Today I taught Bayes’ rule and thought I would collect everything in a blog post. This post was written with the help of an LLM, based on the transcript of a recording.
Bayes’ rule is often introduced through elementary examples involving urns filled with colored balls. We know the composition of each urn, we draw a ball, and we ask how the observation changes the probability that the ball came from one urn rather than another.
In this setting, everything is straightforward. The experiment is specified in advance, all relevant probabilities can be interpreted as fractions of possible outcomes, and Bayes’ rule follows from the elementary laws of probability. The observation tells us something about the urn, but does not require us to reconsider the statistical model describing the experiment.
However, when we move from urns to the much more general problem of updating our beliefs in light of new evidence, something conceptually important happens. We are no longer simply calculating conditional probabilities within a fixed statistical model. We are prescribing how an observer should change their beliefs when new information becomes available.
These two problems look almost identical mathematically, but they are logically different.
Two statements that look the same
Let 
The usual formula reads
This is a mathematical identity. It follows directly from the definition of conditional probability, with the denominator computed, when necessary, using the law of total probability.
Notice, however, that all probabilities appearing in this formula belong to the same probability distribution,
Now suppose that we actually observe 
This is a different statement.
The first equation tells us what the conditional probability of
Nothing in the elementary laws of probability requires these two quantities to coincide. Indeed, the laws of probability constrain a probability distribution at a given time, but do not, by themselves, prescribe how one probability distribution must evolve into another.
The step from the first equation to the second requires an additional principle of rational inference.
What if the observation changes our statistical model?
The distinction may seem artificial when we think about urns. It becomes much more natural when probabilities represent subjective degrees of belief.
Suppose that we are testing a scientific hypothesis using an instrument whose reliability we believe to be known. We observe a surprising result and update our beliefs about the hypothesis using Bayes’ rule.
But what if the result is so surprising that we begin to suspect that the instrument itself is malfunctioning?
In that case, the observation has done more than provide evidence about the original hypothesis. It has also led us to reconsider the statistical model, and in particular the probabilities describing the measurement apparatus.
An even more interesting situation arises when the observation suggests an entirely new hypothesis, one that we had not considered before.
A concrete example appears in a paper by Sylvia Wenmackers and Jan-Willem Romeijn, New Theory about Old Evidence: A Framework for Open-Minded Bayesianism (2016). They describe a food inspector who obtains five consecutive positive results from tests for Salmonella. Initially, the inspector considers only whether the restaurant follows proper hygiene procedures. However, the surprising sequence of results leads her to consider a third possibility: perhaps the testing strips themselves are contaminated.
The problem is no longer simply how to redistribute probabilities among the original hypotheses. The observation has changed the set of hypotheses under consideration.
Of course, if the possibility of a defective instrument had already been included in the original probabilistic model, ordinary Bayesian conditioning could account for it. But this observation does not eliminate the conceptual difficulty. A sufficiently rich model may anticipate many kinds of uncertainty, but the laws of probability alone do not tell us how to revise a model when a previously unconsidered possibility enters the picture.
In other words, Bayesian conditioning is a rule for updating beliefs within a specified probabilistic framework. The justification for applying it as a general rule of learning is a separate question.
Ramsey had already noticed this in 1926
This distinction had been formulated very explicitly already a century ago by Frank P. Ramsey, one of the founders of the subjective interpretation of probability.
In his celebrated essay Truth and Probability, written in 1926 and published posthumously in 1931, Ramsey carefully distinguishes the degree of belief in a proposition 
His explanation is strikingly direct:
"knowledge of q might for psychological reasons profoundly alter his whole system of beliefs."
This is from page 180 of the original publication, which can be found in this electronic edition of Ramsey’s essay.
Ramsey was not suggesting that every psychological change of belief is rational. Rather, he was pointing out that conditional belief and actual belief revision are conceptually different things.
The question is therefore not whether an observation can change our beliefs in ways other than conditioning. Clearly, it can. The question is whether, and under what conditions, such changes are rationally justified.
When is Bayesian conditioning justified?
This question has an extensive history in the philosophy of probability, where the updating prescription is usually called the Principle of Conditionalization.
Richard Jeffrey, in Conditioning, Kinematics, and Exchangeability (1988), identifies two conditions under which ordinary conditioning is appropriate.
The first is certainty: after the observation, the event
The second is sufficiency: learning
Together, these conditions imply the usual Bayesian updating rule.
The crucial point is that the second condition is not imposed by probability theory. It is an assumption about what the observation has, and has not, changed in our state of knowledge.
Jeffrey also developed a more general theory of updating, now known as Jeffrey conditionalization, which allows an observation to change probabilities without making any particular event certain. This theory was subsequently investigated in detail by Persi Diaconis and Sandy Zabell in their paper Updating Subjective Probability (1982).
Richard Bradley, in Radical Probabilism and Bayesian Conditioning (2005), goes even further, examining situations in which the observation changes conditional beliefs themselves.
Thus, the question of when conditioning is appropriate has been recognized for a long time, including within the Bayesian tradition itself.
Why should we follow Bayes’ rule, then?
Once we recognize that conditioning is an additional rule, rather than a mathematical identity, the natural question is how to justify it.
Several answers have been proposed.
One influential approach is based on consistency across time. David Lewis, in Why Conditionalize? (written in 1972 and published in 1999), develops a diachronic Dutch-book argument. Under appropriate assumptions about an agent’s betting dispositions and updating policy, deviating from conditionalization can expose the agent to a sequence of individually acceptable bets that together guarantee a loss.
Another approach is based on epistemic utility. In Justifying Conditionalization: Conditionalization Maximizes Expected Epistemic Utility (2006), Hilary Greaves and David Wallace show that, under specified conditions on epistemic utility, conditionalization is the unique updating rule that maximizes expected epistemic utility.
A third possibility is to invoke a principle of minimum change: when new information arrives, our beliefs should change only to the extent required to accommodate that information, and not more.
For example, suppose that 
For 
namely, ordinary Bayesian conditioning.
The argument extends naturally to Jeffrey updating when the new information specifies an updated marginal distribution rather than a certain event. I have discussed this construction in a separate note on this blog.
All these approaches have something important in common. They do not derive the Bayesian updating prescription from the elementary laws of probability alone. Instead, they identify additional requirements that select conditionalization as the appropriate rule.
Naturally, the strength of each justification depends on its assumptions. A minimum-change argument, for example, requires specifying what counts as a change and which revisions are admissible. A Dutch-book argument relies on assumptions about betting and intertemporal commitments. An epistemic-utility argument requires assumptions about how epistemic success is evaluated.
Such assumptions are not mathematical consequences of the product rule. They are precisely what gives an updating rule its normative content.
The difference matters
One might object that all this is unnecessarily complicated. After all, Bayes’ rule works extraordinarily well in statistical inference, machine learning, and countless scientific applications.
I agree. But its practical success should not obscure the logical distinction between a conditional probability and a prescription for changing beliefs.
The important point is not that Bayesian updating is wrong, nor that every possible reconsideration of a statistical model invalidates it. Rather, the point is that its status as a rule of rational inference requires a justification that goes beyond the elementary probability calculus.
This becomes particularly important when we attempt to generalize Bayesian reasoning beyond classical probability theory, as in quantum Bayesian retrodiction. In that setting, the problem is not merely to write down an analogue of a familiar formula, but to understand which requirements make one among the possible generalizations the appropriate rule of inference.
The lesson, however, is already present in the classical case.
Bayes’ theorem tells us how conditional probabilities are related. Bayesian conditionalization tells us how we should learn.
The former is a theorem of probability theory. The latter is a principle of rational inference.
Confusing the two risks making Bayesian updating appear either more trivial or more universal than its actual justification warrants.
References and further reading
- F. P. Ramsey (1926), Truth and Probability, published in The Foundations of Mathematics and Other Logical Essays (1931), pp. 156–198. See especially p. 180.
- R. Jeffrey (1988), Conditioning, Kinematics, and Exchangeability, reprinted in Probability and the Art of Judgment (1992).
- P. Diaconis and S. L. Zabell (1982), Updating Subjective Probability, Journal of the American Statistical Association 77, 822–830.
- D. Lewis (1972/1999), Why Conditionalize?, in Papers in Metaphysics and Epistemology, pp. 403–407.
- R. Bradley (2005), Radical Probabilism and Bayesian Conditioning, Philosophy of Science 72, 342–364.
- H. Greaves and D. Wallace (2006), Justifying Conditionalization: Conditionalization Maximizes Expected Epistemic Utility, Mind 115, 607–632.
- S. Wenmackers and J.-W. Romeijn (2016), New Theory about Old Evidence: A Framework for Open-Minded Bayesianism, Synthese 193, 1225–1250.
- Bayesian Epistemology, Stanford Encyclopedia of Philosophy, especially Sections 1.2 and 5, for an overview of conditionalization, its justifications, and its limitations.
