Predictive inference with the Bayesian approach

In-depth Articles

Predictive inference with the Bayesian approach

Starting from the posterior, structural inference can be performed (i.e., on the parameters).

Let's now see how to perform predictive inference.

Point estimation

Every summary statistic (expected value, posterior mean, mode, etc.) of the posterior is a Bayesian point estimate.

Interval estimation

CS: Simpler but more imprecise

HPD: More precise but longer

Credible set

CS (credible set): in order to have a reliability pair. The credible set is an interval in the domain of a posterior probability distribution used to obtain interval estimates. The credible set is the analogue of confidence intervals in frequentist statistics.

Highest Posterior Density

HPD (highest posterior density): I fix a positive number \(h > 0\)

I obtain \(S_h = \{ \theta : \pi(\theta | \underline x) \geq h\}\)

If \(h : S_h\) has an associated posterior probability that is less than \(1- \alpha\), then I choose an \(h^| < h\), otherwise I choose an \(h^| > h\)

Hypothesis testing

In the Bayesian framework, I obtain a partition of the support of the random variable \(\theta\), but thanks to the posterior, the parameter space becomes probabilized. Therefore, in hypothesis testing, I can calculate the probability underlying the null hypothesis and that of the alternative hypothesis; the one with the higher probability will be the one I accept.

Hypotheses:

\[H_0 : \theta \in \Theta_0\] \[H_1 : \theta \in \Theta_1\]

Calculating the probabilities conditional on the hypotheses:

\[P(H_0 | \underline x )=P(\theta \in \Theta_0 | \underline x)\] \[P(H_1 | \underline x )=P(\theta \in \Theta_1 | \underline x)\]

The higher one corresponds to the accepted hypothesis.