Bayesian inference

Bayesian inference represents uncertainty about unknown parameters using probability distributions.

Bayes' theorem gives

\[p(\theta\mid y)=\frac{p(y\mid\theta)p(\theta)}{p(y)}.\]

Here \(p(\theta)\) is the prior distribution, \(p(y\mid\theta)\) the likelihood and \(p(\theta\mid y)\) the posterior distribution.

Posterior inference

The posterior can be summarised by means, medians, credible intervals or probabilities of biologically meaningful events.

Prediction

Posterior predictive distributions propagate parameter uncertainty into predictions for future or unobserved quantities.

Key idea. Bayesian inference combines prior information with observed data and expresses the resulting parameter uncertainty through the posterior distribution.