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# Pre-requisites

Be familiar with basic probability and Bayesian calculations.

# Overview

The goal here is simply to point out that everything you want to compute in Bayesian calculations is an integral.

# Examples

Consider inference for a parameter $$\theta$$ from data $$D$$.

The posterior distribution of $$\theta$$ is given by Bayes Theorem

$p(\theta | D) = p(\theta) p(D | \theta)/ p(D)$

First note that the denominator $$p(D)$$ is an integral: $p(D) = \int p(D | \theta) p(\theta) d\theta$.

Now suppose we want to estimate $$\theta$$ by its posterior mean. This is $E(\theta | D) = \int \theta p(\theta |D ) d\theta.$

And if we want to find a 90% posterior credible interval for $$\theta$$ then we want to find $$A$$ and $$B$$ such that $$\Pr(\theta \in [A,B] | D) = 0.9$$. Note that the LHS of this is $\Pr(\theta \in [A,B]|D) = \int I(\theta \in [A,B]) p(\theta | D) d\theta,$ where $$I(E)$$ denotes the indicator function for the event $$E$$, which takes the value 1 if $$E$$ is true and $$0$$ otherwise.

# Examples: discrete

Of course, if $$\theta$$ is discrete then the integrals above all become sums.

For example $E(\theta | D) = \sum_n \theta_n \Pr(\theta=\theta_n | D)$ where $$\theta_1,\theta_2,\dots$$ are the possible values for $$\theta$$.

# Summary

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, Simpsons 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: naive Monte Carlo, and Importance Sampling. Also Laplace approximation.

For higher dimensions we usually resort to Markov Chain Monte Carlo.

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