What's wrong with this illustration of posterior distribution?

Question

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.

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.

Answer 1

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 $$\dfrac{1}{n/\sigma^2 + 1/\tau^2} < \min( \sigma^2/n, \tau^2).$$