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I have the following image which I've been told is an illustration of how the posterior probability distribution is a combination of the prior and likelihood distributions.

enter image description here

I've been told that there's something wrong with the image, namely that the posterior distribution cannot have the form it does given the form of the likelihood function. But I'm struggling to think of what is wrong with the image.

The posterior seems to be the likelihood but pulled to the right by the prior distribution. This matches my understanding of what should happen and makes sense. Does anyone know what might be wrong?

My only thought is that the area under the posterior may be slightly less than the area under the likelihood. This seems like a really picky aspect to bring up though given that the posterior seems a bit fatter than the likelihood.

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>My only thought is that the area under the posterior may be slightly less than the area under the likelihood. And yet it is a shibboleth often murmured that a likelihood function is not a density function. Thus, it is not at necessary that the area under a likelihood function be $1$ the way that it is for all densities. I don't believe a minor difference in area is indicative of anything. –  Dilip Sarwate Jun 9 at 18:53

1 Answer 1

up vote 19 down vote accepted

It looks like the prior and likelihood are normal, in which case the posterior should actually be narrower than either the likelihood or the prior. Notice that if

$$X \mid \mu \sim N(\mu, \sigma^2/n)$$ and $$\mu \sim N(\mu_0, \tau^2),$$

then the posterior variance of $\mu \mid X$ is $$\dfrac{1}{n/\sigma^2 + 1/\tau^2} < \min( \sigma^2/n, \tau^2).$$

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