# How to understand the posterior distribution is the same as likelihood function

So I read this post Why is the posterior distribution the same as the likelihood function when uniform prior distribution is used in Bayesian Analysis, and learned that when we have a uniform prior, the posterior distribution is the same as the likelihood function. However, we also have that What is the reason that a likelihood function is not a pdf, i.e. likelihood is not a pdf. For example, the sum of the likelihood might not be 1.

Given these, how should we understand the posterior distribution is the same as the likelihood function when we have a uniform prior? Does this mean the sum of a posterior distribution doesn't need to be 1?

• It might help to realize that in many cases uniform priors just can't exist. The standard example is any location family of distributions.
– whuber
Nov 30, 2021 at 18:18
• The likelihood function may need to be normalized in order to integrate to one, so it may be that they are the same up to a constant of proportionality. Nov 30, 2021 at 18:21

Furthermore, the likelihood function is parameterisation invariant in that changing the parameterisation of the model does not change the likelihood function, in the sense that $$\ell_\eta(\eta_0)=\ell_theta(\theta_0)$$ if $$\eta=F(\theta)$$ and $$\eta_0=F(\theta_0)$$. This means that the likelihood function does not obey the change-of-variable rule for density functions, i.e., no Jacobian appears in the transform. Therefore, this is a further argument for likelihoods not being pdfs.