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I know that there have already been a lot of questions about why the likelihood is no probability density function and I ve read most of the answers. However, to me the point is still not clear yet why the likelihood is no pdf. There have been several arguments, mainly involving that
- it does not integrate to 1
- it is a distribution over the parameters with the data fixed
However, there has also been an accepted answer that says "it is the probability (density) of the data given the parameter value", which to me sounds like a probability then.
My general confusion and problem about the understanding of the likelihood contains the following items:
1.) The Likelihood is (often) defined as $L(\theta|X)=p(X|\theta)$. But that IS a (conditional) pdf. There is nothing I can do about it to interprete it otherwise (e.g. by assuming to hold any of $X$ or $\theta$ fixed). The above expression means that I have a pdf over the random variable $X$ conditioned on the parameter $\theta$ which in fact is a conditional pdf, no?
2.) One could argue (such as on wikipedia) that the likelihood function is defined as $L(\theta|X)=p(X;\theta)$, i.e., explicitely not as a conditional pdf. However in Bayes theorem the likelihood is always a conditional pdf, as Bayes theorem is in principle only a consequence of the definition of conditional probability (density). Therefore in Bayes theorem I have to inteprete the likelihood as a conditional probability density.
3.) I am also confused about the definition of the likelihood in frequentist and Bayesian framework. In the former one assumes the data to be random variables and the parameters to be fixed unknowns and in the latter one assumes the data to be fixed and the paramters to be random variables. So it seems that the interpretation of the likelihood also depends on the framework I am working in?
4.) The pdf of a given distribution is often written as a conditional probability, e.g., the gaussian is often written as $p(X=x|\mu, \sigma)$ and then treated as a likelihood when using e.g. Bayes theorem. In that case we explicitly assume the likelihood to be a (conditional) pdf then. However, how is this then justified (if the likelihood is not a conditional pdf)?
5.) Why are there many textbooks in applied statistics and machine learning, that just use the likelihood as a conditional pdf just like in point 4.) if this is not correct?
EDIT: The discussion I have looked at involve: