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Be familiar with basic probability and Bayesian calculations.
The goal here is simply to point out that everything you want to compute in Bayesian calculations is an integral.
Consider inference for a parameter \(\theta\) from data \(D\). The posterior distribution of \(\theta\) is given by Bayes Theorem: \[ p(\theta \mid D) = \frac{p(\theta) \, p(D \mid \theta)}{p(D)}. \] First note that the denominator \(p(D)\) is an integral: \[ p(D) = \textstyle \int p(D \mid \theta) \, p(\theta) \, d\theta. \] Now suppose we want to estimate \(\theta\) by its posterior mean. This is \[ E(\theta \mid D) = \textstyle \int \theta \, p(\theta \mid D) \, d\theta. \] And if we want to find a 90% posterior credible interval for \(\theta\), then we want to find \(A\) and \(B\) such that \(\Pr(\theta \in [A,B] \mid D) = 0.9\). The LHS of this is \[ \Pr(\theta \in [A,B] \mid D) = \textstyle \int \mathbb{I}(\theta \in [A,B]) \, p(\theta \mid D) \, d\theta, \] where \(\mathbb{I}(E)\) denotes the indicator function for the event \(E\), which takes the value 1 if \(E\) is true, and \(0\) otherwise.
Of course, if \(\theta\) is discrete then the integrals above all become sums. For example, \[ E(\theta \mid D) = \sum_n \theta_n \Pr(\theta = \theta_n \mid D), \] where \(\theta_1, \theta_2, \dots\) are the possible values for \(\theta\).
Pretty much all the things you want to compute when doing Bayesian inference are integrals (or sums) of one kind or another…
If you are computing 1-dimensional integrals, then numerical methods are often useful. For example, Simpson’s Rule, Gaussian Quadrature. These can also work in 2-dimensions, and maybe even 3 or 4.
Other simple methods that can work for low dimensions include naive Monte Carlo and Importance Sampling. Also Laplace’s method.
For higher dimensions, we resort to other methods such as Markov chain Monte Carlo (MCMC).
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