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.
“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.”
A die is not the model, but the object to be modeled. If you model the die with the fair-die-model, you can compute the change that 5 rolls of the die result in 5 sixes. The model says that out of the 6*6*6*6*6 possible outcomes, there is exactly one with 5 sixes.
“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.”
There are a lot of models possible for a die, the fair-die-model is one, the die-that-always-comes-up-6-model is another one, and the highly contrived die-that-will-show-5-sixes-and-then-1-and-then-2-and-then-3-etc-model is still another one. It is in fact possible to think of an infinite number of possible models, simply because it is always possible to throw the die one more time.
Now, what would an indirect probability mean? Given a set of data, and a number of different theories that all predict this particular set, it should be the change that I pick the right theory from all the possible theories. But there are an infinite number of such theories, and the change of picking the right one is therefore zero.
That makes no sense.
Look at it like this. Throwing 5 sixes will eliminate all theories in which the first 5 throws are not sixes. It will not eliminate any theory predicting the first 5 throws to be sixes, including the theory that the die is a fair die. And that is all it can do.
Sometimes I wonder if all of the discussions of fair dice, isn’t a waste of time.
If I take casino crap tide, and I put it in a bag with some rocks and shake it about. Now it has chipped corners, rough edges, and slightly less than square faces. It would be reasonable to say that it is in all likelihood not-a-fair die. Yet, as I don’t really know how this dice abuse has changed the die, I would still have to say that based on the information that I have, the probability of rolling any number is still 1/6.
The probability of rolling the same number on consecutive rolls may be slightly greater than 1/6.
More thoughts on dice. How do you test to see if dice are fair? Again back to the casino, they take a little T-square and check the dice for any anomalies in shape and balance. You cannot test the fairness of dice by rolling them. If you roll the dice 1000 times, you will have begun to wear down the die. If it wasn’t biased at the beginning your test, it may be biased by the end of your tests.
It is highly probably that trying to follow this line of articles will make my head explode.
Am I a frequentist or a Bayesian?
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I think I am on Senn’s side but might be too pragmatic to give a darn. My die has about 5000 sides. 0 (no failure) is the most probable result if the die (hardware under test) is not broken.
Otherwise you hope that a string of sixes is because physics has biased the die to show six whenever test six fails.
As such I get concerned any time any number shows up even three times in a row if followed by a bunch of zeros. Since this has happened I need a probability going from data to model.
I am not sure what’s wrong with Senn’s direct and inverse probabilities that
involve hypothesis (model) and data. They are the probabilities concerned in the fallacy of the transposed conditional in a previous post.
Is there only one kind of probability? If yes, is this kind of probability numerical? We know that probabilities might be comparative but not quantitative.