# Proportional to Gamma means the posterior is gamma

I'm reading through these lecture notes on posteriors and conjugate priors. https://web.stanford.edu/class/stats200/Lecture20.pdf

In particular, it asserts that: "This is proportional to the PDF of the Gamma(s + α, n + β) distribution, so the posterior distribution of Λ must be Gamma(s + α, n + β)." on page 20-4.

Why is this allowed? Does this just generally work for data drawn from poisson with a Gamma prior?

• This refers to the concept of a kernel of a distribution, which determines its shape. All that is missing is a proportionality constant that ensures that the density integrates to one. So the argument works for any proper density. Commented Oct 19, 2022 at 10:07
• Thanks. So when calculating posteriors, is it simply enough then to find p(x|theta)p(theta) as long as I pick a conjugate prior? Since I know there will be an analytical solution to p(x), as in, it will integrate to some constant that can be ignored. Commented Oct 19, 2022 at 20:04
• Yes. It should also work even if you do not pick a conjugate prior, although there will then be relatively few cases in which analytical solutions are available. Commented Oct 20, 2022 at 4:36

If a probability density $$f$$ is known up to a multiplicative constant, $$f(x)\propto \tilde f(x)\qquad\forall x\in\mathfrak X$$ meaning that there exists a constant $$c>0$$ such that $$f(x)=c\tilde f(x)$$, the constant $$c$$ is determined by the constraint that $$f(\cdot)$$ is a probability density: $$c^{-1} = \int_\mathfrak X \tilde f(x)\, \text dx$$