2.1 Summary

The ingredients of Bayesian inference

The likelihood: $f(x\mid\theta)=\prod_{i=1}^{n}f(x_{i}\mid\theta)$: is a function of $\theta$ .
The prior: $\pi(\theta)$: is the probability of $\theta$ prior to data collection.
The joint distribution: $h(x;\theta)$: of $\theta$ and $x$ , factorized as $h(x;\theta)=f(x\mid\theta)\pi(\theta)$ .
The posterior distribution, $h(\theta\mid x)$: is the probability of the unknown upon consideration of the current data.
The marginal likelihood: $m(x)$: or evidence can be obtained by integrating out $\theta$ from the joint distribution $m(x)=\int_{\theta}\pi(\theta)f(x\mid\theta)d\theta$ .
The prior predictive distribution: $f(x^{\star}\mid x)$: is the probability of a future observation, $x^{\star}$ , before the data is looked at.
The posterior predictive distribution: $f(x^{\star}\mid x)$: is the probability of a future observation, $x^{\star}$ , given the data in hand, $x$ .