Every probability problem has the form Pr(Q|E), where Q is the proposition of interest and E the evidence, premises, or “data” probative (or not) of E. Change the evidence, change the probability of Q. That is, unless E1 is, given Q, logically equivalent to E2, Pr(Q|E1) will not equal Pr(Q|E1).
For instance, Q = ‘A 6 shows’, E1 = ‘This is an n-state machine with states labeled 1-n, and one state must show and this is a state’ and E2 = ‘This is a 2*n-state machine, etc.’. Thus Pr(Q|E1) = 1/n and Pr(Q|E2) = 1/(2n).
This is why there is no “probability of Q” without evidence; i.e. there is no such thing as implicit probability. If somebody says, as for example Richard Dawkins recently said, “There’s a very interesting reason why a prince could not turn into a frog — it’s statistically too improbable”, he must have reasoned he and his listener agreed on the E, the evidence used in deducing that curious probability. For there could not be a probability of a Q = ‘This prince turned into a frog’ without it.
Part of the evidence, incidentally, is tacit understanding of the words and grammar used in Q and E. Presumably Dawkins did not (here) mean by frog “A Frenchman.”
The proviso “given Q” in the first paragraph is interesting. For example, let E1 = ‘This is an n-state machine with states labeled 1-n, and one state must show and this is a state’ and E2 = ‘This is an n-state machine with states labeled 1-n, and one state must show and this is a state and the machine is blue’. Given Q, these two statements of evidence are equivalent. The color of the machine is of no interest in calculating the probability of Q. But given (say) R = ‘The machine is blue’, E1 and E2 are no longer logically equivalent. In this way, the probability equation “works in both directions.”
Since all probability is conditional, every experiment you have ever have or ever will hear of is conditional on the premises of that experiment. Data are part of these premises, as are the characteristics of that experiment.
Unfortunately, we must also assume no errors in calculation are made, including cheating. These aren’t as rare as one hopes. But worse are mistakes in reasoning. Above I said that a tacit premise is grammar. A frequent and flagrant error is to abuse grammar, for instance in the epidemiologist fallacy. This is where the researcher says he has calculated the probability of Q given E, but where he actually gives the probability of some R given (usually some modified) E. Click that link for examples.
Let’s take a test case. A set of questions is asked of a handful of college kids at an earnest American university. Are the answers these kids give representative of every other human being, of all time? Consider that the questions are to confirm some psychological, sociological, or biological theory.
One implicit premise is that the participants are human beings. Don’t laugh. When bench scientists conduct “exposure” experiments on “animal models”, this premise, that the results apply to human beings, is much shakier than when a sociology professor asks questions. Another premise is that the kids go to this college. Another is that so many wore dark socks and so many light ones. You can think of many more: indeed, you must. Are these probative of Q or aren’t they?
If you want the results to apply for people outside this university and for people who wear socks of different color, or indeed wear no socks at all, you must say (or hope) no. If being at this university, which is a premise, is probative of Q, then moving to human beings at another university changes the premises, and thus changes the probability of Q. Which direction? Who knows?
There are a host of other attributes about the experiment and the individuals involved, like the sock color, that are all premises. Everything that was there at the experiment is a premise, down to the last quark. The announced probability of Q might be dependent on all of these, but not all of these “shift” Q in any particular direction. Remember that probability is not the language of causality. We’re not saying whether any premises caused Q, only how it relates to the probability of Q.
If we want to say how our experiment is applicable to other people, then it is our duty to clearly identify the premises thought (but usually not proven) to be most probative of Q. For instance, sex of the participants is a premise. Removing it from the list of premises might shift the probability of Q, but if the shift is trivial or minor,
This is only a sketch of this topic. More to come. Enough’s enough on a Saturday.