ABRUPT END OF THE GAME - DO WE NEED VERIFIABILITY?

There is another interesting lesson that we can learn from our six box toy experiment.After some time the game has to come to an end, and the audience expects that I finally show the composition of box

. Instead, I take it, put it back together with the others and shuffle all them well. As you might imagine, the reaction to this unexpected end is surprise and disappointment. Disappointment because it is human to seek the `truth'. Surprise because they didn't pay attention, or perhaps didn't take me seriously, when I said at the very beginning that “we are forbidden to look inside the box, as we cannot open an electron and read its mass and charge in a hypothetical label.”

The reason for this unexpected ending of the game is twofold. First, because scientists (especially students) have to learn, or to remember, that when we make measurements we remain in most cases in a condition of uncertainty. And not only in physics. Think, for example, of forensics. How many times judges and jurors will finally know with Certainty if the defendant was really guilty or innocent?(We know by experience that we have to distrust even so-called confessed criminals!)

The second reason is related to the question of the verifiability of the events about which we make probabilistic assessments. Imagine, that during our toy experiment we made 6 extractions, getting White twice, as for example in the following simulation in R. (Note that if you run the lines of code as they are, deleting ri immediately after it is used in the second line, you will never know the true composition! If you want to get exactly the probability numbers of the last two outputs shown below, without having to wait to get x equal 2, as it resulted here, then just force its value.)
> N=5; i=0:N; pii=i/N; n=6
> ri = sample(i, 1)
> ( s=rbinom(n,1,pii[ri+1]) ); rm(ri)
[1] 0 0 1 1 0 0
> ( x=sum(s) ) # nr of White
[1] 2
( PBi = pii^x * (1-pii)^(n-x) / sum( pii^x * (1-pii)^(n-x) ) )
[1] 0.00000000 0.34594595 0.43783784 0.19459459 0.02162162 0.00000000
> sum( pii * PBi )
[1] 0.3783784
At this point we have 44% belief to have picked and only 2.2% ; and 38% belief to get White in a further extraction. And these degrees of belief should be maintained, even if, afterwards, we lose track of the box.This is like when we say that a plane was at a given instant in a given cube of airspace with a given probability. Or, more practically (16), imagine you are in a boat on the sea or on a lake, not too far from the shore, so that you are able, e.g. using Whatsapp on your smartphone, to send to a friend your GPS position, including its accuracy. The location is a point, whose accuracy is defined by a radius such that “there is a 68% probability that the true location is inside the circle.”This is a statement that normal people, including experienced scientists, understand and accept without problems and which our mind uses to form a consequent degree of belief, the same as when thinking of the probability of a white ball being extracted blindly from a box that contains 68 white and 32 black balls. And practically nobody has concerns about the fact that such an event cannot be verified. Exceptions are, to my knowledge, strict frequentists and strict definettians (but I strongly doubt that they do not form in their mind a similar degree of belief, although they cannot `professionally' admit it.) In fact, for different reasons, it is forbidden to scholars and practitioners of both schools to talk about probability of hypotheses in the most general case, including probability that true values are in a given interval. For example neither of them could talk of the probability that the mass of Saturn is within a given interval, as instead it was done by Laplace, to whom was perfectly clear the hypothetical character of the so called coherent bet. As they would not accept talking about the most probable orbit (“orbitam maxime probabilitatem”), or the probability that a planet is at given point in the sky, as instead did Gauss when he derived his way the normal distribution from the conditions (among others) that i) all points were a priori equally likely (“ante illas observationes [...] aeque probabilia fuisse”); ii) the maximum of the posterior (“post illas observationes”) had to be equal to the arithmetic average of the observations (17).