# Is the likelihood in Bayes theorem a probability? [duplicate]

First some notation and definitions: In the Bayes formula as written for machine learning applications, $$p(\theta|D) = \frac{ p(D|\theta) p(\theta) }{ p(D) }$$ commonly $$p(\theta)$$ is labeled the prior, $$p(D|\theta)$$ is called the likelihood, and $$p(D)$$ is called the evidence (or marginal likelihood I think).

Elsewhere (in various books), the likelihood is defined as $$p(D|\theta)$$ viewed as a function of $$\theta$$, not of $$D$$. But $$\int p(D|\theta) d\theta$$ does not necessarily integrate to one, and therefore is not a probability density. As support, in the Bishop Pattern Recognition & Machine Learing book p.22, "Note that the likelihood is not a probability distribution over w, and its integral with respect to w does not (necessarily) equal one."

On the other hand, I believe $$\int p(D|\theta) d D$$ does integrate to one. But this is the integral of a conditional probability, not a likelihood.

Finally the question: is the likelihood in the Bayes formula really a likelihood (since a likelihood is not a probability)? If not, why is it called a likelihood? If so, how can a non-probability appear in this formula that calculates the posterior probability?

(In fact I have a theory about what the answer is, but since I am self learning I hope to hear a better and complete explanation rather than an answer "yea that's right", from which I would learn nothing)

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The likelihood is a so-called conditional density. It is a probability density function on the data space (for $$D$$) given any parameter $$\theta$$ that we pass over to the function. When integrating over it with respect to $$D$$ we obtain the conditional distribution of the data given a certain parameter. This conditional distribution is a Markov Kernel.
Maybe this helps: The fundamental Problem that is to be solved in any kind of parametric statistics is: There is some underlying parameter $$\theta^*$$. We do not know this parameter. We obtain a sample $$D \sim p(\cdot|\theta^*)$$. Given this sample $$D$$, we now want to identify $$\theta^*$$. Frequentist statistics uses estimators, and hypothesis tests. Bayesian statistics uses posterior measures. Hence the likelihood is the density from which the data is sampled.