but only in Statistics. (I promise, this is the last Bayesian post, at least for a while.)
I've always thought that 'probability' and 'likelihood' were synonyms, but yesterday I learned that in Statistics they have distinct and complementary meanings. Unfortunately it's hard to define either of them without using the other, so I'll use the word 'chance' and clarify with examples.
Consider that you have some data and that you have a hypothesis about the reality that produced this data. For example, the data could be that plating the same volume of bacteria on a novobiocin-agar plate and a plain-agar plate gave 43 and 321 colonies respectively, and your hypothesis about reality is that 15.0% of the cells in the culture are able to grow in the presence of novobiocin (are NovR).
Likelihood (as defined for statistical work) is the chance that a real culture with 15.0% NovR cells would have given these numbers of cells when that volume was plated. More generally, it's the chance that the reality you've hypothesized (often your 'null hypothesis') could have produced the particular data you got. This is what classic 'frequentist' statistical methods deal with. Phylogenetic methods using 'maximum likelihood' presumably take this approach.
Probability (as defined for statistical work) reasons the other way around. It's the chance that the culture really has 15.0% NovR cells, given that your plating experiment produced 43 NovR colonies out of 321 total colonies. More generally it's the chance that the reality you're considering is true, given the data you have. This is what Bayesian methods deal with. The phylogenetic software 'Mr. Bayes' presumably takes this approach.
For now I'm not going to worry about why this might matter.
Subrahmanyan Chandrasekhar: A study in fortitude and rigor
1 day ago in The Curious Wavefunction