# Definition of Likelihood in Bayesian Statistics [duplicate]

Can the likelihood be defined as the probability of the rate parameter given a range of data. Or as the probability of the data, given a range of rate parameters?

• You need to give us more details. In general, likelihood is defined as p(data | parameters).
– Tim
Jan 14, 2020 at 17:50
• It were the two plausible alternatives on a multiple choice exam today. Jan 14, 2020 at 18:02

## 2 Answers

I think I understand your confusion. Typically, Bayes' rule is written as:

$$p(\theta |y) = \frac{ p(y|\theta) p(\theta)}{p(y)}$$

where $$p(\theta |y)$$ is the posterior for the observed data $$y$$ given unknown parameters $$\theta$$, $$p(\theta)$$ is the prior distribution, and $$p(y)$$ is the marginal distribution of $$y$$. As far as Bayes' rule is concerned, $$p(y)$$ is a constant since it doesn't depend on the unknown parameters, so this simplifies to:

$$p(\theta |y) \propto p(y|\theta) p(\theta).$$

Now some would refer to $$p(y|\theta)$$ as the likelihood function. Technically, the likelihood is a function of $$\theta$$ for fixed data $$y$$, say $$L(\theta |y)$$. However, the liklelihood is proportional to the sampling distribution, so $$L(\theta |y) \propto p(y|\theta)$$.

In other words, $$p(y|\theta)$$ isn't technically the likelihood, but it is proportional to it, and as far as applying the Bayesian methodology is concerned, the distinction is not important. Hence why it is often referred to as the likelihood.

• Could you explain in detail why there is "Because the likelihood is proportional to the probability of the data given the hypothesis" in Etz, thanks! Jul 14, 2023 at 3:24
• @wangzhe That discussion you link to is in the context of maximum likelihood estimation, which as a Bayesian statistician, is not something I would feel confident trying to explain, sorry. But I suspect the explanation is very similar to what I have described above. Since you want a detailed answer, I suggest starting a new question. Jul 17, 2023 at 23:56

The likelihood function $$L(\theta|\mathbf x)$$ is defined as a function of $$\theta$$ indexed by the realisation $$x$$ of a random variable with density $$f(\mathbf x|\theta)$$: \begin{align}L\,&:\Theta\longmapsto \mathbb R\\ &\ \ \ \ \theta\longmapsto f(\mathbf x|\theta)\end{align}

• This "definition" and notation are in the middle of nowhere. Clearly, such a function of $\theta$ should be simply denoted $L\left( \theta \right)$. But it is instead denoted $L\left( {\left. \theta \right|{\text{x}}} \right)$ while the notation "${\left. \theta \right|{\text{x}}}$" has not been defined yet and makes sense only in the Bayesian framework, not it the likelihoodist one. Fisher himself used to denote his likelihood function $L\left( {\theta {\text{;x}}} \right)$ that makes more sense in his framework.
– user193726
Feb 2, 2020 at 16:11
• Therefore the notation $L\left( {\left. \theta \right|{\text{x}}} \right)$ is actually a bad mix between Fisher's likelihood function $L\left( {\theta {\text{;x}}} \right)$ and the Bayesian posterior probability distribution $p\left( {\left. \theta \right|{\text{x}}} \right)$. Hence the confusion. There is nothing like the likehood function within the Bayesian framework, only the "direct probability distribution” $p\left( {\left. {\text{x}} \right|\theta } \right)$ according to H. Jeffreys.
– user193726
Feb 3, 2020 at 5:32