Is that true If I say posterior = prior multiply likelihood?

Question

I see a formula that we should also have marginal distribution as denominator. But some tutorial said posterior = prior multiply likelihood?

Answer 1

It is not equal to, but it is proportional to:

$$ \text{posterior} = \frac{\text{likelihood} \times \text{prior}}{\text{marginal}} \propto \text{likelihood} \times \text{prior}. $$ The reason is that we are interesting in the parameter, say $\theta$, and this parameter only appears in the likelihood and prior, but not the marginal, which is purely a function of the data $x$. In other words, the marginal is just a normalisation quantity that allows the posterior to be a proper probability.

Answer 2

It's possible your tutorial was referring to posterior and prior odds rather than posterior and prior distributions, although the equation should then be posterior odds = prior odds multiplied by likelihood ratio.

If $\ A\ $ is the event whose prior and posterior odds you're interested in, and $\ d\ $ is the evidence, or data, on which the posterior odds are to be calculated, then \begin{align} \text{posterior odds}&=\frac{P(A|d)}{P(\tilde A\,|d)}\\ &=\frac{P(A\, \&\, d )}{P(\tilde A \,\&\, d)}\\ &=\frac{P(A)}{P(\tilde A)}\frac{P(d|A)}{P(d|\tilde A)}\\ &=\text{prior odds}\times\frac{P(d|A)}{P(d|\tilde A)}\ , \end{align} the quantity $\ \frac{P(d|A)}{P(d|\tilde A)}\ $ being the ratio of the likelihood of $\ A\ $ to the likelihood of its complement.