I'm interested in likelihood $L(\theta | x) = P( x | \theta)$ seen as the function of $\theta$ for a given $x$.

I have been reading in books and other posts that $L$ is generally not a probability function (with respect to $\theta$) because it may not integrate to 1 and, moreover, there my be even no $\sigma$-field defined over the $\theta$ space.

In the Bayesian approach, we define a prior probability $P(\theta)$ and so the $\sigma$-field had to be defined.

What other properties we need to be able to interpret $L(\theta|x)$ as the conditional probability $P(\theta|x)$? Perhaps normalization to integrate to 1? What else? (whuber mentions some regularity conditions in this post Wikipedia entry on likelihood seems ambiguous. What would these be?)


After some searching, I believe the answer is in Fisher's fiducial distribution. See Casella&Berger(2002, sec. 6.3.1), the original Fisher's papers (1933,1935) or this question: What is the fiducial argument and why has it not been accepted?


Now that I know the key-word is fiducial, there is a fairly good explanation in the answer to this question: What does "fiducial" mean (in the context of statistics)?

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    $\begingroup$ I don't think this is a duplicate. This goes beyond the question and discussion in 'what is the difference between likelihood and probability'. I know they are not the same, the differences are clear, no need to explain that. I'm asking: what would be the conditions (possibly some regularity conditions and need to normalize) so that the likelihood could be interpreted as a probability (of the parameter given the data)? $\endgroup$ – mag May 19 '17 at 17:59
  • $\begingroup$ Indeed, not a duplicate, see my EDIT in the question. $\endgroup$ – mag Jun 14 '17 at 16:02
  • $\begingroup$ See also stats.stackexchange.com/questions/224037/… $\endgroup$ – Tim Jun 20 '17 at 9:50
  • $\begingroup$ @Tim - Thanks. Yes, I've seen this discussion before. It indeed explains well the standard view of likelihood function (not a prob distrib of $\theta|x$) and the bayesian interpretation. I will read a bit more on the fiducial argument of Fisher who was known to be an anti-Bayesian yet, I believe, in this notion (the fiducial distribution) wanted to capture the possibility of treating theta as r.v. I'll update my post accordingly if I find a neat answer to my question. $\endgroup$ – mag Jun 21 '17 at 11:04