In this post, I take Bayesianism to be the following normative epistemological claim: “You should treat your beliefs like probabilities, and reason according to the axioms of probability theory.”
Here are a few reasons why I support this claim:
I. Logic is not enough
Reasoning deductively from premises to conclusion is a wonderfully powerful tool when it can be applied. If you have absolute certainty in some set of premises, and these premises entail a new conclusion, then you can extend your certainty to the new conclusion. Alternatively, you can state clearly the conditions under which you would be granted certain belief, in the form of a conditional argument (if you were to convince me that A and B are true, then I would believe that C is true).
This is great for mathematicians proving theorems about abstract logical entities. But in the real world, deductive inference is simply not enough to account for the types of problems we face. We are constantly reasoning in a condition of uncertainty, where we have multiple competing theories about what’s going on, and we seek evidence – partial evidence, not deductively complete evidence – as to which of these theories we should favor.
If you want to know how to form beliefs about the parts of reality that aren’t clear-cut and certain, then you need to go beyond pure logic.
II. Probability theory is a natural extension of logic
Cox’s theorem shows that any system of plausible reasoning – modifying and updating beliefs in the presence of uncertainty – that is consistent with logic and a few minimal assumptions about normative reasoning is necessarily isomorphic to probability theory.
The converse of this is that any system of reasoning under uncertainty that isn’t ultimately functionally equivalent to Bayesianism is either logically inconsistent or violates other common-sense axioms of reasoning.
In other words, probability theory is the best candidate that we have for extending logic into the domain of the uncertain. It is about what is likely, not certain, to be true, and the way that we should update these assessments when receiving new information. In turn, probability theory contains ordinary logic as a special case when you take the limit of absolutely certainty.
III. Non-Bayesian systems of plausible reasoning result in inconsistencies and irrational behavior
Dutch-book arguments prove that any agent that is violating the axioms of probability theory can be exploited by cleverly capitalizing on logical inconsistencies in their beliefs. This combines a pragmatic argument (non-Bayesians are worse off in the long run) with an epistemic argument (non-Bayesians are vulnerable to logical inconsistencies in their preferences).
IV. You should be honest about your uncertainty
The principle of maximizing entropy mandates a unique way to set beliefs given your evidence, such that you make no presumptions about knowledge that you don’t have. This principle is fully consistent with and equivalent to standard Bayesian conditionalization.
In other words, Bayesianism is about epistemic humility – it tells you to not pretend to know things that you don’t know.
V. Bayesianism provides the foundations for the scientific method
The scientific method, needless to say, is humanity’s crowning epistemic achievement. With it we have invented medicine, probed atoms, and gone to the stars. Its success can be attributed to the structure of its method of investigating truth claims: in short, science is about searching theories for testable consequences, and then running experiments to update our beliefs in these theories.
This is all contained in Bayes’ rule, the fundamental law of probabilistic inference:
Pr(theory | evidence) ~ Pr(evidence | theory) · Pr(theory)
This rule tells you precisely how you should update your beliefs given your evidence, no more and no less. It contains the wisdom of empiricism that has revolutionized the world we live in.
VI. Bayesianism is practically useful
So maybe you’re convinced that Bayesianism is right in principle. There’s a separate question of if Bayesianism is useful in practice. Maybe treating your beliefs like probabilities is like trying to do psychology starting from Schrödinger’s equation – possible in principle but practically infeasible, not to mention a waste of time.
But Bayesianism is practically useful.
Statistical mechanics, one of the most powerful branches of modern science, is built on a foundation of explicitly Bayesian principles. More generally, good statistical reasoning is incredibly useful across all domains of truth-seeking, and an essential skill for anybody that wants to understand the world.
And Bayesianism is not just useful for epistemic reasons. A fundamental ingredient of decision-making is the ability to produce accurate models of reality. If you want to effectively achieve your goals, whatever they are, you must be able to engage in careful probabilistic reasoning.
And finally, in my personal experience I have found Bayesian epistemology to be infinitely mineable for useful heuristics in thinking about philosophy, physics, altruism, psychology, politics, my personal life, and pretty much everything else. I recommend anybody whose interest has been sparked to check out the following links:
- Arbital guide to Bayes’ rule (if you’re only going to check out one of the links, make it this one)
- E.T. Jaynes’ full-length textbook Probability Theory: The Logic of Science
- Stanford Encyclopedia of Philosophy entry on Bayesianism
- The blog SlateStarCodex, often a source of good applied Bayesian thinking – especially this and this