# Why the Definition of Probability Matters

Mises Daily: Wednesday, January 26, 2011 by

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Few debates in the social sciences are ostensibly more boring and pointless than those involving the definitions of words. If one group of people chooses to define a word one way, while another group of people chooses to define that same word in a completely different way, why would anyone possibly care?

It turns out, however, that the definitions of words can often have profound implications for social science. Examples abound where the definitions of single words have made important and lasting impacts on all subsequent thought in the disciplines where they were adopted. The words "inflation," "capitalism," "institution," "socialism," "right," "liberty," and even the word "science" itself have been defined in countless and sometimes conflicting ways; and each definition has profoundly influenced the thinkers who adopted it.

In this article, I focus on the definition of one word in particular: probability. I attempt to demonstrate that the definition we choose to adopt for this similarly innocuous-looking word has important methodological and epistemological implications for social science.

In order to accomplish this task, I first offer a discussion of the two most general definitions of probability that are available for us to adopt: the so-called objective and subjective definitions. I then turn to an examination of some of the methodological and epistemological implications for social science that inevitably flow from the adoption of one or the other of these definitions. I conclude with a brief defense of the subjective definition for probability, which, I argue, is the definition we are forced to adopt given the nature of the real world.

## The Two General Definitions of Probability

At the most general and abstract level, there are two definitions for probability that are open for social scientists (and natural scientists, for that matter) to adopt. On the one hand, we could choose to define probability as a measure of an objective, real, physical property of the world. According to this objective definition, we would conceive of probability as being something "out there" in the world to be measured, recorded and analyzed in the same manner as we do with, for example, the hardness of different metals. Just as aluminum and cobalt have real physical differences that we can observe and measure, so too do things and events have different and real physical probabilities that we can observe and measure.

On the other hand, we could choose to define probability, not as an objective physical property in the world, but rather as a measure of man's subjective beliefs about what will happen in the world. According to this subjective definition, there exists no such thing as physical probabilities "out there" in the world, because things in the world are all governed by the principle of "causality" — which is to say that nothing that happens in the world is random or accidental.[1]

In the natural world, an event occurs because some force or forces caused it to happen. Leaves do not simply fall off of trees for no reason, and neither does any other event in the world occur for no reason whatsoever. Similarly, in the realm of human action, an action occurs because the actor values acting more than not acting. There is no such thing as an action that would occur in the world for no reason whatsoever. There is thus a causal explanation for everything that occurs in the world, according to the subjective definition, and probability is a way for man to measure and try to overcome his ignorance of those causal explanations.

The social scientist must decide which of these two general definitions of probability he will adopt. If he is honestly seeking to understand the world in which he lives, he must adopt the definition for probability that accurately describes to the phenomena he is seeking to define and study.

He is free, of course, to adopt a definition for probability that bears no relation to the real world, just as he is free to adopt a definition for "hardness" that would have no relation to the real world. He does so, however, at the peril of abandoning the realm of science and entering the realm of fantasy. The man engaged in true science seeks to describe and understand the real world, not make-believe concepts and castles in the sky.

## Implications of the Objective Definition of Probability

There are several important methodological and epistemological implications that necessarily flow from the adoption of an objective definition for probability by social scientists. First on the epistemological side is the implication that there exists only one "correct" probability for any given event or phenomenon in the world; this probability is to be more or less perfectly measured by scientists. Just as there is only one "correct" and "objective" hardness of the element cobalt, so too would there exist only one "correct" and "objective" probability of rain tomorrow in Denver. One would scarcely have an objective definition for probability, after all, if there were several different and competing probabilities out there for any given event.

The goal of the scientist investigating problems of this sort would thus to be to search out methods capable of revealing, however imperfectly, the one correct and objective probability for any given event at any specific moment in time.[2] An important corollary to this is that scientists are likely to conceive of there being only one legitimate method for the accurate measurement of objective probabilities. Just as there are standardized and universally recognized methods for the measurement of lengths and weights, so too is there only one "correct" and "scientific" way of measuring probability.[3]

Another important implication of an objective definition for probability is that probability will necessarily be conceived as a property that can only be known a posteriori, through measurement. Just as we cannot know the length or the hardness of some object before measuring it, so too would we have no inkling about the probability of some event occurring until we actually went out and measured it. The obvious methodological implication of this is that scientists are bound to focus exclusively on past data to generate probabilities, regardless of the specific methods they might choose to employ. At root, then, the defender of an objective definition for probability is an empiricist.

Moreover, by treating probability as a property that can only be known a posteriori, consistent social and natural scientists are bound to treat events for which there exist no past data as completely outside the purview of probability. Again, just as we are in no position to say what the length of a steel rod is until we actually go out and measure it, so too are we incapable of saying what the probability of some future event is unless we have some a posteriori measurements of similar phenomena to use as a guide. Hence, if we adopt an objective definition for probability, there can be no statements of probability for completely singular and unprecedented events.[4]

A broader and more important epistemological and methodological consequence of treating probability as an objective property of things in the world is that probability is likely to be conceived as a concern of the natural sciences alone. If things have objective, physical probabilities "in" them, in the same way that things have objective and inherent weights and lengths, then the measurement of this physical property is in principle no different than measuring any other objective physical property in the world. It is simply a property to be identified and measured by dispassionate natural scientists in the same way that natural scientists measure the weights of hippopotamuses, the intensity of hurricanes, or the average height of dwarfs. This is not to say that the social scientist cannot make use of objective probability measurements in his empirical research, but it does mean that in gathering data of this sort the social scientist is not acting as a social scientist per se.

A proper analogy here is of the social scientist that takes blood-pressure measurements in the course of his research. In measuring blood pressure, the social scientist is not acting as a social scientist per se; rather, he is simply adopting the methods for measuring blood pressure that have been developed in the natural sciences and then utilizing the data he subsequently generates for his social research. No one would ever confuse the specific methods of measuring blood pressure with a specially designed machine as a concern of the social sciences, and, if probability is conceived as an objective physical property "in" things in the world, then neither is anyone likely to view the measurement of objective probabilities as a concern of the social sciences.

## Implications of the Subjective Definition of Probability

The implications for social science that flow from the adoption of a subjective definition for probability are radically different from those that flow from the adoption of an objective definition. In the first place, a subjective definition for probability does not imply that scientists must conceive of there being only one "correct" or "objective" probability "out there" for any given event.

On the contrary, because the subjective definition of probability means that probability is a measure of man's subjective beliefs about the likelihood of the occurrence of some event, this means that the number of probabilities for the same event that could conceivably exist in the world at any given moment is only limited by the number of people on the planet. Each of these different probabilities would represent a measure of some man's subjective beliefs about the likelihood of the event's occurrence, based upon evidence he deems to be relevant to the event's outcome.

"Probability in the natural world is thus merely a measure of man's uncertainty about the causal factors at work in the natural world."

This is not to say that all of these competing probabilities must be treated as equally valuable predictors of the event's outcome, however. Some men have developed and will develop methods that prove to be very reliable predictors of certain events, and their methods and the probabilities they generate will no doubt be accorded respect that other men's probabilities will not. But a subjective definition of probability opens up the field of probability to a wide array of different methods and approaches for generating probabilities — something that is precluded by the adoption of an objective definition.

The foregoing observation has a fortuitous implication for social science in that it focuses the attention of the scientist on accurately predicting future events, rather than obsessively searching for and perfecting the one "correct" method to be employed on every occasion. With no definitional barriers in his way dictating that he only ever employ one method for generating probabilities, the social scientist is free to use any methods he can either borrow or devise to predict outcomes. He is likely to seek out and use methods that have a proven track record of accurate predictions in similar situations, but even if he opts to employ a new and untested method, this method will be subject to real-world testing as it either succeeds or fails to accurately predict outcomes. Either way, the social scientist cannot cloak his predictions with preternatural authoritativeness, simply because he has utilized a particular method.

The subjective definition for probability also undercuts the idea that only a posteriori data can ever legitimately be used to generate probabilities. According to the subjective definition, probabilities are merely measures of man's beliefs about the likelihood that something will or will not occur in the world, so there is no reason why scientists cannot legitimately make use of a priori information, expert opinions, indirect information, or anything else that they might deem relevant to predicting the event's outcome.

Again, with a subjective definition for probability the only important measure of a method's usefulness is the accuracy with which it predicts future outcomes that man was previously uncertain about. The subjective definition does not tie the social scientist's hands with a dogmatic prescription that only a posteriori data be used to measure his uncertainty. If anything, the subjective definition admonishes the social scientist to go out and find any information out there that he can use in order to better predict what he and others are uncertain about.

Another critically important implication of the subjective definition for probability is that it opens the door to generating probabilities for singular events and phenomena.[5] The objective definition, as was seen above, was predicated on the use of a posteriori data from similar cases, which precluded the generation of probabilities for singular cases with no past precedent.

Because the subjective definition does not require the use of a posteriori data, however, it does not similarly preclude and condemn the generation of probabilities for singular cases. All phenomena in the world are thus fair game for probabilistic measurement, because man is ignorant or uncertain to greater or lesser degrees about virtually everything he encounters in the world. The subjective definition invites man to go out and measure his uncertainty everywhere he finds it.

The implication of this is that the subjective definition for probability, unlike the objective definition, does not exile probability to the natural sciences. Because the subjective definition opens the door to generating probabilities for singular events and phenomena, this obviously includes generating probabilities for singular human actions for which there are no precedents and no past data.

The subjective definition thus welcomes the use of probability in all fields of science, not just the natural sciences. This is fortuitous, because man is often just as uncertain about whether his neighbor will take his trash out tomorrow as he is about the weather this afternoon, and the subjective definition for probability invites him to measure his uncertainty in the former case just as much as he does in the latter.

## Which Definition Is Correct?

An analysis of the implications of possible definitions of probability does not, in itself, help us to determine which one is correct. The implications of the definitions are only useful insofar as they highlight that the question of which definition we adopt is not inconsequential or trivial. The question that remains, then, is, which one is "correct"?

The answer, I would suggest, is that the subjective definition is the correct definition for the world in which we live, because we live in a world where events and actions never occur without any reason whatsoever.[6] The events and phenomena that occur in the natural world around us always occur for a causal reason. Things fall to the ground for a reason. Things catch fire for a reason. Things grow for a reason. All natural events that occur in the world occur for a causal reason, and natural science itself is predicated on the idea that man can discover and understand the causal factors that govern the world.

Because this is so, natural things do not possess a mystical property of "probability" inside them; rather, they behave according to the causal laws of the natural world. Probability in the natural world is thus merely a measure of man's uncertainty about the causal factors at work in the natural world.

If man knew every causal factor affecting any given event in the natural world, he would know the outcome beforehand. He would never need or use probability.

Similarly, in the realm of human action, actions always occur for a causal reason. Man drives to the grocery store for a reason. Man drinks alcohol for a reason. Man paints his house for a reason. Every human action occurs because the actor subjectively prefers acting to not acting, and the study of human action (praxeology) is itself predicated on the idea that man can discover and understand the logic that governs the realm of human action.

Because this is so, we know that human beings do not possess a mystical property of "probability" inside them; rather, they always act on their subjective beliefs and values. Probability in the human world is thus merely a measure of man's uncertainty about the subjective beliefs and values that influence the actions of other men.

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If man knew every subjective factor affecting any given action, he would know the outcome beforehand. He would never need or use probability.

If, contrary to fact, actions and events occurred in the world for no reason whatsoever, there might be justification for saying that things had probabilities "inside" them that man could measure. But the world is not so constituted. Everything that occurs in the world occurs for a reason, and if man knew all the factors affecting any action or outcome, he would know the outcome in advance and for certain.

Man is not omniscient, however, which is why he uses probability to help him predict outcomes — by measuring his uncertainty about the outcome. The uncertainty resides in himself, however, not "out there" in the world; so probability is a subjective measure of human uncertainty, not a measure of something in the world.

##### Notes

[1] On the principle of causality, see, for example, Ludwig von Mises, Theory and History (Auburn, Ala.: Ludwig von Mises Institute, 1985), p. 74, Ibid., Human Action 4th ed. (Irvington-on-Hudson, NY: Foundation for Economic Education, 1996), p. 22; and Hans-Hermann Hoppe, Economic Science and the Austrian Method (Auburn, Ala.: Mises Institute, 1995), pp. 77–78.

[2] An obvious analogy to the point being made here can be seen in the realm of inferential statistics, where the researcher attempts to estimate the "true" value of a given constant (e.g., μ) from a sample of the population. Even though the researcher is never in a position to know the actual value of μ, he nevertheless attempts to estimate this value as best he can, given the data available to him. Similarly, if probability is conceived as a "real" physical property of things, then the goal of the researcher is to estimate or discover this value as best he can, given the data available to him.

[3] For the most dogmatic statement of this idea, see Richard von Mises, Probability, Statistics and Truth (New York: Chapman and Hall, 1988).

[4] For a statement of this idea, see Ludwig von Mises, Human Action, p. 113.

[5] For more on this particular point, see Mark R. Crovelli, "On The Possibility of Assigning Probabilities to Singular Events, or: Probability is Subjective Too!" Libertarian Papers 1, 26 (2009).

[6] For a more thorough defense of the subjective definition of probability, see Crovelli, "On the Possibility."