When a Bayesian posterior is derived from a likelihood function that does not integrate (or sum) to unity the posterior function is simply re-scaled to make that integral (or sum) equal one in order that the posterior can be a 'proper' probability distribution.

The use of a uniform prior makes the scaled likelihood function into the posterior and so it might make one wonder whether such a Bayesian approach offers anything beyond a pure likelihood approach. See these little books if you are interested in likelihood-based inference: https://www.goodreads.com/book/show/735705.Likelihood https://www.routledge.com/Statistical-Evidence-A-Likelihood-Paradigm/Royall/p/book/9780412044113

When a Bayesian posterior is derived from a likelihood function that does not integrate (or sum) to unity the posterior function is simply re-scaled to make that integral (or sum) equal one in order that the posterior can be a 'proper' probability distribution.

The use of a uniform prior makes the scaled likelihood function into the posterior and so it might make one wonder whether such a Bayesian approach offers anything beyond a pure likelihood approach. See these little books if you are interested in likelihood-based inference:

• Likelihood by A.W.F. Edwards
• Statistical Evidence, A Likelihood Paradigm by Richard Royall.