Conditioning on a Finite Partition

Partitions as Finite Information

Often we do not know the exact outcome of an experiment, but we do know which one of a finite collection of cases occurred. For example, a test result may reveal “positive” or “negative” without revealing whether the person has the condition; a report may say that a count is low, medium, or high without giving its exact value. A finite partition records these possible information outcomes. Conditioning on a cell means calculating a probability or average under the knowledge that the outcome lies in that cell.

Work on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). A finite partition \(\{A_1,\ldots,A_r\}\) of \(\Omega\) consists of measurable events that are pairwise disjoint and whose union is \(\Omega\). Thus each outcome belongs to exactly one cell. The cells need not have equal probabilities. If a cell has positive probability, probabilities conditional on that cell are defined by the event-conditioning formula. If its probability is zero, that ratio is undefined, so ordinary conditioning on that cell is not available.

The second quantity is a number, not a new random variable: it is the average of \(X\) among outcomes in \(A_i\), with probabilities renormalized to sum to one on that cell. Integrability, \(\mathbb{E}[|X|]<\infty\), ensures the expectation in the numerator is finite. When \(X\) has a discrete distribution, the numerator can be computed by summing each possible value of \(X\) times its probability and the indicator of the cell. A conditional average may be negative, just as an ordinary average may be negative.

These formulas apply cell by cell. They do not require that the partition itself be generated by a random variable, though a recorded category often does give such a description. When information tells us only which cell occurred, the cell is the event on which we condition; we do not replace it by a more detailed observation that was not given.

Two Partition Calculation Rules

Proof. For the probability identity, the events \(B\cap A_i\) are pairwise disjoint and their union is \(B\), because the cells partition \(\Omega\). Therefore \(\mathbb{P}(B)=\sum_{i=1}^r\mathbb{P}(B\cap A_i)\). If \(\mathbb{P}(A_i)=0\), then \(B\cap A_i\subseteq A_i\), so \(\mathbb{P}(B\cap A_i)=0\). For every \(i\in I\), the definition gives \(\mathbb{P}(B\cap A_i)=\mathbb{P}(A_i)\mathbb{P}(B\mid A_i)\). Substituting these expressions proves the first identity. For the average identity, the indicators of the cells satisfy \(\sum_{i=1}^r\mathbf{1}_{A_i}=1\) at every outcome. Hence \(X=\sum_{i=1}^r X\mathbf{1}_{A_i}\), and linearity of expectation gives \(\mathbb{E}[X]=\sum_{i=1}^r\mathbb{E}[X\mathbf{1}_{A_i}]\). If \(\mathbb{P}(A_i)=0\), then \(X\mathbf{1}_{A_i}=0\) almost surely, so its expectation is zero. For \(i\in I\), the definition of \(m_i\) gives \(\mathbb{E}[X\mathbf{1}_{A_i}]=\mathbb{P}(A_i)m_i\). Substitution proves the second identity. \(\square\)

The identities provide useful checks: combine cell-specific probabilities or averages using the probabilities of the cells as weights. In particular, the cell averages should not usually be combined with equal weights. The identities also explain why a partition describes information: once a cell is known, calculations use probabilities and averages restricted to that event; before learning which cell occurred, the cell probabilities determine how those results are combined.

Denominators, Zero-Probability Cells, and Information

The denominator in a conditional probability or average is the probability of the cell on which the calculation is made. It is not the probability of the target event, nor the probability of some larger set unless that larger set is the actual conditioning event. For a discrete model, an efficient calculation is to list the outcomes in the cell, retain their original probabilities, and divide by the total probability of the cell. This normalization makes the conditional probabilities within that cell add to one.

If \(\mathbb{P}(A_i)=0\), the displayed ratios have denominator zero. One must not assign a value by direct substitution. In some more advanced settings, conditional quantities are defined up to almost-sure equivalence, so their values on zero-probability events are immaterial; that convention does not make the elementary ratio on a null cell defined. For the finite-cell calculations here, use only cells with positive probability. If there are no positive-probability cells, the probabilities of all cells would sum to zero, contradicting that they form a partition of a probability space.

Common mistake. A frequent error is to divide the joint probability by the probability of the wrong event. For the positive-test cell above, the correct disease probability is \(\mathbb{P}(D\cap A_+)/\mathbb{P}(A_+)=\tfrac{8}{100}/\tfrac{17}{100}=\tfrac8{17}\). The incorrect calculation \(\mathbb{P}(D\cap A_+)/\mathbb{P}(D)=\tfrac{8}{100}/\tfrac{1}{10}=\tfrac45\) answers a different question: it is the probability of a positive test given disease. The numerator is the same, but the conditioning event determines the denominator.

A second error is to average cell-specific results without weighting by how likely the cells are. In the spinner example, the unweighted mean of the two conditional averages is \(\tfrac12(\tfrac32+\tfrac{11}{3})=\tfrac{31}{12}\), not the overall average \(\tfrac{25}{8}\). The cells have probabilities \(\tfrac14\) and \(\tfrac34\), so the correct combination is \(\tfrac14(\tfrac32)+\tfrac34(\tfrac{11}{3})=\tfrac{25}{8}\). The weights represent the relative frequency with which each information outcome occurs.