Probabilistic inference

What it is

Probabilistic inference treats belief as a matter of degree and gives an exact rule for updating those degrees on new evidence. What you should believe after seeing the evidence depends on three things: how likely the evidence would be if the claim were true, how likely it would be anyway, and how likely you took the claim to be before.

The third of those is the one people forget, and forgetting it is the source of nearly every mistake made with this method.

How it works

State your prior — how likely the claim was before the evidence. State how likely the evidence would be if the claim were true, and how likely if it were false. Combine them to get the posterior.

The rule itself is arithmetic and not in dispute. What is in dispute is where the prior comes from, and every objection to the method is some version of that question.

The form of the inference

1. Begin with a prior degree of belief in H
2. Observe evidence E
3. Weigh how much more likely E is if H holds than if it does not
4. Revise the belief in proportion to that weight and to the prior
5. The result is the posterior degree of belief in H

What it is good for

Any situation with a known base rate and a fallible test, which covers medical screening, quality control, intelligence analysis and legal evidence. It is also the only method on this list that tells you how much to change your mind rather than merely that you should.

Where it fails

The priors have to come from somewhere. Where there is a real base rate, as above, the method is unanswerable. Where there is not — and in most interesting philosophical and scientific cases there is not — the prior is a judgement, and two honest people with different priors will reach different posteriors from identical evidence.

Defenders answer that repeated evidence washes out the difference in priors, which is true in the long run and not much help in the cases that matter, where evidence is scarce and the disagreement is now.

Common misconceptions

That it is a way of getting objectivity out of uncertainty. It is a way of being consistent about uncertainty, which is a different and lesser thing.

Also that the base rate is a technicality. It is usually the dominant term, as the worked case shows.

Where it came from

Thomas Bayes's essay was published in 1763 after his death, by Richard Price. Laplace independently developed the same result and did far more with it. The philosophical programme of treating confirmation this way is largely twentieth-century.

What became of it

Now the dominant framework in statistics, machine learning and philosophy of science, after a long period in which frequentist methods held the field.

Lasting influence

It reshaped how evidence is understood across every empirical discipline, and it is the formal machinery behind most of what is currently called artificial intelligence.

Why it still matters

As high as anything on this list, and rising.

Put another way

Adjusting a set of scales you were already holding, rather than weighing from empty.

Texts

Where to start

Work the screening example above by hand with a hundred thousand imaginary people. Doing it with counts rather than percentages is what makes it stick.

The surprising bit

The single most common error in interpreting medical tests is not a failure of arithmetic but the omission of a number nobody thinks to ask for.

In their own words

Who held it

Richard Price, Bruno de Finetti

Connections

Relies on

Induction

Parallel to

Abduction

The verdict

The best account we have of reasoning under uncertainty, resting on a starting point it cannot supply.

Sources

• Stanford Encyclopedia of Philosophy: Bayes' Theorem