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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.
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”.
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 \((30+a,70+b)\).
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 \[q|D \sim \text{Beta}(n_1+a, n_0+b).\]
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”.
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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