We’re almost done. Only one more after this.
There are examples without number of the proper use of Bayes’s Theorem: the probability you have cancer given a positive test and prior information is a perennial favorite. You can look these up yourself.
But be cautious about the bizarre lingo, like “random variable”, “sample space”, “partition”, and other unnecessarily complicated words. “Random variable” means “proposition whose truth we haven’t ascertained”, “sample space” means “that which can happen” and so on. But too often these technical meanings are accompanied by mysticism. It is here the deadly sin of reification finds its purchase. Step lightly in your travels.
Let’s stick with the dice example, sick of it as we are (Heaven will be the place where I never hear about the probability of another dice throw). If we throw a die—assuming we do not know the full physics of the situation etc.—the probability a ‘6’ shows is 1/6 given the evidence about six-sided objects, etc. If we throw the same die again, what is the probability a second ‘6’ shows? We usually say it’s the same.
But why? Short answer is that we do so when we cannot (or will not) imagine any causal path between the first and second throws.
Let’s use Bayes’s theorem. Write for the standard premises about dice (“Six-sided objects, etc.), means “A ‘6’ on the first throw”, and means “A ‘6’ on the second throw”. Thus we might be tempted to write:
In this formula (which is written correctly), we know and say . Thus is must be (if this formula holds) . This says given what we know about six-sided objects and assuming we saw a ‘6’ on the first throw, the probability of a ‘6’ on the second is the same as the probability of a ‘6’ on the first toss assuming there was a ‘6’ on the second toss. Can these be anything but 1/6, given ? Well, no, they cannot.
But there’s something bold in the way we wrote this formula. It assumes what we wanted to predict, and as such it’s circular. It’s strident to say . This assumes, without proof, that knowledge of the first toss does not change our knowledge of the second. Is that wrong? Could the first toss change our knowledge of the second. Yes, of course.
There is some wear and stress on the die from first to second throw. This is indisputable. Casinos routinely replace “old” dice to forestall or prevent any kind of deviation (of the observed relative frequencies from the probabilities deduced with respect to ). Now if we suspect wear, we are in the kind of situation where we suspect a die may be “loaded.” We solved that earlier. Bayes’s Theorem is still invoked in these cases, but with additional premises.
Bayes as we just wrote it exposes the gambler’s fallacy: that because we saw many or few ‘6’s does not imply the chance of the next toss being a ‘6’ is different than the first. This is deduced because we left out, or ignored, how previous tosses could have influenced the current one. Again: we leave this information out of our premises. That is, we have (as we just wrote) the result of the previous toss in the list of premises, but does not provide any information on how old tosses affect new ones.
This is crucial to understand. It is we who change to the evidence of plus that which indicates a die may be loaded or worn. It is always us who decides which premises to keep and which to reject.
Think: in every problem, there are always an infinite number or premises we reject.
If it’s difficult to think of what premises to use in a dice example, how perplexing is it in “real” problems, i.e. experiments on the human body or squirrel mating habits? It is unrealistic to ask somebody to quantify their uncertainty in matters which they barely understand. Yet it’s done and people rightly suspect the results (this is what makes Senn suspicious of Bayes). The solution would be to eschew quantification and rely more on description until such time we have sufficient understanding of the proper premises that quantification is possible. Yet science papers without numbers aren’t thought of as proper science papers.
Conclusion: bad uses of probability do not invalid the true meaning of probability.
Next—and last—time: the Trials of Infinity.