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Pointwise multiplication of, say, two density functions gives the likelihood function of two independent random variables with these densities. This function is
by Mithriil 6y ago
Pointwise multiplication of, say, two density functions gives the likelihood function of two independent random variables with these densities. This function is not a density function, as you say, but is still an important object, as it is used for inference based on the likelihood principle.
- H8crilA 6y agoI'm sorry, but I'm lost, are we talking about this likelihood function? I'm guessing no, since that's an object defined in a statistical space, not in a probabilistic space: https://en.m.wikipedia.org/wiki/Likelihood_function https://en.m.wikipedia.org/wiki/Likelihood_function
- lalaithion 6y agoOftentimes, a “likelihood function” is generalized from that definition to refer to any non-normalized function which is still treated like a probability distribution. Any function with a finite integral over the real line can be turned into a probability distribution easily, and there are other functions that can be used in places where you would normally use a probability function, even when they don’t integrate to a finite value over the real line. (For example, improper priors in Bayesian analysis.) “Function that can in some contexts be treated like a probability distribution” is just too long.
- moultano 6y agoYes, that's the right likelihood function. To get probabilities distributions out of the products in the post, all you have to do is normalize them. That would be equivalent to applying bayes rule and combining independent evidence about a parameter with an uninformative prior. Normalizing doesn't change the shapes, so I left out the details.
- srean 6y agoBut then the term would be P(x1) P(x2) and the correct summation/integration to use would be over the cartesian product of the sample space. Under such summation/integration its a valid density