Just after the introduction, Senn starts his argument by claiming an “important distinction between two types of probabilities: direct and inverse.”
The distinction is simply explained by an example. The probability that five rolls of a fair die will show five sixes is an example of a direct probability—it is a probability from model to data. The probability that a die is fair given that it has been rolled five times to show five sixes is an inverse probability: it is a probability from data to model.
If we accept this distinction and example as written, we are already lost; all the standard confusions are there.
If probability is all of one sort, then there is no distinction between “direct” and “inverse” kinds. Our candidate is logical probability, in which, as in just-plain-logic, there is only evidence (equivalently, premises), a proposition to be considered with respect to that evidence, and a probability this proposition is true deduced from the evidence.
Let’s begin by rewriting the examples. The evidence is what? Trouble starts with the words “fair die”. This is taken to mean that we have a real, physical, tangible object which must, when tossed, results in equal chance of any side face up. This is asserted and not proved. It is a dictate. It sets in the mind a view of an actual die, of the kind that cannot (or at least does not) exist. Once this die is imagined, objections immediately arise: what if it isn’t “fair”? Can real dice be “fair”? What about imperfections? The confusion between asserting a probability and wondering whether the asserted probability equals the “real” probability, i.e. the long-run frequency of tosses, is already ineradicable. It becomes impossible to keep in mind what the real question is.
Start over rewriting all as a logical argument. “We have a six-sided (logical) object, just one side of which is labeled ‘1’, just one side of which is labeled ‘2’, and so on up to ‘6’, which when tossed must show just one of these sides.” No physical, real die is implied, though because of the ubiquity of dice-like examples, people usually think one is. So if you find yourself unable to imagine a logical, i.e. non-physical, six-sided object, change it to a six-state Martian bleen, a device which is activated by tentacle and displays each time it is activated on a screen one and only one of the figures (translated into English) ‘1’, ‘2’, etc. There is no hint—as in no hint—of the workings of this device. All—as in all—we know is that the device when activated can show one of ‘1’ through ‘6’; how it does so is a mystery.
I stress again (and again) that since there are no Martians, there are no bleens. Any imperfections we imagine in a bleen are our own creations and are not part of the evidence supplied. The key to LPB is that we must—as in must—use only the evidence supplied, and all of it, in our deductions of probability. What is not directly implied from the given evidence must—as in, well, you get the idea—be ignored.
Now using the statistical syllogism (which itself can be deduced from simpler principles), we deduce the probability a ‘6’ shows on one activation of a bleen, just as we can deduce the probability of five ‘6’ activations. Or we can deduce anything which can happen in any (for now stick to finite) number of activations.
We are done with the first example which ends with at a conditional probability; i.e. a probability deduced from given, fixed evidence. All probability is likewise conditional. If you think not, see the series linked above for examples, or see Part III tomorrow for more on this.
Notice that I do not use the word “model”. It isn’t needed. Not here, and in far fewer cases than usually thought.
Senn’s second (“inverse”) example is also confusing. This asks the probability the following proposition is true: “This die is ‘fair’.” The only written evidence is “This die has been rolled five times and has showed five ‘6’s.” That we are dealing with a real, physical die is implied from the words, but it is never stated. But suppose this is wrong and Senn meant a logical die or a breen: then where would we be?
Right where we started. If this is the logical “die” or breen, then we start by knowing the chance each number is displayed is 1/6. We end there, too. We have deduced “fairness.”
So we must be talking of a physical, rea-life die. Our task is to interpret this proposition with regard to the given observations.
This evidence is easy and means just what it says: five rolls, five ‘6’s of some real die. The proposition is less clear. The subject makes sense: “This die” means some real, actual physical die. The difficulty is with the verb: “is fair.”
Ah, fairness. From youth we are told that there is nothing finer! Indeed, fairness is so fine that we discuss it next time.