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?
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1$\begingroup$ You need to give us more details. In general, likelihood is defined as p(data | parameters). $\endgroup$– Tim ♦Jan 14, 2020 at 17:50
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$\begingroup$ It were the two plausible alternatives on a multiple choice exam today. $\endgroup$– Roelof WagnerJan 14, 2020 at 18:02
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2 Answers
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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.
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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}
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$\begingroup$ 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. $\endgroup$– user193726Feb 2, 2020 at 16:11
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$\begingroup$ 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. $\endgroup$– user193726Feb 3, 2020 at 5:32