1. Dependence can disappear when we condition
Use rain as a hidden variable in the coats-and-accidents example. See marginal association appear even though coats and accidents are independent within each weather condition.
Open activity2. Coin MLE: likelihood you can see
Flip a coin, watch the likelihood curve change, locate the MLE, compare 5 vs. 50 flips, and calculate how many samples Hoeffding’s inequality asks for.
Open activity3. Gaussian MLE and the variance bias
Estimate μ and σ² from continuous samples, inject an outlier, then repeat the experiment hundreds of times to see why the MLE variance is biased.
Open activity4. MLE vs. MAP: prior meets data
Move a Beta prior, add heads and tails, and see prior, likelihood, and posterior together. Compare MLE and MAP and watch the prior get washed out.
Open activity5. Bayes rule and the base-rate effect
Recreate the diagnostic-test example using expected counts. Then add a conditionally independent follow-up test and see why two different tests can be much more informative.
Open activity6. Why the Naïve Bayes assumption helps
Compare parameter counts with and without conditional independence, then assemble a posterior feature-by-feature and see what happens when evidence is duplicated.
Open activity7. Text Naïve Bayes & Laplace smoothing
Use the Chinese/Tokyo/Japan example from the lecture, edit the test document, turn smoothing on and off, and verify that word order does not matter in the bag-of-words model.
Open activity8. Gaussian Naïve Bayes
Move a continuous 2-D query point through two class models and inspect the per-feature Gaussian likelihoods that combine into the class posterior.
Open activity