Why is the posterior distribution the same as likelihood function when uniform prior distribution is used in Bayesian Analysis?

Question

While studying bayesian analysis, I am told that the posterior distribution is the same as the likelihood function if we use a uniform prior distribution. I am having some difficulty to understand why it is so. I am referencing a lecture on the Intenet and the link is as follows:

http://www.sumsar.net/blog/2017/02/introduction-to-bayesian-data-analysis-part-one/

The lecturer shows Bayes' Theorem to show the calculation for [pior * likelihood] done in the video but I cannot find when [pior * likelihood] is done in the video. What am I missing here?

Answer 1

The posterior is prior$\,\times\,$likelihood$\,\times\,$constant; the uniform density is simply a constant and gets absorbed in the other constant term.

Take as an explicit example the prior $\mathrm{uniform}(0,1)$; then, since the prior pdf is $f(\theta) = 1$, prior$\,\times\,$likelihood = 1$\,\times\,$likelihood = likelihood.

Answer 2

The intuition, I think, is that with the prior you nudge the distribution of the parameter values of the model (i.e. the posterior) in the direction you think are more likely. With a uniform prior you give equal weight to all possible values, that is, you are not nudging in any direction. Consequently, the prior has no effect and you end up with just the likelihood.