STAT 101
M09 · L01
Module 9 · Lesson 1
The Bayesian Framework
Treat probability as a degree of belief that evidence updates. The parameter is what we are uncertain about; the data is what changes our mind.
01 / 08
STAT 101
M09 · L01
Two readings of probability
Frequentist vs. Bayesian
- Frequentist — parameter fixed, data random; long-run frequencies
- Bayesian — data fixed, belief updates; probability = degree of belief
- The Bayesian answers the natural question: given my data, how likely is my hypothesis?
02 / 08
STAT 101
M09 · L01
The three ingredients
Prior, Likelihood, Posterior
- Prior P(θ) — belief before the data
- Likelihood P(D∣θ) — how well θ explains the data
- Posterior P(θ∣D) — updated belief; the output
03 / 08
STAT 101
M09 · L01
Tying them together
Bayes’ Rule
Posterior
P(\theta \mid D) = \dfrac{P(D \mid \theta)\,P(\theta)}{P(D)}
P(D) is just a normalising constant — it makes the posterior integrate to one.
04 / 08
STAT 101
M09 · L01
The updating engine
Posterior ∝ Likelihood × Prior
Proportional form
P(\theta \mid D) \propto P(D \mid \theta)\,P(\theta)
Strong data overwhelms a weak prior; with enough data the prior washes out entirely.
05 / 08
STAT 101
M09 · L01
When updating is just arithmetic
Conjugate Priors
Beta-Binomial
\text{Beta}(\alpha,\beta) \to \text{Beta}(\alpha+s,\beta+f)
A conjugate prior keeps the posterior in the prior’s family. Beta-Binomial: add successes to α, failures to β.
06 / 08
STAT 101
Summary
Recap
What you learned
- Bayesian = probability as degree of belief, updated by data
- Prior + Likelihood → Posterior via Bayes’ rule
- Posterior ∝ likelihood × prior; P(D) just normalises
- Conjugate priors (Beta-Binomial) make updating closed-form
08 / 08