Continuing with the probability theory, a quick myth-busting. I touched on this last time, but it comes up often enough to deserve its own post. Recall that rationality requires us to calculate the probability of our theory of interest T given everything we know K. We saw that it is almost always useful to split up our knowledge into data D and background B. These are just labels. In practice, the important thing is that I can calculate the probabilities of D with B and T, so that I can calculate the terms in Bayes’ theorem,
Something to note: in this calculation, we assume that we know that D is true, and yet we are calculating the probability of D. For example, the likelihood . The probability is not necessarily one. So do we know D or don’t we?!
The probability is not simply “what do you reckon about D?”. Jaynes considers the construction of a reasoning robot. You feed information in one slot and, upon request, out comes the probability of any statement you care to ask it about. These probabilities are objective in the sense that any two correctly constructed robots should give the same answer, as should any perfectly rational agent. Probabilities are subjective in the sense that they are relative to what information is fed in. There are no “raw” probabilities . So the probability asks: what probability would the robot assign to D if we fed in only T and B?
Thus, probabilities are conditionals, and in particular the likelihood represents a counterfactual conditional: if all I knew were the background information B and the theory T, what would the probability of D be? These are exactly the questions that every maths textbook sets as exercises: given 10 tosses of a fair coin, what is the probability of exactly 8 heads? We can still ask these questions even after we’ve actually seen 8 heads in 10 coin tosses. It is not the case that the probability of some event is one once we’ve observed that event.
What is true is that, if I’ve observed D, then the probability of D given everything I’ve observed is one. If you feed D into the reasoning robot, and then ask it for the probability of D, it will tell you that it is certain that D is true. Mathematically, p(D|D) = 1.