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I'm reading this online book and there is something unclear to me in this table where the posterior probability of each model computed as:

$$\ P(model \ | \ data) = \frac{P(data \ | \ model) \times P(model)}{P(data) } $$

I understand $\ P(model) $ is a prior, and $\ P(data \ | model ) $ is just a binomial (probability of observing such data given the prior) distribution but what exactly is $\ P(data) $ ?

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It's just a normalizing constant which makes the posterior a valid density. In practice, we don't care so much about it. It should be noted that

$$p(x) = \int p(x\vert \theta) p(\theta) \, d\theta$$

So it is as if you are averaging the likelihood over the prior.

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  • $\begingroup$ Ok so is it as if I would say that $\ P(model | data) $ proportional to $\ P(data | model) \times P(model) $ ? If i understand correctly then I just take the $\ P(data \ | \ model) \times \ P(model) $ and divide it by the sum of all $\ P(data | model)P(model( $ to get the posterior probability of each? $\endgroup$
    – bm1125
    Commented Jan 16, 2020 at 16:56
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    $\begingroup$ That's right. Often times, we just write bayes rule as a proportionality rather than an equality because the normalizing constant can be tricky to compute. $\endgroup$
    – Demetri Pananos
    Commented Jan 16, 2020 at 16:58
  • $\begingroup$ Thanks a lot Demetri! $\endgroup$
    – bm1125
    Commented Jan 16, 2020 at 17:01
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I agree with @Demetri. Baysian inference is based on cause and effect relationship.
$P(effect|cause)$ is usually known. Therefore we try to infer $P(cause|effect)$.

Remember effect is observable, hence we try to infer the cause given the effect.

$P(data)$ is just a normalizing factor which is total $P(cause)$.

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