William M. Briggs

Statistician to the Stars!

Probability’s Empirical Bias

Big Jule's dice have no spots. But probabilities are still possible.

Big Jule’s dice have no spots. But probabilities are still possible.

Bit of logic (a favorite example) from our teacher Lewis Carroll: given “All cats understand French and that some chickens are cats”, what can we say about the proposition “Some chickens understand French”?

We can say that it follows, that its probability is 1, that it is proved, that it is deduced, true. True conditionally, true given the premises, locally true.

But we also know it is false. Universally false, false because we observe that no cats understand French and that no chickens are cats. Universally or necessarily false, but conditionally true. And also conditionally false, conditional on different premises (such as “No cats understand French” and “No chickens are cats” and, also necessary, “No non-human animals, and very few human animals, understand French”).

Conditional truth is no trouble for logic, because logic is merely the study of the relation between propositions. Logic has nothing to say about the propositions themselves, not formally. Not about where they come from, their utility, their practicality or their lyricism. This is why the premises do not need to be empirical for logic to “work”.

For instance, given “All Martians wear hats and George is a Martian” it is conditionally true that “George wears a hat”—even though we know there are no Martians. Examples abound. See this site for a wealth of Carroll’s fun logic puzzles.

Probability, part of logic, is also concerned only with the relation between propositions and not the origin or usefulness of those propositions. We deduced “George wears a hat” had probability 11, just as we deduce the probability of “George wears a hat” is 3/4 given the premise “Three-quarters of Martians wears hats and George is a Martian.”

Probability, like logic, has no difficulty with non-empirical propositions. This is how we know it is (conditionally) true that “One creature must come out” given the premises “A gnome, fairy, and Godzilla are in a room and only one must come out”, and it is how we know the probability of “Godzilla comes out” is 1/3 given those same premises. We deduced the certainty of both conclusions (we are certain the probability is 1/3 in the second case).

Probability got its start answering empirical questions, mostly about gambling and people’s guilt (see Jim Franklin’s The Science of Conjecture: Evidence and Probability Before Pascal). Aristotle also relied on many empirical questions when delineating logic, but somehow logic came to be seen to be general, whereas probability hasn’t. Probability is empirical to many.

The empirical bias in probability is strong. It accounts for the frequentist fallacy that no probability exists except limiting frequencies of empirical events. A frequentist must remain mute about hat-wearing Martians, and about every other non-empirical proposition, including counterfactual ones. For example, historians often debate about whether there would have been a World War II in Europe if America had not joined World War I. Germany would have won WWI, these historians say, thus ensuring a kind of peace, or at least a Germany victory would not have created the conditions necessary for the National Socialists in that country and the International Socialists in Russia to come into existence.

Now those are perfectly understandable (and debatable) propositions, as are most “What if?” questions. And probability (and logic) can handle them, but no system which is empirically based can. Given (Stove’s example) “Bob is a winged horse” it follows that “Bob is a horse”, a statement which makes sense in logic and probability-as-logic, but a frequentist must pretend it is incomprehensible because there is no way to construct any empirical relative frequency.

More subtly, the empirical bias accounts for de Finetti’s and Ramsey’s error of defining probability as coherent gambles. This is backward. That coherent gambles are good decisions is result of probability, not its definition. (See this about Dutch Books.) There will be no “payoffs” for “events” which will never occur (or never not occur).

Besides, coherence is a weak criterion and mixes up consequences of evidence with evidence itself. A subjectivist can say the probability Godzilla comes out is 0.015764 (or any 0-1 number). As long as he also insists the probability Godzilla does not comes out is 1 – 0.015764, he is coherent. And you cannot prove his probability is wrong if probability is a gamble. No empirical evidence will ever be available.

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1In measure theory “with probability 1” has a technical meaning which I do not here use. I mean this phrase in plain English.

13 Comments

  1. Seeing is believing. For most folks, anyway.

  2. I’m going to disagree a bit and say that it’s not probability’s empirical bias, but human nature’s empirical bias in the realm of probability.

    Have you ever noticed how hard it is for people to accept premises arguendo and not get all riled up? Even asking “What if CO2 wasn’t a major driver of climate change?” will drive people into apoplectic fits. I think we are hard-wired, in some respects, to attach assumptions/statements to the observable first. It’s harder to skip that step and think hypothetically.

  3. Frequentism’s contortions originating in — or (if I may add to your remarks) even more accurately and strongly, dictated by — its imagining that probability is ’empirical’ rather than a subset of logic: I don’t remember your making this particular point as cogently before. This is a fine addition to the teaching of the meaning of probability as logical probability.

  4. Gary and James,
    I think Kahneman would say the hard wiring manifests itself in the form of heuristics humans use to describe/interpret the observable. The fallibility of these heuristics leads to what Gary describes as “human nature’s empirical bias.”

    It does require a conscious effort to overcome this bias. Logic is one of the tools that humans use to do this. Still it is so deeply ingrained that anyone can fall prey to the self-deception.

  5. Your lines on chickens and cats talking French reminded me of a similar scene in “Huckleberry Finn”, when Jim responds to a fallacious argument by Huckleberry.
    “Is a chicken a man? Is a cow a man? Is a Frenchman a man? Then why don’t he TALK like a man?”

  6. Quote from an old Nero Wolfe mystery by Rex Stout:

    “Looking at the two pictures side by side with Wolfe, I would have made it thirty to one that he wasn’t. Now, two to one, or maybe even money, and I would take either end.”

    Evidently, De Finetti’s concept of probability seemed intuitive to this novelist.

  7. Gary and SteveE,

    Thanks for the comments. I have more reading for my list!

  8. What implications, if any, are there to the -observable- fact that human beings can reason about things that can never be experienced or observed?

    For my part, I think this shows or at least suggests that human beings are not the result entirely of a process that is limited to what is actually experienced.

  9. ““Bob is a winged horse” it follows that “Bob is a horse….”

    But what if Bob is seahorse? Or a sawhorse? Or a wooden horse?

    Ah, nothing is easy….

  10. Bob,

    Shout that long enough and you may very well become hoarse.

  11. Briggs

    March 28, 2015 at 11:49 am

    DAV, BOB

    Given “Bob is a hoarse horse”, the probability that “Bob is a horse” is 1. Given “Bob is a winged saw-horse”, the probability “Bob is a saw-horse” is 1. Etc.

  12. I think this is related to a misunderstanding about the principle of explosion, “from a contradiction, anything follows”. It isn’t that a contradiction implies that any state of affairs can exist, but rather that no additional proposition can further restrict (or expand) the allowed states of affairs, once there is a contradiction. In other words, the additional proposition is satisfied (trivially) by all of the rows of the truth table that also satisfy the contradiction. The principle of explosion happens by implosion.

    Thus, despite that logic allows for contradictions (and this is necessary, otherwise it would not be possible to use propositional logic to analyze a logical system to determine if it contained contradictions), we are still entitled to use contradiction as a basis for ruling out models of reality. For if the model is contradicted by observation, then there is no row of a truth-table that will satisfy both the model and the observation, and thus it cannot be the case that the model applies to the actual state of affairs.

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