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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?

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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.

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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.

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    $\begingroup$ Although I like the simple intuitive nature of this answer (hence the :+1:), I would warn that without caution this idea is dangerous; the prior does not nudge the posterior away from the likelihood, the likelihood nudges the posterior away from the prior (since the prior comes before your observed data, it must be the thing that is being nudged). I say dangerous because it can fuel misinformed critiques of Bayesian inference and/by leading to a fundamental misunderstanding of the project of Bayesian inference in the manner hinted in my last sentence. $\endgroup$
    – duckmayr
    Aug 17 '20 at 16:58
  • $\begingroup$ Does "Nudging" mean "affecting the posterior"? $\endgroup$
    – StoryMay
    Sep 2 '20 at 14:57

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