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I am confused about the difference between the prior/posterior density and distribution. In my lecture notes the posterior density is:

$$ f(\theta | x_1, \ldots, x_n) = \frac{f(\theta) L(\theta)}{\int f(s)L(s)ds}, $$

where $L$ is the likelihood function and the prior density is $f(\theta)$.

But my notes also say that the posterior distribution is the same as above, with the prior distribution being $f(\theta)$.

So is the prior/posterior distribution and density calculated the same way? Are they the same thing? If not, can someone elaborate on this a bit?

Waldir Leoncio
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Brian Quach
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  • Distribution is the general name, then you can have both discrete (probability mass function) and continuous (probability density function) distributions. https://en.wikipedia.org/wiki/Probability_distribution – PhDing Nov 12 '16 at 07:31
  • There are also probability distributions are neither discrete nor continuous. – Math1000 Nov 12 '16 at 08:50
  • So the posterior density = f(θ|x1,…,xn)=f(θ)L(θ)∫f(s)L(s)ds, is the same as the posterior distribution = f(θ|x1,…,xn)=f(θ)L(θ)∫f(s)L(s)ds, (the same thing?) – Brian Quach Nov 12 '16 at 14:32

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