Bayesian inference combines prior belief $\pi(\theta)$ with likelihood $p(x\mid\theta)$ to obtain the posterior $\pi(\theta\mid x) \propto p(x\mid\theta)\pi(\theta)$. The posterior is the basis for estimation, prediction, and uncertainty quantification.

Common summaries include the posterior mean, median, mode, credible intervals, posterior predictive distributions, and Bayes factors. The choice of summary depends on the decision problem and loss function.

For action $a$ and parameter $\theta$, the loss is $L(a,\theta)$. The posterior expected loss is $\rho(a\mid x)=\mathbb{E}[L(a,\theta)\mid x]$. The Bayes action minimizes $\rho(a\mid x)$.

Exact posteriors are often intractable. Markov chain Monte Carlo, variational inference, Laplace approximation, expectation propagation, and sequential Monte Carlo are central computational tools.