Probability, propensity and (relative) frequency

A curious myth is that physical probability, or propensity, has “only a frequentist interpretation” (and therefore “physicists must be frequentist”, as ingenuously stated by the Roman PhD student quoted on the first page). But it seems to me to be more a question of education, based on the dominant school of statistics in the past century (and presently),rather than a real logical necessity.

It is a matter of fact that (relative) frequency and probability are somehow connected within probability theory, without the need for identifying the two concepts.

• A future frequency $f_n$ in $n$ independent `situations' (not necessarily `trials'), to each of which we assign probability $p$, has expected value $p$ and `standard uncertainty' decreasing with increasing $n$ as $1/\!\sqrt{n}$, though all values 0, $1/n$, $2/n$, ..., 1 are possible (!). This a simple result of probability theory, directly related to the binomial distribution, that goes under the name of Bernoulli's theorem, often misunderstood with a `limit', in the calculus's sense. Indeed $f_n$ does not “tend to” $p$, but it is simply highly improbable to observe $f_n$ far from $p$, for large values of $n$. In particular, under the assumption that a system has a constant propensity $p$ in a large number of trials, we shall consider very “unlikely to observe $f_n$ far from $p$.” Reversing the reasoning, if we observe a given $f_n$ in a large number of trials, common sense suggests that the `true $p$' should lie not too far from it, and therefore our degree of belief in the occurrence of a future event of that kind should be about $f_n$.
• More precisely, the probability of a future event can be mathematically related, under suitable assumptions, to the frequency of analogous events $E^{(i)}$ that occurred in the past. For example, assuming that a system has propensity $p$, after $x$ occurrences (`successes') in $n$ trials we assign different beliefs to the different values of $p$ according to a probability density function $f(p\,\vert x,n,I)$, whose expression has been reported in Footnote 24. In order to take into account all possible values of $p$ we have to use Equation (14), in whose r.h.s. we recognize the expected value of $p$. We get then the famous Laplace rule of succession (and its limit for large $n$ and $x$),
  

  $$P(E\,\vert\,x,n,I) = \mbox{E}[p\,\vert\,x,n,I] = \int_0^1\! p\,\frac{(n+1)!}{x!\,(n-x)... ...-x} \mbox{d}p = \frac{x+1}{n+2} \ \ \longrightarrow \ \frac{x}{n} = f_n\,, \ \$$ (15)

  
  
  which can be interpreted as follows. If we i) consider the propensity of the system constant; ii) consider all values of $p$ a priori equally likely (or the weaker condition of all values between 0 and 1 possible, if $n$ is `extraordinary large'); iii) perform a `large' number of independent trials, then the degree of belief we should assign to a future event is basically the observed past frequency. Equation (15) can then be seen as a mathematical proof that what the human mind does by intuition and “custom” (in Hume's sense) is quite reasonable. But the formal guidance of probability theory makes clear the assumptions, as well as the limitations of the result. For example, going back to our six box example, if after $n$ extractions we obtained $x$ White, one could be tempted to evaluate the probability of the next White from the observed frequency $f_n=x/n$, instead of, as probability theory teaches, firstly evaluating the probabilities of the various compositions from Equation (6) and then the probability of White from (10). The results will not be the same and the latter is amazingly `better'(1).

There is another argument against the myth that physical probability is `defined' via the long-term frequency behavior. If propensity $p$ can be seen as a parameter of a physical system, like a mass or the radius of the sphere associated with the shape of an object, then, as other parameters, it might change with time too, i.e. in general we deal with $p(t)$. It is then self-evident that different observations will refer to propensities at different times, and there is no way to get a long-term frequency at a given time. At most we can make sparse measurements at different times, which could still be useful, if we have a model of how the propensity might change with time.