$~$\\mathcal L(H \\mid e) < 0.05$~$ also doesn't mean "$~$H$~$ is less than 5% likely", and the student still needs to learn to keep "probability of $~$e$~$ given $~$H$~$" and "probability of $~$H$~$ given $~$e$~$" distinctly separate in their heads\. However, likelihood functions do have a simpler interpretation: $~$\\mathcal L(H \\mid e)$~$ is simply the probability of the actual data $~$e$~$ occurring if $~$H$~$ were in fact true\. No need to talk about experimental design, no need to choose a summary statistic, no need to talk about what "would have happened\." Just look at how much probability each hypothesis assigned to the actual data; that's your likelihood function\.

Have I gone mad, or do you mean "L(H|e) is simply the probability of H given that the the actual data e occurred"?

## Comments

Nate Soares

If those are the only two options, then you've gone mad :-)

L(H|e) is defined to be P(e|H) (which, yes, was a confusing and bad plan).

Reporting "the probability of H given the actual data e" would not work, because that requires mixing a subjective prior into the objective likelihoods. That is,

everyonecan agree "this sequence of coin tosses supports 'biased 55% towards heads' over 'fair' by a factor of 20 to 1", but we may still disagree about the probability that the coin is biased given the evidence. (For example, you may have started out thinking it was 100 : 1 likely to be fair, while I started out thinking it was 20 : 1 likely to be biased. Now our posteriors are very different.)The reason humanity currently uses p-values instead of Bayesian statistics is because scientists don't want to bring subjective probabilities into the mix; the idea is that we can solve that problem by reporting P(e | H) instead of P(H | e). The objective measure of P(e | H) is written L(H | e).