In Reichenbach’s view, statements about probability are empirical statements about actual relative frequencies. Probability theory, in the version he developed, was the theory of conditional relative frequencies ‘in the long run’, that is, in infinitely long sequences of events.
But Reichenbach too understood that language has a pragmatic dimension, and that probability judgements play a crucial role in decision making. So he added that a person’s probability judgments are to be understood as estimates of relative frequency.
For example, that it seems more likely than not to me that it will rain in next July he understood as an estimate of the proportion of days on which it will rain, in next July (namely, of being more than 50%).
Just putting it like this, it fits well enough with probabilism in epistemology. But there is a difference in the probabilist’s view, that would be a constant thorn in Reichenbach’s flesh.
From the point of view of subjective probability, an estimate is an expectation value. An agent, as conceived in probabilism, would have a probability of rain for each coming day in July. If then asked for the estimate of the relative frequency of rainy days in July, the answer would be the average of those probabilities. This follows by a very quick theorem, De Finetti’s Law of Small Numbers. (See appendix below.)
But how would this sound to someone like Reichenbach? He might say: you are not talking about real relative frequencies in the world, you are just explaining a connection between subjective probabilities and subjective expectation values:
“A nice logical point! But are not going to make any claims about how your judgments relate to what really happens in July?”
This is challenge to the agent to engage in self-assesment. Does s/he, in addition to expressing that subjective expectation value judgment, make or imply any claim about how the estimate will relate to what will actually happen?
You can imagine the agent’s very non-plussed stare! Having said something like
“my estimate is that it will rain 3/5 of the days in July”,
s/he is now asked,
“Yes, but what are you claiming about
the actual proportion of rainy days in July?”
Is there any sensible way to understand, or answer, this question?
There is one way to answer this, that should gain some sympathy from someone like Reichenbach. Indeed, it consists in a theorem all of us share: the Law of Large Numbers. But the way in which probabilists understand it, differently from the frequentist, is just what will again elicit a sense of orgulity on our part.
The probabilists’ approach to the Law of Large Numbers:
A given agent’s opinion is represented by a probability function P that assigns subjective probabilities to a family of propositions (or events, if you like). Let us think of this family as the set of prpositions about possible outcomes of an experiment.
To discuss the relation between those subjective probabilities and what will be the real, actual outcomes, the agent may then imagine an infinitely long series of Bernoulli trials, that is a sequence of performances properly and independently carried out again and again.
If the agent is now asked about how his probability for outcome A will compare to the frequency with which outcome A will occur in that infinite sequence, what can s/he say? The short answer is: by the strong law of large numbers s/he will be sure (with subjective probability 1) that the relative frequency of the occurrence of A in the long run equals probability P(A).
How is this answer demonstrated? To make the self-assessment in question, the agent must model that long run of repetitions of the experiment. There are infinitely many different possible ways this long run could be. In detail:
Let the set of possible outcomes of the experiment be K, and let the propositions describing those outcomes form the family F of subsets of K. The agent’s probability function P assigns to a proposition A in F a number, P(A).
Then the model of the series of possible long runs that are Bernoulli trials will have as elements the items in this infinite product space:
K* = the set of all sequences s(1), s(2), s(3), …. of items in K
F* = the least Borel set that contains all the sets A(1) x A(2) x A(3) x … ,
where A(1) etc. are all members of F
P* is the product: P*(A(1) x A(2) x A(3) x …) = P(A(1)).P(A(2)).P(A(3))……
Defining the relative frequency of A in sequence s in the obvious way, the theorem then establishes that the following set :
the set of sequences s in K* in which the relative frequency of A equals P(A)
has P* probability 1.
In other words, with subjective probability function P* on the possible series of long runs, the agent is sure that the relative frequency will match his probability.
What more is there to be said? Roughly as in the case of the example of the Law of Small Numbers, the agent can present this as meaning: my probability for an event is my estimate of its relative frequency, if the same conditions are [or were] indefinitely repeated.
Our imaginary Reichenbach furrows his brow! “There you go again!” he says. That P* is just your own subjective opinion writ large!
“You have done nothing to make a claim about how your probabilities relate to actual frequencies, not even for the case in which real conditions are indefinitely repeated. All you have done, once again, is to make a little logical point about how various parts of your opinion hang together.”
The agent is non-plussed, and asks: So by what probability measure should I assess the relation of my opinion to what can happen? Someone else’s, maybe?
A new voice may weigh in here (Deborah Mayo’s, perhaps): what is important is not your confidence in your own opinion, underwritten by your own opinion! What is important is prudence with respect to real error. What is the chance that a hypothesis would be borne out by the experiment, if it is false? And what is the chance that it would not be borne out by the experiment, if it is true?
That is frequentist humility, in stark contrast to your orgulity!
Maybe this was a standoff? But just wait, some striking theorems are just around the corner.
APPENDIX. De Finetti’s Law of Small Numbers
To represent rainy days consider a quantity defined this way: it has value 1 on days when it rains and value 0 on days when it does not rain. The actual number of days in July when it rains is just the sum of the values of this quantity on the days in July.
For such a two-valued quantity, its expectation value for, say, July 1, just equals the probability that it rains on that day. And expectation value is additive, so the expectation value of that quantity on the whole of July is just the sum of its expectation values on the individual July days, which is, therefore, just the sum of the probabilities of rain on those days.
So the expectation value of the relative frequency of rain in July is that number divided by 31 (the number of July days), and therefore, precisely the average of the probabilities of rain on those days.
Cool! And it is irrelevant to the argument whether the events in question are statistically independent or not, or exchangeable, or whatever.
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