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# Overview

This vignette introduces the idea of “conjugate prior” distributions for Bayesian inference for a continuous parameter. You should be familiar with Bayesian inference for a binomial proportion.

# Conjugate Priors for binomial proportion

## Background

In this example we considered the following problem.

Suppose we sample 100 elephants from a population, and measure their DNA at a location in their genome (“locus”) where there are two types (“alleles”), which it is convenient to label 0 and 1.

In my sample, I observe that 30 of the elephants have the “1” allele and 70 have the “0” allele. What can I say about the frequency, $$q$$, of the “1” allele in the population?

The example showed how to compute the posterior distribution for $$q$$, using a uniform prior distribution. We saw that, conveniently, the posterior distribution for $$q$$ is a Beta distribution.

Here we generalize this calculation to the case where the prior distribution on $$q$$ is a Beta distribution. We will find that, in this case, the posterior distribution on $$q$$ is again a Beta distribution. The property where the posterior distribution comes from the same family as the prior distribution is very convenient, and so has a special name: it is called “conjugacy”. We say “The Beta distribution is the conjugate prior distribution for the binomial proportion”.

## Details

As before we use Bayes Theorem which we can write in words as $\text{posterior} \propto \text{likelihood} \times \text{prior},$ or in mathematical notation as $p(q | D) \propto p(D | q) p(q),$ where $$D$$ denotes the observed data.

In this case, the likelihood $$p(D | q)$$ is given by $p(D | q) \propto q^{30} (1-q)^{70}$

If our prior distribution on $$q$$ is a Beta distribution, say Beta$$(a,b)$$, then the prior density $$p(q)$$ is $p(q) \propto q^{a-1}(1-q)^{b-1} \qquad (q \in [0,1]).$

Combining these two we get: $p(q | D) \propto q^{30} (1-q)^{70} q^{a-1} (1-q)^{b-1}\\ \propto q^{30+a-1}(1-q)^{70+b-1}$

At this point we again apply the “trick” of recognizing this density as the density of a Beta distribution - specifically, the Beta distribution with parameters $$(30+a,70+b)$$.

## Generalization

Of course, there is nothing special about the 30 “1” alleles and 70 “0” alleles we observed here. Suppose we observed $$n_1$$ of the “1” allele and $$n_0$$ of the “0” allele. Then the likelihood becomes $p(D | q) \propto q^{n_1} (1-q)^{n_0},$ and you should be able to show (Exercise) that the posterior is $q|D \sim \text{Beta}(n_1+a, n_0+b).$

## Summary

When doing Bayesian inference for a binomial proportion, $$q$$, if the prior distribution is a Beta distribution then the posterior distribution is also Beta.

We say “the Beta distribution is the conjugate prior for a binomial proportion”.

# Exercise

Show that the Gamma distribution is the conjugate prior for a Poisson mean.

That is, suppose we have observations $$X$$ that are Poisson distributed, $$X \sim Poi(\mu)$$. Assume that your prior distribution on $$\mu$$ is a Gamma distribution with parameters $$n$$ and $$\lambda$$. Show that the posterior distribution on $$\mu$$ is also a Gamma distribution.

Hint: you should take the following steps. 1. write down the likelihood $$p(X|\mu)$$ for $$\mu$$ (look up the Poisson distribution if you cannot remember it). 2. Write down the prior density for $$\mu$$ (look up the density of a Gamma distribution if you cannot remember it). 3. Multiply them together to obtain the posterior density (up to a constant of proportionality), and notice that it has the same form as the gamma distribution.

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