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I am struggling with the Wikipedia entry on Likelihood.

In an example it mentions $L(P_H = 0.5 |HH) = 0.25$

It mentions that

Bayes' theorem implies that the posterior probability is proportional to the likelihood times the prior probability.

I am trying to understand, in our scenario, what the prior and the post should be. I thought of

post = prior * likelihood = 0.5 * 0.25 = 0.125

This seems way too small. How is the "proportional" calculated?

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    $\begingroup$ There is a normalizing factor you've forgotten. $\endgroup$
    – Demetri Pananos
    Commented Jan 25, 2022 at 3:33
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    $\begingroup$ Have you read the Wikipedia entry on Bayes' Theorem? $\endgroup$
    – Sycorax ♦
    Commented Jan 25, 2022 at 3:38
  • $\begingroup$ @Sycorax thanks I am reading en.wikipedia.org/wiki/Bayes%27_theorem $\endgroup$
    – Kirsten
    Commented Jan 25, 2022 at 4:04
  • $\begingroup$ I cant see any mention of a normalizing factor. $\endgroup$
    – Kirsten
    Commented Jan 25, 2022 at 4:19
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    $\begingroup$ In $P(A|B) = P(B|A) P(A)/P(B)$ the denominator $P(B)$ is the normalizing constant $\endgroup$
    – Tim
    Commented Jan 25, 2022 at 8:02

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When you multiply a prior probability distribution by the likelihood function you can get a distribution that has an integral of more or less than one. It is not a probability distribution until you scale it to get that integral back to one.

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