Chapman-Kolmogorov Checks For Estimated Matrices

Why Check Composition?

A transition matrix describes the probabilities of occupying each destination state at a later time, conditional on the state occupied now. If two intervals are adjacent, the transition over their combined duration can be calculated by considering every possible state at the point where the intervals meet. This relationship is the Chapman–Kolmogorov identity.

For estimated matrices, the identity gives a practical coherence check: multiply the matrix estimated for the first interval by the matrix estimated for the second, then compare the product with the matrix estimated directly over the combined interval. The comparison can reveal incompatible calculations, differing definitions, or ordinary variation between separately obtained estimates. Agreement alone, however, does not establish that the underlying assumptions are correct.

Definition and Conditions

For a Markov process, the state occupied at the intermediate time \(u\) contains the information needed to describe the distribution of the future, conditional on the past. The process may be time-inhomogeneous: its transition probabilities may depend on calendar time, so \(P(s,u)\) and \(P(u,t)\) need not depend only on interval lengths.

Each matrix is stochastic: its entries are nonnegative and each row sums to one. To check estimated matrices, use the same state definitions and ordering, the same time origin, and exactly adjacent intervals: \([s,u]\) followed by \([u,t]\). Estimates should also refer to compatible populations, covariate conditions, and estimation procedures. If these conditions do not align, a discrepancy may reflect a comparison of different quantities rather than a failure of the identity.

Why the Identity Holds

Condition on the state \(k\) occupied at time \(u\). To be in state \(j\) at time \(t\), a process starting in state \(i\) must first be in some intermediate state \(k\) at time \(u\), and then move from \(k\) to \(j\) during the second interval. The mutually exclusive intermediate states partition all possibilities. The law of total probability therefore gives

The Markov property removes the dependence on the earlier state \(X(s)=i\) from the first conditional probability. The two terms then equal \(P_{kj}(u,t)\) and \(P_{ik}(s,u)\), respectively. Thus the sum is \(\sum_k P_{ik}(s,u)P_{kj}(u,t)\), which is precisely the \((i,j)\) entry of the matrix product. This derivation also explains why the multiplication order matters: the first interval's matrix is on the left and the second interval's matrix is on the right under the row-state convention.

Performing an Estimated-Matrix Check

Let \(\hat P(s,u)\), \(\hat P(u,t)\), and \(\hat P(s,t)\) be estimated matrices. Form the composition \(\hat P(s,u)\hat P(u,t)\), then calculate the residual matrix

A zero residual means exact agreement. Otherwise, inspect individual entries, the largest absolute entry, and—if useful—a summary such as the Frobenius norm, the square root of the sum of squared entries. These summaries describe the size of the numerical discrepancy; they are not, by themselves, significance tests. There is no universal cutoff that makes a residual acceptable in every application. Its interpretation depends on rounding, the estimation design, and the precision needed for the intended use.

Interpreting Agreement and Diagnosing Failure

When all three matrices are formed from the same estimated transition system, with the same event-time conventions and a split at \(u\), the product-integral construction has a composition property: splitting the ordered updates into those before and after \(u\) gives the product of the two interval-specific matrices. A discrepancy beyond numerical rounding in this setting points first to a computational or bookkeeping problem. For example, check multiplication order, state ordering, and whether events at \(u\) were assigned consistently.

If the matrices were estimated separately, exact equality is not generally expected. The direct estimate and the product of two interval estimates can vary because they are different functions of the observed data. In that situation, a nonzero residual is a diagnostic to investigate, not proof of a violated Markov assumption. Similarly, matrices can compose perfectly while sharing a flawed state definition or an unsuitable estimation procedure. Composition checks algebraic coherence; they do not verify the full conditions needed for valid estimation or establish that the chosen state representation is adequate.

Useful checks include:

• State labels and dimensions: each row and column must refer to the same state in all matrices.
• Time alignment: \(u\) must be the endpoint of the first interval and the start of the second, with consistent boundary conventions.
• Probability structure: negative entries or row sums far from one indicate a basic construction or reporting problem.
• Residual pattern: residuals should have row sums near zero when both the direct estimate and product are stochastic. Large localized residuals can identify which transitions need investigation.
• Precision: multiply using unrounded values where possible. Rounded entries can create small apparent discrepancies.

The main alternative to comparing a direct combined-interval estimate with the product is to construct the combined matrix directly from the underlying estimated transition updates and compare it with the split-interval product. This is especially informative when the same estimation procedure is intended for all intervals: it isolates computational consistency. A direct-versus-composed comparison is more useful when estimates were produced separately and the goal is to assess how much their results differ. Neither comparison substitutes for assessing the assumptions behind the estimation.

Common Mistake