As discussed before, frequentism suffers from some conceptual problems as an objective theory of probability. One problem we have not discussed, however, is that posed by one time events. Consider, for example, betting odds on a sporting event. I estimate that the probability the Aggies will beat the Longhorns in their upcoming game is 1/3, and bet accordingly. Now, this game is a one time, irrepeatable event, can we make objective sense of such a probability assignment? (If the fact that the Aggies and the Longhorns have met many times in the past is throwing you, consider this: on each meeting, the teams have had different players, different coaches, and different records for the season; thus, these are not repeats of the same event as with the tossing of a fair coin.)
Another popular example of one time probabilistic events is radioactive decay: when speaking of the probability that a lump of uranium will emit an α-particle within some time period t, we cannot be referring to the frequency of the outcome of a process (what process? something internal to the uranium? ~ but uranium emits particles "spontaneously," surely if there is such a process it is unobservable).
One solution to these worries is to interpret probability in terms of propensity: to say the Aggies only have a 1/3 chance of beating the Longhorns is to speak of something about the Aggies (the makeup of the team, the strategies they use, the quality of the coaching, etc.) which objectively determines their chances of winning this one time event. In the case of the uranium, we can say it has the propensity to decay at a certain rate. In the case of a coin, we can say a coin is "fair" if it has a propensity to come up heads with probability 1/2. Here, it makes sense to speak of the coin (or, more specifically, the mechanism of the toss) as being "fair" or not (as exhibiting a certain structure) even before the coin has been tossed a single time ~ no notion of a hypothetical infinity of trials is needed.
However, there are conceptual problems with the propensity interpretation as well. In particular, propensities cannot themselves be probabilities. The symmetry in the probability calculus which allowed us to derive Bayes' Rule is not exhibited by propensities (as pointed out in Humphries, 1985):
The point can be illustrated by means of a simple scientific example. When light with a frequency greater than some threshold value falls on a metal plate, electrons are emitted by the photoelectric effect. Whether or not a particular electron is emitted is an indeterministic matter, and hence we can claim that there is a propensity p for an electron in the metal to be emitted, conditional upon the metal being exposed to light above the threshold frequency. Is there a corresponding propensity for the metal to be exposed to such light, conditional on an electron being emitted, and if so, what is its value? Probability theory provides an answer to this question if we identify conditional propensities with conditional probabilities. The answer is simple-calculate the inverse probability from the conditional probability. Yet it is just this answer which is incorrect for propensities and the reason is easy to see. The propensity for the metal to be exposed to radiation above the threshold frequency, conditional upon an electron being emitted, is equal to the unconditional propensity for the metal to be exposed to such radiation, because whether or not the conditioning factor occurs in this case cannot affect the propensity value for that latter event to occur. That is, with the obvious interpretation of the notation, Pr(R/¬E) = Pr(R/E) = Pr(R). However, any use of inverse probability theorems from standard probability theory will require that P(R/E) = P(E/R)P(R)/P(E) and if P(E/R) ≠ P(E), we shall have P(R/E) ≠ P(R). In this case, because of the influence of the radiation on the propensity for emission, the first inequality is true, but the lack of reverse influence makes the second inequality false for opensities.
To take another example, heavy cigarette smoking increases the propensity for lung cancer, whereas the presence of
(undiscovered) lung cancer has no effect on the propensity to smoke, and a similar probability calculation would give an incorrect result. Many other examples can obviously be given.
The point here is just that probabilities are symmetric with respect to causal order, this is the trick which allowed us to derive Bayes' Rule. Yet propensities cannot be made sense of in this way; they are asymmetric with respect to causal order.
Nevertheless, Suppes, 1987 and 2002 has argued that in particular cases one can derive a probability calculus from propensities. This is why we speak of propensity theories: there can be no one unified derivation of the probability calculus from a general theory of propensity.
Now that we've got some different perspectives on the table, let's see how they deal with a couple of examples.
next: "priors" and the fair coin revisited