baysian inference

Bayesian Inference

Treats parameters as random variables with a prior distribution $P(\theta )$. Updates beliefs using Bayes Theorem:

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

The posterior $P(\theta \mid X)$ represents our updated beliefs about parameters after seeing data.

Computing it often requires intractable integrals, leading to approximation methods:

• Variational inference - approximate posterior with simpler distribution
• Markov chain Monte Carlo - sample from posterior distribution
• Laplace approximation - Gaussian approximation around posterior mode

bayesianism