Module Review
Congratulations on completing the Bayesian Methods module! You now have a powerful framework for reasoning about uncertainty and learning from data.
Key Takeaways
- Belief Updating: Bayes’ Theorem is the mathematically correct way to update beliefs when new evidence is encountered.
- The Three Pillars: The Prior (what we thought before), the Likelihood (what the data says), and the Posterior (what we think now).
- Rare Events: When the Base Rate (Prior) is low, even strong evidence might not lead to a high Posterior probability.
- Conjugacy: Conjugate Priors (like Beta for Binomial) allow us to perform Bayesian updates using simple addition (“Pseudo-Counts”) instead of complex integration.
- Hardware Reality: We use Log-Probabilities to avoid arithmetic underflow when multiplying many small numbers.
- Deep Learning: L2 Regularization (Weight Decay) is mathematically equivalent to MAP Estimation with a Gaussian Prior.
1. Interactive Flashcards
Test your knowledge by flipping the cards below.
Prior P(A)
The probability of an event or hypothesis before seeing any new evidence. Represents our initial belief.
Likelihood P(B|A)
The probability of observing the evidence B, assuming hypothesis A is true. It measures how well the data supports the hypothesis.
Posterior P(A|B)
The updated probability of hypothesis A after seeing evidence B. This is the result of Bayesian Inference.
Conjugate Prior
A prior distribution that, when multiplied by the Likelihood, results in a Posterior distribution of the same family. (e.g., Beta Prior + Binomial Likelihood = Beta Posterior).
MAP vs MLE
MLE maximizes Likelihood (ignores Prior). MAP maximizes Posterior (Likelihood × Prior). MAP is safer for small datasets.
Log-Sum-Exp
A numerical trick used to normalize probabilities in log-space without causing underflow or overflow.
2. Bayesian Cheat Sheet
| Concept | Formula / Rule | Note |
|---|---|---|
| Bayes’ Theorem | P(A|B) = P(B|A)P(A) / P(B) |
Fundamental rule for inversion. |
| Posterior | Posterior ∝ Likelihood × Prior |
We often ignore P(B) and normalize later. |
| Log-Space | log(P) = log(L) + log(Prior) |
Turn multiplication into addition to avoid underflow. |
| Beta Update | α → α + successes, β → β + failures |
“Pseudo-count” interpretation. |
| Beta Mean | μ = α / (α + β) |
Expected value of the parameter. |
| Laplace Smoothing | (k + 1) / (n + 2) |
Equivalent to Beta(1,1) (Uniform) prior. |
3. Next Steps
Now that you understand the basics of Bayesian reasoning, you are ready to explore Information Theory, where we will learn how to measure the “surprise” or information content of these probability distributions.