Conditioning on Null Events

Elementary conditional probability is defined by a ratio:

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)},P(B)>0$$

The proviso is not a technicality that can be waived. When $P(B)=0$ the ratio is $0/0$ and the conditional probability is simply undefined.

Yet conditioning on null events is unavoidable. Any statement about a continuous quantity taking an exact value — "given that the particle is at position $x$", "given that the parameter equals $\theta$", "given that the point lies on this great circle" — conditions on an event of probability zero, since in a continuous distribution every individual value has probability zero. Statistics does this constantly, and it is not obviously incoherent.

The rigorous treatment resolves the technical problem and leaves a philosophical one: conditioning on the same null event, approached differently, gives different answers.

The Borel–Kolmogorov paradox

Take a uniform distribution over the surface of a sphere. Condition on the point lying on a specified great circle. What is the conditional distribution along that circle?

The intuitive answer is uniform. By symmetry, all great circles are alike and the distribution is uniform over the sphere, so it should be uniform along any of them.

But the answer depends on how the great circle is specified.

As a line of longitude. Parameterise by longitude $\varphi$ and latitude $\theta$. The uniform density on the sphere is proportional to $\cos \theta d\theta d\varphi$. Conditioning on a fixed value of $\varphi$ leaves density proportional to $\cos \theta$ — not uniform. It concentrates toward the equator.

As the equator. Conditioning on $\theta =0$ leaves density proportional to a constant in $\varphi$ — uniform.

Both are great circles. Both are null events. And the conditional distributions differ. Since a sphere can be re-coordinatised so that any great circle becomes the equator or a meridian, the "same" conditioning event yields different answers depending on the coordinate system used to specify it.

The explanation is that the two specifications embed the circle in different families of conditioning events. A meridian is a limit of thin lunes, which are wide at the equator and narrow at the poles; the equator is a limit of thin bands of constant width. The limiting procedures differ, so the limits differ. Conditioning on a null event is really conditioning on a limit of a family, and the family is part of the specification even though the notation hides it.

The rigorous treatment

Measure-theoretic probability handles this by refusing to define conditioning on individual null events at all. Conditional expectation is defined relative to a $\sigma$-algebra, not an event.

Given a probability space and a sub-$\sigma$-algebra $\mathcal{G}$, the conditional expectation $E[X\mid \mathcal{G}]$ is the $\mathcal{G}$-measurable random variable satisfying

$${\int }_{G}E[X\mid \mathcal{G}]dP={\int }_{G}XdP\text{for all }G\in \mathcal{G}$$

Its existence follows from the Radon–Nikodym theorem, and it is unique up to sets of measure zero.

That last clause is where the paradox lives. Since the conditional expectation is only determined almost everywhere, its value on any particular null set — such as a single great circle — is not determined at all. One can modify it arbitrarily on a null set and still have a valid conditional expectation. So the mathematics does not merely fail to answer the question; it tells us the question has no answer as posed.

A regular conditional distribution provides a consistent choice of conditional distribution for every value of the conditioning variable at once. It exists under mild hypotheses, and it resolves the paradox by making the dependence explicit: the conditional distribution is defined relative to the random variable being conditioned on — latitude or longitude — not relative to the event. Different variables give different disintegrations, both correct.

The philosophical moral

The paradox is often presented as a curiosity. It is better understood as the clearest instance of this section's recurring theme, arriving in a place where the modelling choice is completely invisible in the notation.

Writing "$P(A\mid X=x)$" looks like conditioning on an event. It is not: it is conditioning on a variable, and the variable carries a family of neighbouring events with it. Two variables can pick out the same event at a particular value — $\{X=x\}$ and $\{Y=y\}$ may be the same set — while inducing different disintegrations. The notation suppresses this entirely.

Three consequences.

Protocol matters. What one learns is not merely that a null event obtained but how it was ascertained. A measurement of latitude and a measurement of longitude that both place a point on the same great circle are different pieces of evidence, because they are the limits of different families of imprecise measurements. Since all real measurements have finite precision, the limiting family corresponds to something physically real: the shape of the measurement error.

"Conditional on a point value" is an idealisation. Actual conditioning is always on a set of positive measure — an interval of measurement precision. The null-event case is a limit, and which limit depends on how precision shrinks. The paradox is what happens when an idealisation is taken to hide a choice that the unidealised situation makes explicitly.

It is not a defect in the axioms. Kolmogorov's framework behaves correctly throughout: it declines to define what is not determined and specifies exactly the further structure needed. The failure is again in the passage from an English sentence to a model.

Related cases

Improper priors (priors and indifference) generate marginalisation paradoxes with the same structure: two valid routes to a posterior disagree, because a non-normalisable measure does not determine a unique conditional.

Conglomerability fails for merely finitely additive probabilities: the unconditional probability of an event can lie outside the range of its conditional probabilities across a partition. The mathematics of finite additivity sets this out, and it is a further respect in which conditioning behaves badly once countable additivity is dropped.

Regularity (probabilism) is the proposal that no contingent proposition receive probability zero, precisely to keep conditioning always defined. In continuous spaces it requires infinitesimal probabilities, and the non-standard treatment is the formal route. Even granting infinitesimals, the Borel–Kolmogorov problem is not obviously solved: the ratio becomes defined, but its value still depends on the infinitesimal structure assigned, which is the family-dependence in another guise.

Where this sits

The Borel–Kolmogorov paradox shows that a piece of notation used constantly — conditioning on an exact value — conceals a modelling choice. The rigorous theory does not eliminate the choice; it makes it explicit by requiring a $\sigma$-algebra or a disintegrating variable.

This matters for the puzzles that follow. Several of them — Sleeping Beauty above all — turn on what exactly an agent conditions on, and disagreements that look substantive often trace back to different implicit specifications of the conditioning structure.

The next page takes up a case where the ambiguity is not in the conditioning event but in the class to which an individual is assigned: the reference-class problem.