimation-the-pdf-cdf-and-quantile-function monthly 0.5 2019-08-21 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428132438353-XBH68PBMIY4HJMTPRV0Z/beta-distribution Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function The Beta distribution is very useful for estimating unknown probabilities https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428132512961-5358TRZASSUBU2J61W6P/cumulative-distribution-function Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function The Cumulative Distribution Function can be used to quickly estimate precentiles https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428132557621-LW9GOUUUKWC9ZHKDD1LR/estimating-the-median Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function The CDF allows for quick and accurate estimates of the median (and other quantiles!) https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428132613506-EH3W3P5L6KRSZD1C7H7Z/visual-integration Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function Integration has never been so easy! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428132722094-FFJ6O1WOTCBJIFHIR7S1/cdf-confidence-interval Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function CDFs make estimating confidence intervals accuracy much easier than PDFs https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1428132844145-DBC4TFT3AGWQEBGGI8NR/quantile-function Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function Simply turn your CDF sideways and you get the Quantile function https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Parameter Estimation - The PDF, CDF and Quantile Function http://www.countbayesie.com/blog/2015/3/19/expectation-and-variance-from-high-school-to-grad-school monthly 0.5 2019-01-22 http://www.countbayesie.com/blog/2015/3/17/interrogating-probability-distributions monthly 0.5 2019-06-17 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1426570502246-W29M6NOZRP2Y4GIAPT8Q/image-asset.png Explore Probability with Count Bayesie - Working with Probability Distributions https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1426570537843-C3ERSMTQ6PZ99IWX2AGJ/image-asset.png Explore Probability with Count Bayesie - Working with Probability Distributions https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1426570560539-GE72XG6KJ4V0GLLJUKDI/image-asset.png Explore Probability with Count Bayesie - Working with Probability Distributions https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1426570586385-XQBDDHFISUAGWM34G9Q8/image-asset.png Explore Probability with Count Bayesie - Working with Probability Distributions http://www.countbayesie.com/blog/2015/3/3/6-amazing-trick-with-monte-carlo-simulations monthly 0.5 2020-09-28 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425422635138-Q4KXF81SNSX35ZEYFI8C/monte-carlo-integration.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R In blue is the area we wish to integrate over https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425422953934-K8RKDQT21OCCL8O76EXZ/basic-circle.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R Basic properties of a circle https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425423073377-E2QATXP5SYN4RUZJ82D2/image-asset.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R Sampling 100,000 points inside and outside of a circle https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425423195071-GCAPWBM1HI9U0SGRQIZQ/two-beta-distributions.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R Visualizing the possible overlap between two Beta distributions https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425425938194-G5ICDHB93CJ4UA5BLN7G/monte-carlo-simulation-histogram.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R By simulating samples between two distributions we can see the average improvement. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425425634292-YTC6GJ8OFZ9CHRKKG8IF/probability-spinner.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R The great spinner of probability! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425425906670-FD74H0DB4BJDGYW5R90V/bayz-stock-simulation.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R BAYZ Stock Ticker https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Monte Carlo Simulations in R http://www.countbayesie.com/blog/2015/2/18/hans-solo-and-bayesian-priors monthly 0.5 2019-08-21 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424373943054-0332LHQQROV9C3RTVXHL/Hansoloprofile.jpg Explore Probability with Count Bayesie - Han Solo and Bayesian Priors From the Wookiepedia https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424374977733-JX7OTBRKTN7JXCJQK73J/C-3PO_TPM.png Explore Probability with Count Bayesie - Han Solo and Bayesian Priors From the Wookiepeida https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424932227465-XTE0O9S8MD8AN4ZETDD3/image-asset.png Explore Probability with Count Bayesie - Han Solo and Bayesian Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424755166389-SQ8IM938HAL1O244I6K0/image-asset.jpeg Explore Probability with Count Bayesie - Han Solo and Bayesian Priors Consequences of not having a Bayesian Prior https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424932262004-SZX8GCA3H0BEIU5FLHNL/image-asset.png Explore Probability with Count Bayesie - Han Solo and Bayesian Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424932356350-CA3WDWF72J0KNH8I6WO2/image-asset.png Explore Probability with Count Bayesie - Han Solo and Bayesian Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424758230390-JHF0J62WOJHJ6TIQ772M/image-asset.jpeg Explore Probability with Count Bayesie - Han Solo and Bayesian Priors https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Han Solo and Bayesian Priors http://www.countbayesie.com/blog/2015/2/18/one-in-a-million-and-e monthly 0.5 2016-04-04 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1425930484061-H7OQKLACY9PQP9ABZXKB/image-asset.png Explore Probability with Count Bayesie - One in a Million and e http://www.countbayesie.com/blog/2015/2/21/variance-co-variance-and-correlation monthly 0.5 2020-09-27 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424583067430-9FGSPN5QEMWSFHTBJBIN/random-variable-robot.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation Sampling Robot and the Expectation of a Random Variable. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424593257081-JP3BOZFWWZQJ6JAE6ZZD/variance-squaring-machine.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation In it's most general form Variance is the effect of squaring Expectation in different ways. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424594630902-W5JGROGMPXDAWCS0YW96/simple_random_variable.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation A simple random variable for a 3 color spinner https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424594885892-3PYKW0ERG0NXYWAO2Y6H/robot-variance-visualization.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation Variance is the difference of squaring out Random Variable at different points when we calculate Expectation. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1601219763215-4CXLGWZVQ75W2J81M6E5/Screen+Shot+2020-09-27+at+11.15.19+AM.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424598424308-X3EFI12EFZ9QV5QSER3Q/happy-sampler.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation Correlation between different Random Variables produce by the same event sequence https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Understanding Variance, Covariance, and Correlation Order your copy of Bayesian Statistics the Fun way! http://www.countbayesie.com/blog/2015/2/20/random-variables-and-expectation monthly 0.5 2016-06-19 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424459299557-8WFSUINCCP4LE8EVMI9C/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424460019997-M3KMZQ19YE1RHXAGKEQ6/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424460276937-U02JII430QYDO0EO15QO/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424461953418-EOG54C9ZWBKTZWJI2P75/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424468330458-00IX1FEX2MLXLISEO744/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! Wow! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424480393838-8KRY26BYVXITCI94KHYE/CodeCogsEqn+%284%29.gif Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424480167542-153AN2KXH7X0I5NSV0PB/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424473080438-X2JOL6H1ZT9MOPVGG0DI/image-asset.gif Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424476576161-HSXAN7D8K2I49Z9D0MUM/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424477383098-CKRS6U6AJHE1B1NJAA30/image-asset.png Explore Probability with Count Bayesie - Random Variables and Expectation with Robots and Stuff! http://www.countbayesie.com/blog/2015/2/18/bayes-theorem-with-lego monthly 0.5 2019-08-21 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424319898083-TEZXUWXX340Z3KY36ZG8/lego-probability-space.png Explore Probability with Count Bayesie - Bayes' Theorem with Lego Lego Brick Probability Space https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1462144413630-7EATEYWU8LUNWCUYGGZP/visualizing-bayes%27-theorem.jpg Explore Probability with Count Bayesie - Bayes' Theorem with Lego Visualizing Bayes' Theorem: Solving "Probability of yellow given red" with Lego. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1424323206953-C3LO360QONPLOPQPPXRZ/probability-of-yellow-given-red.png Explore Probability with Count Bayesie - Bayes' Theorem with Lego Try to work out the probability that if you are on a yellow brick, there's a red brick underneath. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1563719197896-391TQ6PV9S54WW2FCDSN/Bayesian-statistics-the-fun-way.png Explore Probability with Count Bayesie - Bayes' Theorem with Lego Learn more about Bayesian Statistics! http://www.countbayesie.com/blog/tag/bayesian monthly 0.5 http://www.countbayesie.com/blog/tag/basics monthly 0.5 http://www.countbayesie.com/blog/tag/Integration monthly 0.5 http://www.countbayesie.com/blog/tag/Markov+Chains monthly 0.5 http://www.countbayesie.com/blog/tag/October+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/hypothesis+testing monthly 0.5 http://www.countbayesie.com/blog/tag/March+2016 monthly 0.5 http://www.countbayesie.com/blog/tag/September+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/April+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/variance+and+expectation monthly 0.5 http://www.countbayesie.com/blog/tag/Machine+Learning monthly 0.5 http://www.countbayesie.com/blog/tag/July+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/June+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/August+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/Stochastic+Processes monthly 0.5 http://www.countbayesie.com/blog/tag/November+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/May+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/parameter+estimation monthly 0.5 http://www.countbayesie.com/blog/tag/March+2015 monthly 0.5 http://www.countbayesie.com/blog/tag/Measure+Theory monthly 0.5 http://www.countbayesie.com/blog/tag/popular monthly 0.5 http://www.countbayesie.com/blog/tag/January+2016 monthly 0.5 http://www.countbayesie.com/blog/tag/February+2015 monthly 0.5 http://www.countbayesie.com/all-posts daily 0.75 2016-03-16 http://www.countbayesie.com/about daily 0.75 2020-09-11 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1548957238480-26G0G5YFEFD2DI9DPSSA/will-kurt.JPG About http://www.countbayesie.com/books daily 0.75 2022-08-07 https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1559911855596-WJXCSQ3PKOMPOJLFW5W8/BayesianStats_front_V1_placeholder.jpg Count Bayesie's Recommended Books in Probability and Statistics https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1559909352013-JYK07XBJPZ4U6S4O0JV6/BayesianStats_front_V1_placeholder.jpg Count Bayesie's Recommended Books in Probability and Statistics My book on probability and statistics is a great way to learn more! https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/9213a17e-c479-44e5-a0fe-3ea4f0059c42/bernoulli_s_fallacy.png Count Bayesie's Recommended Books in Probability and Statistics - Make it stand out Whatever it is, the way you tell your story online can make all the difference. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1559910079324-4VIR15H1E4QC5D8H2QXE/9781482253443.jpg Count Bayesie's Recommended Books in Probability and Statistics - Statistical Rethinking By Richard McElreath So, unlike most of my recommendations, I actually haven’t gotten a chance to read this yet, but it’s absolutely next on my list. I’m currently working through the lectures online and everything so far seems really excellent. Everyone who has read this book has told me it’s amazing and I really think this is the next logical step after “Bayesian Statistics the Fun Way”. The amount of supporting materials that McElreath has on the linked site is phenomenal and I know it has an update coming soon. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/78934676-96eb-4fad-93d8-dd5cc407e33a/bayesian_modeling_cover.jpg Count Bayesie's Recommended Books in Probability and Statistics - Bayesian Modeling and Computation in Python By Osvaldo Martin, Ravin Kumar and Junpeng Lao Here’s my official review for this book I sent to CRC: “By far one of the biggest challenges in the practical (and academic) application of Bayesian Statistics is that practitioners need both a strong understanding of the mathematics of Bayesian statistics as well as fairly sophisticated programming ability. This book does a consistently great job of teaching both of these simultaneously…One great example of this is that way in which practical advice, drawing from both academic experience and software engineering experience, is placed throughout the learning process. Pointing out tools to help avoid errors in your model, along with common libraries that make the process easier, really help the reader feel that they are being onboarded by an experienced, kind and helpful team of Bayesian Practitioners. This book is the advanced, practical Bayesian statistics book that is currently missing from my bookshelf.’ https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1559910910069-YHINT1KQA5I77VCV1UEZ/jaynes.jpg Count Bayesie's Recommended Books in Probability and Statistics - Probability Theory: The Logic of Science By E.T. Jaynes This is the book on Bayesian analysis. I really recommend getting a strong foundation in probability and statistics before diving in, only because you'll enjoy it that much more. Jaynes doesn't assume that Bayesian analysis is just an evolution of Classical statistics, but rather starts from first principles and builds it up as a form of logic. This is one of the most important books I have read, period. It is also in that category of books that are never truly "finished" because you could easily spend a life time on a single chapter. https://images.squarespace-cdn.com/content/v1/54e50c15e4b058fc6806d068/1559911101615-XN19B6BD650NBG3WYIFJ/bda_cover.jpg Count Bayesie's Recommended Books in Probability and Statistics - Bayesian Data Analysis By Andrew Gelman, et al. This is a tremendous work on theoretical statistics if, as Andrew Gelman phrased it, “theoretical statistics was the theory of applied statistics”. This book used to be recommend by anyone doing Bayesian analysis because it was really the only major, comprehensive work on the subject. This book is brilliant, but it is also fairly challenging. Everyone doing Bayesian stats should have a copy of this on their desk. I use mine very frequently. That said, this is not a book you sit and read cover to cover easily. McElreath’s and Martin’s books are better places to get introduce into serious Bayesian stats. However nothing changes this books place as a true classic! http://www.countbayesie.com/subscribe daily 0.75 2020-09-27