Physical Probability versus Evidential Probability

There is a very important divide between two different types of probabilities. These are:

(1) physical probability and
(2) evidential probability.

The nature of (2) evidential probability seems to be very important indeed to the issue of uncertainty in economic life.

For if one cannot be certain of something, especially of a particular event in the future, then one must face risk, uncertainty, or degrees of uncertainty.

I can’t claim to have any great expertise in mathematics or probability theory, so I have quite likely made mistakes below.

I discuss the types of probability, as follows.

(1) Physical probabilities (or objective or frequency probabilities)
These exist within a strict domain of certain random events, phenomena or processes, such as (non-fraudulent and fair) games of chance like roulette and rolling of dice, and the states of physical systems in both classical and quantum mechanics.

In any such phenomena, over a long period of time, any particular possible event or state of the system supposedly tends to occur at a persistent rate or “relative frequency.”

The current widely-used set of axioms for probability were formulated by Andrei Nikolaevich Kolmogorov in 1933.

Like me, I suspect most people are familiar with physical probability from standard courses on discrete mathematics. A real physical probability (or objective/frequency probability) is limited to strict conditions:

(1) it must be random, in the sense that, when an outcome occurs, it is from a set of possible known outcomes, and one such outcome is sure to occur and it is impossible to predict with certainty what that outcome will be. E.g., in tossing a fair coin, we know that the result will be heads or tails and can denote the set of outcomes by {heads, tails}. The latter is the sample space.

(2) it must have a sample space (or set of outcomes) that is known, in the sense that there are finitely many outcomes of the random process and these can be stated. (An event is defined as a subset of the sample space.)

(3) All events are equally likely to occur.

Simply stated, the probability P(E) of any event E in a finite sample space S, where all outcomes are equally likely, is the number of outcomes for E divided by the total number of outcomes in S.

But the philosophical interpretation of physical probabilities are disputed, and have been interpreted in the following ways:

(1) in the Classical interpretation (of Laplace);

(2) in a frequentist sense (as argued by Venn, Reichenbach and Richard von Mises) (also called aleatory probability), or

(3) in a propensity sense (as argued by Popper, Miller, Giere and Fetzer).

Note that Richard von Mises was the brother of Ludwig von Mises!

(2) Evidential probability (or Bayesian or subjectivist probability)
An evidential probability is distinct from a physical probability and can be assigned to any proposition, even when the process is not within the strict domain of physical probability. It is sometimes defined as a method of representing the subjective plausibility of an event and the degree to which a statement p can be supported by available evidence, or the degree of belief one can have that an outcome will happen.

An evidential probability is a type of conditional probability, denoted by the expression P(A, B), which is to be read as “the probability of A, given B.”

Examples of evidential probability statements could be:

(1) The extinction of the dinosaurs was caused by an asteroid hitting the earth.

(2) Napoleon did not die by poisoning.

(3) The rate of unemployment in the UK in 2014 will be 3%.

The precise nature of evidential probability is disputed, and there are four interpretations as follows:

(1) Classical interpretation (of Laplace);

(2) the subjective interpretation (de Finetti and Savage);

(3) the epistemic or inductive interpretation (Ramsey, Cox) and

(4) the logical interpretation (Keynes and Carnap).

There is a fundamental division between objective and subjective theories of probability, where subjective interpretations identify probabilities with degrees of belief of an individual, while objective theories see probabilities as indicative of the objective behaviour of the real world, such as relative frequencies or propensities.

The subjectivist view of evidential probability seems to be associated with Bayesianism or advocates of epistemic probability, who regard evidential probabilities as having a subjective status by which they measure the “degree of belief” of the individual assessing the probability of an event or outcome.

Precisely what position Keynes took in the Treatise on Probability on evidential probability may be disputed, but (for what it is worth) this is what Wikipedia says:

“One of the main points of disagreement lies in the relation between probability and belief. Logical probabilities are conceived (for example in Keynes’ Treatise on Probability) to be objective, logical relations between propositions (or sentences), and hence not to depend in any way upon belief. They are degrees of (partial) entailment, or degrees of logical consequence, not degrees of belief. (They do, nevertheless, dictate proper degrees of belief, ...) Frank P. Ramsey, on the other hand, was skeptical about the existence of such objective logical relations and argued that (evidential) probability is "the logic of partial belief" ("Truth and Probability", 1926, p. 157). In other words, Ramsey held that epistemic probabilities simply are degrees of rational belief, rather than being logical relations that merely constrain degrees of rational belief.”
http://en.wikipedia.org/wiki/Probability_interpretations

For example, the truth of many propositions inferred from a body of evidence is not strictly necessary (or entailed), but merely probable. We call these types of arguments inductive reasoning. Keynes may well have called this inductive argument “partial entailment.”

Keynes’s logical interpretation of evidential probability takes the latter to be a measure of the logical relation between a proposition and the evidence adduced for it, but this relation is not deductive (that is, an inference is not strictly entailed by evidence). The logical interpretation of evidential probability also seems to be an objectivist one. In contrast, Bayesianism is a subjectivist view of evidential probability, and probabilities are contingent and not logical.

Today almost all philosophers reject the logical interpretation of probability as flawed (Lyon 2010: 112)

Conclusion
It should be apparent that physical probability is highly restricted in its usefulness to special domains, and useless for many real world assessments of probability.

While Bayesianism seems to be an important modern theory of evidential probability, a more interesting question is: do economic agents, say, the average consumer or capitalist making an investment decision, all or generally behave as required by Bayesian probability theory?

If not, it follows that Bayesian probability has far less relevance to the theory of decision making in economic life than imagined in neoclassical economics.

Furthermore, even if some agents do use Bayesian probability, are their probability calculations really providing useful measures of the probability of future events affecting economies?

Other Questions
Some questions that I do not have answers to:

(1) In probability theory, there are (1) “absolute probabilities” (also known as “unconditional probabilities”) and (2) “conditional probabilities.”

Are “absolute probabilities” to be identified with “physical probabilities,” and “conditional probabilities” to be identified with “evidential probability”?

(2) Some thinkers (such as Alfred Rényi and Popper) have developed a theory of probability that takes “conditional probabilities” as the primitive notion. Has there been any resolution of this in modern theory?

(3) Did Keynes change his mind on probability theory after he read Frank Ramsay’s the critique of A Treatise on Probability?

(4) Is the model of decision making by the rational agent in neoclassical economics basically derived from Frank Ramsay’s Bayesian and subjectivist theory probability?

BIBLIOGRAPHY
Lyon, A. 2010. “Philosophy of Probability,” in F. Allhoff (ed.), Philosophies of the Sciences: A Guide. Wiley-Blackwell, Chichester, UK and Malden, MA. 92–127.