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I have been racking my brains trying to understand Bayes theorem. So, the way I have understood is that the likelihood is the probability of observing the particular outcome given a set of parameter values. The prior describes our belief probabilities over the different parameter values. The denominator acts like a normaliser ensuring that the posterior is a valid distribution.

I have a couple of things I am not clear about:

1: The likelihood is a probability and the prior is a probability as well. So, why is the numerator already not normalised between 0 and 1? Assume I have not ignored any multiplicative constants when computing the likelihood.

2: One way I think of this denominator now is that if we have a bunch of numbers and we would like to normalize them between 0 and 1, one thing we can do is divide each of these numbers by their sum. Is this what the denominator in the Bayes theorem is affectively doing?

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From a technical point of view, here is the argument:

For densities (but the argument is analogous in the discrete case), we write $$ \pi \left( \theta |y\right) =\frac{f\left( y|\theta \right) \pi \left(\theta \right) }{f(y)} $$ The norming constant can be obtained as, by writing a marginal density as a joint density and then writing the joint as conditional times marginal, with the other parameter integrated out, \begin{align*} f(y)&=\int f\left( y,\theta \right) d\theta\\ &=\int f\left( y|\theta \right) \pi \left(\theta \right)d\theta \end{align*} It ensures integration to 1 because \begin{align*} \int \pi \left( \theta |y\right) d\theta&=\int\frac{f\left( y|\theta \right) \pi \left(\theta \right) }{\int f\left( y|\theta \right) \pi \left(\theta \right)d\theta}d\theta\\ &=\frac{\int f\left( y|\theta \right) \pi \left(\theta \right) d\theta}{\int f\left( y|\theta \right) \pi \left(\theta \right)d\theta}\\ &=1, \end{align*} where we can "take out" the integral in the denominator because $\theta$ had already been integrated out there.

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  • $\begingroup$ Thanks. Now, I see how the denominator acts as a normaliser. $\endgroup$
    – user42140
    Jun 15 '15 at 5:58
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    $\begingroup$ Also it is important to stress that likelihood is not a probability in case of continuous variables. $\endgroup$
    – Tomas
    Jun 15 '15 at 8:08
  • $\begingroup$ It's always confusing when $\theta$ isn't $\theta$; consider renaming one of the dummy variables. $\endgroup$
    – user41979
    Jun 15 '15 at 8:41
  • $\begingroup$ @Hurkyl: Which bits are you referring to? $\endgroup$
    – user42140
    Jun 15 '15 at 18:44

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