Probability and Uncertainty

There are two fundamental types of probability with subcategories:

(1) Physical/Objective probabilities (class probabilities), divided into:

(i.) A priori probabilities (mathematical/Classical probabilities)
(ii.) Relative frequency probabilities (or a posteriori/empirical/experimental probabilities), and

(2) Subjective probability (or evidential/Bayesian probability).

These are discussed below, with the issue of uncertainty in section (3).

(1) Physical/Objective Probabilities
Again, these are divided into:

(i.) A priori probabilities (mathematical/Classical probabilities), and
(ii.) Relative frequency probabilities (or a posteriori/empirical/experimental probabilities).

Objective probabilities are either in practice or in theory quantifiable with a numerical value (or numerical coefficient of probability). The numerical value that describes the likelihood of an occurrence or event can range from 0 (impossibility) to 1 (certainty).

A priori probabilities can be calculated from antecedent information and before the experiment or the event in question, such as probabilities of coin tosses.

Relative frequency probabilities, on the other hand, are derived from the empirical data of a sufficiently representative, random sample. Usually a reference class and attribute of interest are involved, and in theory the probability can be expressed as a numerical value, calculated as a fraction where the denominator is the number of members in the reference class and the numerator is the number of members of the reference class who have the attribute involved. Probabilities are assigned to events on the basis of available evidence or sample, and therefore may be different when people have different sized data.

But is also possible to view a priori probabilities as relative frequency probabilities: hence the probability of heads in a fair coin toss at 0.5 can be conceived as the relative frequency of that outcome in repeated experiments of coin tosses over many instances, and as the repeated experiments approach infinity supposedly the numerical value will approach 0.5.

Advocates of the frequentist interpretation of probability might contend that a priori probabilities do not exist, but are ultimately explained by relative frequencies. It is interesting that Ludwig von Mises referred to objective probabilities as “class probabilities,” under the influence of his brother Richard von Mises (1883–1953), a proponent of the frequentist interpretation of physical probabilities.

I assume (I could be wrong) that if Bayesian probability uses frequency probabilities, it may also be able to yield objective probabilities.

A fundamental point is that risk (as opposed to uncertainty) is associated with objective probabilities, either a priori probabilities or relative frequency probabilities, when a numerical value can be assigned, as Frank Knight argued (though Knight’s terminology was potentially misleading as he also called risk “measurable uncertainty”).

Post Keynesians would argue that risk is not the relevant concept in many entrepreneurial investment decisions, but uncertainty.

(2) Subjective Probability (or evidential probability/Bayesian probability)
In instances where probabilities of events cannot be analysed in terms of relative frequencies or because the events are unique and cannot be included in a reference class, probability theory has been developed that measures “degrees of belief,” and that can be termed “subjective probability.” The usual procedure for this is some form of Bayesian probability theory.

In neoclassical economics, subjective probability theory was developed from the work of John von Neumann, Oskar Morgenstern, Frank Ramsey, Bruno de Finetti, and Leonard J. Savage, the latter of whom (drawing on Bayesian probability theory as well) formulated a formal model of decision-making where optimal decisions are made to maximise expected utility, and probability distributions are given by subjective evaluations.

But even here uncertainty is seen as a state of the mind, not as a state of the world, and ultimately Walrasian general equilibrium theory in its various forms requires real, objective probabilities to actually exist for events in economic decision making, and for the subjective probabilities of agents to converge towards these objective probabilities over time.

Curiously, though being subjectivists, Austrian economists reject the expected-utility representation of decision making under uncertainty in neoclassical economics (Langlois 1994: 118). We should also note that Ludwig von Mises’s “case probability” is not really the same thing as Bayesian subjective probability. Case probability is a purely subjective form of probability and Mises argued that “case probability is not open to any kind of numerical evaluation” (Mises 1998: 113). By contrast, Bayesianism does give numerical values to evidential probabilities, even if these are deemed subjective, but are updated and revised in light of new evidence.

(3) Uncertainty
In understanding uncertainty, the distinction between ergodic and non-ergodic processes is important. For neoclassical theory, reliable knowledge of the future requires the assumption of the ergodic axiom. Ergodicity is a property of some process or phenomenon in which time and/or space averages or attributes of that system either coincide for an infinite series or converge as the finite number of observations increases (Dunn 2012: 434). Thus a sufficient sample of the past can be said to reveal the future in an ergodic process.

But, for Post Keynesians, the complications involved in assessing the ergodic or non-ergodic nature of an economic process might be considerable, especially as there exist:

(1) genuinely ergodic economic processes/phenomena;

(2) genuinely non-ergodic economic processes/phenomena;

(3) economic processes/phenomena that appear ergodic for short periods of calendar time, but may change. (Dunn 2012: 435).

For example, non-stationarity and Shackle’s “crucial decision” concept in decision making are sufficient conditions for non-ergodicity, but not necessary conditions (Dunn 2012: 435–436). Future events or processes that are created by human agency are, above all, candidates for non-ergodicity.

Events where objective probabilities exist imply an ergodic world or a justified use of the ergodic axiom. Information from past and present data series should allow a probability estimate that approaches the objective numerical value as the data increases, even for future events.

Keynesian uncertainty (in the sense of Keynes and Post Keynesianism) stresses the unknowable nature of the future and the inappropriateness or profound limitations of probability theory.

Although there is not an exact equivalence between all the various concepts below (and perhaps some important differences), these concepts of uncertainty are roughly similar to Keynesian uncertainty:

(1) Knightian (unmeasurable) uncertainty;

(2) Misesian case probability;

(3) G. L. S. Shackle’s radical uncertainty;

(4) Ludwig Lachmann’s radical uncertainty;

(5) Austrian “structural uncertainty” (Langlois 1994: 120);

(6) Loasby’s partial ignorance, and

(7) O’Driscoll and Rizzo’s genuine uncertainty.

When the idea of fundamental uncertainty is understood as a crucial one for economic science, the next question is: how do economic agents act and make decisions under uncertain conditions?

George L. S. Shackle developed a theory of decision making under uncertainty that dispensed with probability theories in describing such behaviour, and this was a project derived from the work of Frank Knight and Keynes. In contrast, as we have seen, mainstream neoclassical economics via Arrow adopted the use of subjective probability in decision making theory, and effectively denied the (1) risk versus (2) Knightian/Keynesian uncertainty distinction.

Neoclassical theory was influenced by the work of Frank Ramsay and Leonard J. Savage and essentially went down the path of subjective probability theory with a Bayesian flavour.

I conclude by posing some other questions that seem important to me:

(1) what is the contribution and value of Gilboa and Schmeidler’s non-additive probability approach to decision-making under uncertainty?

(2) to what extent did Ludwig von Mises follow the frequentist interpretation of probability of his brother Richard von Mises?

(3) Knight made a distinction between “statistical probability” and “estimated probability.” Is “estimated probability” more or less “subjective probability”?

(4) What is the significance of Daniel Kahneman and Amos Tversky’s critiques of standard economic decision making theory, and that of Daniel Ellsberg in Risk, Ambiguity and Decision (2001)?

BIBLIOGRAPHY
Copi, Irving, Cohen, Carl and Kenneth McMahon. 2011. Introduction to Logic (14th edn.). Prentice Hall, Boston, Mass. and London.

Dunn, S. P. 2012. “Non-Ergodicity,” in J. E. King (ed.), The Elgar Companion to Post Keynesian Economics (2nd edn.), Edward Elgar, Cheltenham, UK and Northampton, MA. 434–439.

Langlois, R. 1994. “Risk and Uncertainty,” in Peter J. Boettke (ed.), The Elgar Companion to Austrian Economics. E. Elgar, Aldershot. 118–122.

Mises, L. 1998. Human Action: A Treatise on Economics. The Scholar's Edition. Mises Institute, Auburn, Ala.

Runde, Jochen. 2000. “Shackle on Probability,” in Stephen F. Frowen and Peter Earl (eds.), Economics as an Art of Thought: Essays in Memory of G. L. S. Shackle. Routledge, New York.

Skyrms, B. 2010. “Probability, Theories of,” in Jonathan Dancy, Ernest Sosa, and Matthias Steup (eds.), A Companion to Epistemology (2nd edn.). Wiley-Blackwell, Oxford. 622–626.

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.