โš ๏ธ This is an AI-generated test article for theme demonstration/testing only. All content is synthetic.

The prior

Bayesโ€™ rule lets us update a belief after observing data. The prior encodes our starting belief.

$$ P(\theta \mid D) = \frac{P(D \mid \theta) P(\theta)}{P(D)} $$

The posterior

As data accumulates, the posterior sharpens around the most plausible parameter values.

Conjugate families

Choosing a conjugate prior keeps the algebra closed-form, which makes computation simple.

1
2
3
4
5
6
7
8
# simple conjugate Beta-Binomial update in R
library(ggplot2)
alpha <- 2; beta <- 2
successes <- 5; failures <- 3
posterior <- data.frame(
  theta = seq(0, 1, length.out = 200),
  prior = dbeta(seq(0, 1, length.out = 200), alpha, beta)
)

When to use it

Bayesian methods shine when we want interpretable intervals and the ability to inject domain knowledge.