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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?

MoneyPrinting
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    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. – Christoph Hanck Oct 19 '22 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. – MoneyPrinting Oct 19 '22 at 20:04
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    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. – Christoph Hanck Oct 20 '22 at 04:36

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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: $$c^{-1} = \int_\mathfrak X \tilde f(x)\, \text dx$$

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Xi'an
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