Bayesian Programming

Probability as an extension of logic


Objectivism versus subjectivism controversy and the “mind projection fallacy”

There are two extreme positions in the epistemology of probability: the objectivist and the subjectivist ones.

On the one hand, the objectivists consider that probability is a mathematical tool to model the real world.

On the other hand, the subjectivists consider probabilities as a way to model the reasoning of a given subject about the world.

It is considered by many as only philosophical quibbling, but they are wrong as it has very important practical consequences in the way probabilities are used and in how the obtained results may be interpreted.

For the objectivists, one should use probabilities to build models of the world as objective as possible, meaning that these models should depend on the observable data and only on them and should be independent of their own knowledge of any possible observers. It is a praiseworthy goal, a direct heritage of the idea that science can provide an objective, an exact, or even a true description of the world.

The subjectivists consider that probability is a model of reasoning. As such, the subject who is reasoning and who’s own knowledge about the world is central for the model; is at least as important as the data he collects by making observations. Subjectivists even deny the possibility of building a model of the world independent of any preliminary knowledge to interpret these data. They propose probability as an alternative and an extension of logic to formalize rational reasoning when information is incomplete and uncertain. Preliminary knowledge plays the same role for probability that axiomatic plays for logic. Starting from “wrong” axioms (not true in the world) will lead to “wrong” conclusions (not describing, explaining, or predicting the behavior of the world) even with exact logical inferences. Starting from “wrong” preliminary probabilistic knowledge will also lead to “wrong” conclusions whatever the data and even with perfectly valid probabilistic calculus. The “objectivity” of the subjectivist approach then lies in the fact that two different subjects with same preliminary knowledge and same observations will inevitably reach the same conclusions. A quite different meaning of objectivity than the one adopted by the objectivists.

The Laplace succession law controversy which has made for an exciting debate for the last 150 years is an interesting example of the two different points of view. Laplace proposed to model a series of experiments using the following law:

where nx is the number of times the x value appears in the series, Ω is the cardinal of variable X, and n is the total number of observations in the series. If the observed series is the life of an individual and the variable X stands for “the individual survives this year,” then Ω = 2 and we get for a 14-year-old boy a probability of surviving one more year equal to 15/16 when for his 75-year-old grandfather we get a probability of 76/77.

Using this kind of argument, the objectivists have been making fun of Laplace and his succession law, saying that they were both stupid.

The subjectivists’ position is that the Laplace succession law is just one element of the reasoning subject to preliminary knowledge, and if the obtained result is in contradiction with “common sense” it just means that “common sense” has more information to make its judgment than solely this rule. Adding the knowledge that human life has an upper limit, indeed easily solves the question of the probability of survival of a given individual.

Edwin T. Jaynes in his book Probability Theory: The Logic of Science [Jaynes, 2003] presents a fervent plea for the subjectivist point of view. He warns us again of what he calls the “mind projection fallacy”:

Common language — or at least, the English language — has an almost universal tendency to disguise epistemological statements by putting them into a grammatical form which suggests to the unwary an ontological statement. A major source of error in current probability theory arises from an unthinking failure to perceive this. To interpret the first kind of statement in the ontological sense is to assert that one’s own private thoughts and sensations are realities existing externally in Nature. We call this the “Mind Projection Fallacy,” and note the trouble it causes many times in what follows. But this trouble is hardly confined to probability theory; as soon as it is pointed out, it becomes evident that much of the discourse of philosophers and Gestalt psychologists, and the attempts of physicists to explain quantum theory, are reduced to nonsense by the author falling repeatedly into the Mind Projection Fallacy.

Probability Theory: The Logic of Science, Edwin T. Jaynes [2003]

You can find in Jaynes’ book many examples of misuses of probability due to an objectivist interpretation and especially a review of apparent paradoxes that can be easily solved with a subjectivist point of view.

Of course we presented here only the two extreme positions, when a lot of intermediary approaches exist. For instance, a usual definition for “Bayesianism” refers to probabilists that accept the use of priors as reasoning subjects’ knowledge. Even this position has been largely attacked by objectivists with endless discussions on the relevance of the used priors. From a subjectivist position, the subject is free and takes his own risks when using a given prior. If he makes a wrong choice then he will get an inappropriate model.

In this work we went much further in the subjectivist direction. We do not only use priors but “preliminary knowledge.” Priors are limited to the specification of a few parametric forms to summarize subject preliminary knowledge, when, in contrast, preliminary knowledge is made of the specification part of the Bayesian program made of

  • (i) the choice of the relevant variables,
  • (ii) the choice of the decomposition assuming conditional independences,
  • and (iii) the choice of the parametric forms for each of the distributions appearing in the decomposition.

A major contribution of all these works is precisely this formalization of the preliminary knowledge which, we hoped, has been proved in these exemples to be general and generic enough to model a lot of different problems.

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