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Bayes's Theorem states that

$p(H|D) = \frac{p(D|H)}{p(D)}p(H)$

where H is the hypothesis and D is the new data. It can also be written as

$posterior = \frac{likelihood}{evidence} * prior $

p(D) is sometimes called the "marginal likelihood", because it is the likelihood of the data regardless of whether the hypothesis (or anything else) is true or not, and is sometimes referred to as the "normalising constant".

But why is it sometimes called "the evidence"?

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  • $\begingroup$ Do you have any refs for that terminology? $p(D|H)$ are indeed often called evidence and marginal likelihood. I’ve never seen them applied to $p(D)$. $\endgroup$
    – innisfree
    Oct 11 '20 at 9:45
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It is called the marginal likelihood because: this constant value $p(D)$ is what you obtain when you integrate over all possible values of $H$, leaving you with the probability of observing $D$ under your model.

It is called the normalizing constant because: this constant value $p(D)$ normalizes $p(D|H)p(H)$, making it a proper distribution that integrates to one.

It is called the (model) evidence because: this constant $p(D)$ can act as a measure of quality of fit of your model. Informally, in some sense it maps your model (your choice of likelihood and prior) and observations to a single value that describes the probability of your observation. The higher the more suitable your model is for the data, hence "model evidence". It acts as some evidence supporting your claim that the data is generated under your model.

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Notice that you can multiply both sides by $p(D)$:

$$p(H|D)p(D) = p(D|H)p(H)$$

This directly corresponds to the graphical proof of Bayes' Theorem - if all your variables are independent, you get the same slice of the pie (total probability) no matter in which order you make the cuts (conditioning on one variable).

In other words, $p(D|H)$ is the space of events supported (with various probabilities) by the hypothesis, and the entire RHS just normalizes that fraction relative to how much of the space of all hypotheses under consideration is occupied by the hypothesis that can generate your data.

Similarly, $p(H|D)p(D)$ is the space of all considered hypotheses supported by the data, normalized to the probability of observing that data.

Calling $p(D)$ 'evidence' is therefore a bit of a mental shortcut; more accurately, it would be 'the probability of seeing the evidence', but that's a bit of a mouthful.

$D$ would be a better match for the term - it's the set of events that supports or refutes the hypothesis, which is, more or less, the intuitive, real-world definition of evidence.

For completeness' sake - please note the 'all's in the bolded parts - the formulation of the theorem you've used, while correct, can only be naively applied in a binary comparison. Otherwise, you need to sum/integrate over all the components - that's what had been holding large-scale Bayesian inference back for a long time.

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I probably wouldn't call it "evidence", however I think it means "all the information in the data", which in Bayesian statistics is codified as $P(D)$ marginalised over all hypotheses/distributions deemed possible.

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