STAT 101
M09 · L02
Module 9 · Lesson 2
Bayesian Estimation
The posterior is the answer. Estimation is summarising it — and the credible interval finally means what everyone always wanted a confidence interval to mean.
01 / 08
STAT 101
M09 · L02
The output of Bayes’ rule
The Posterior Is the Answer
P(θ∣D) is a full distribution over the parameter — it already holds everything we know, uncertainty included. We just summarise it.
02 / 08
STAT 101
M09 · L02
A single number
Point Estimates
Posterior mean
\hat{\theta} = \mathbb{E}[\theta \mid D]
- Mean — minimises squared error
- Median — robust to skew
- MAP (mode) — = MLE under a flat prior
03 / 08
STAT 101
M09 · L02
An interval
The Credible Interval
95% credible interval
P(\theta_L \le \theta \le \theta_U \mid D) = 0.95
A 95% posterior probability that θ lies inside. The highest-density version is the shortest such interval.
04 / 08
STAT 101
M09 · L02
The payoff
Credible ≠ Confidence
Module 5 forbade “95% probability the parameter is in this interval” for a confidence interval. For a credible interval that is the definition — a probability statement about the parameter, not the procedure.
05 / 08
STAT 101
M09 · L02
Where belief enters
Choosing a Prior
- Flat prior — let the data speak
- Informative prior — fold in real knowledge
- Sensitivity analysis — report how much the prior drives the result
Gaussian → Ridge, Laplace → Lasso: priors are regularisation.
06 / 08
STAT 101
Summary
Recap
What you learned
- Summarise the posterior: mean, median, or MAP (= MLE under a flat prior)
- A credible interval holds a stated fraction of posterior probability
- Credible ≠ confidence — it means what Module 5 forbade for CIs
- Flat vs. informative priors; a sensitivity analysis keeps it honest
- Bayesian regression: a posterior per coefficient; priors = regularisation
08 / 08