Bayesian Likelihood function range

Cross Validated Asked by SHAMM on October 29, 2020

I have given a prior which has a range of data between[0.5,1.5] and a normal distribution N(0.9,1) which should truncate at the above range and normalize.

also I have the likelihood distribution in a form of errors distributed normally N(0,1)

when I try to write the likelihood function should I take the range [0.5,1.5] itself? or I should take the range from [0,inf]?

Direct answer: The likelihood as described in your second paragraph is defined on the whole of [-inf,inf], and that's the domain on which it should be.

Meta-answer: It doesn't matter for any practical inference purpose. Outside of [0.5,1.5], the likelihood will always be multiplied by the zero value of the prior, so you can assign the likelihood any set of values you like - Gaussian, zero, the square root of a giraffe - and it won't make any difference to your results.

Meta-meta-answer. That prior is a bad idea, for one philosophical reason and one practical reason. The philosophical reason is that that prior expresses the certainty that the value of the parameter is not outside [0.5,1.5]. But the prior represents what you believe before you've seen the evidence, and you should never express certainty of anything (or at least, anything logically contingent) before you've seen the evidence. (Unless, of course, that prior is actually the posterior from an inference process you've already done based on some previous data set.) The practical reason is that that prior has no gradient outside [0.5,1.5], which will make a lot of computational integration and optimization methods very (perhaps infinitely) inefficient.

Correct answer by Daniel Hatton on October 29, 2020

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