In Bayes theorem of a parameter $\theta$ with data $D$, we have:

$$P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)}$$

where I know $P(D)$ as the marginal likelihood. Is it true that the marginal likelihood is referred to as evidence in Bayesian statistics? If not what is commonly refered to as evidence?

  • 4
    $\begingroup$ The "evidence" would be $D$ itself and not its marginal distribution $P(D)$. $\endgroup$
    – dsaxton
    Sep 9 '16 at 16:05
  • 2
    $\begingroup$ It is correct that the term evidence is sometimes used as a substitute for marginal likelihood, or marginal density. See for instance Skilling (2006). $\endgroup$
    – Xi'an
    Sep 10 '16 at 8:23

The term are loosely defined.

In your question, We use the term data ($D$) and parameter ($\theta$).

In other literature, people use another set of terms: evidence ($E$) and hypothesis ($H$), which are exactly the same thing to your $D$ and $\theta$.

Using the evidence and hypothesis combo, following notation is also widely used.

$$P(H|E) = \frac{P(E|H)P(H)}{P(E)}$$

Check the wikipedia page on Bayesian Inference, you can see the formula.

  • $\begingroup$ It seems the answer needs editing, see comment by @Xian above. $\endgroup$
    – tomka
    Sep 10 '16 at 8:50

$P(D)$ is the model evidence, unfortunately "model" is often dropped. The model evidence is also referred to as marginal likelihood.

Wikipedia calls the data $D$ the evidence.

The model evidence is defined as:


It is called the model evidence, since the larger its value, the more apt the model is generally fitting the data.


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