Checking Numerical Transition Matrices

Introduction

A numerical transition matrix can look plausible and still fail to represent probabilities. One entry may be negative, a row may not sum to one, or a calculation for a zero-length interval may not produce the identity matrix. These checks are quick, but they are meaningful only when the state space, time interval, and matrix convention are clear.

This tutorial uses the row-to-column convention: the entry in row \(i\), column \(j\) is the probability of being in state \(j\) at the end of an interval, given state \(i\) at its start. The checks below can identify structural problems in a numerical result. Passing them does not, by itself, prove that the approximation is accurate or that the model assumptions are appropriate.

Basic Probability Checks

For a transition matrix on a complete finite state space, every entry must be between zero and one, inclusive, and every row must sum to one. The row condition follows because, conditional on starting in state \(i\), the events “the process ends in state \(j\)” over all states \(j\) are mutually exclusive and exhaustive. Their probabilities must therefore total one.

A row sum below one is not automatically acceptable merely because the entries are nonnegative. It may indicate an omitted state, a computation error, or a model that allows probability to leave the listed state space. If the intended model has a complete state space, the matrix is not a valid transition matrix as reported. A model that deliberately removes probability, such as one that stops tracking a process after a specified event, needs to be described accordingly; its matrix over the retained states may be substochastic rather than a transition matrix on a complete state space.

Similarly, do not assume that a displayed row sum of \(0.999\) must be wrong. If entries were rounded for presentation, a small discrepancy may be due to rounding. Check the precision and the size of the possible rounding error. If the calculation itself is under review, use unrounded values where possible. A row sum check is not permission to change an entry to force the total to one.

Boundary Behavior: Zero Time and Absorbing States

When the interval has zero duration, the process has no time to change state. Therefore, starting in state \(i\) implies that the ending state is \(i\) with probability one. The transition matrix at that boundary must be the identity matrix, whose diagonal entries are one and whose off-diagonal entries are zero.

This is a boundary check, not a claim that the diagonal entries of a positive-duration transition matrix must equal one. During a positive interval, the process may leave its initial state and may later return; the same-state probability is then determined by the model and duration.

If a state is absorbing, its row must have a one in the column for that state and zeros elsewhere, for every interval under consideration. This condition applies only when the state is truly absorbing in the specified model. A state that is “usually terminal” but permits recovery or re-entry is not absorbing.

Products Across Consecutive Intervals

Suppose \(P_1=P(s,u)\) describes the first interval and \(P_2=P(u,t)\) the next interval, with \(s\le u\le t\). Under the Markov assumption stated above, the transition matrix over the combined interval is \(P_1P_2\) in the row-to-column convention. The entry in row \(i\), column \(j\) is found by summing over each possible state \(k\) at the intermediate time.

The Markov condition matters. Conditioning on the intermediate state always gives a sum involving \(P(X_t=j\mid X_u=k,X_s=i)\). To replace that conditional probability with \(P_{kj}(u,t)\), it must not depend on the earlier starting state \(i\), given \(X_u=k\). Without this property, the product of the two matrices need not give the correct combined transition probabilities.

There is also a useful numerical check on a product. If both factors are nonnegative and have row sums one, then their product has those properties too. For a row \(i\), \(\sum_j(P_1P_2)_{ij}=\sum_k P_{1,ik}\sum_j P_{2,kj}=\sum_k P_{1,ik}=1\). This algebraic fact checks the product’s probability structure; the Markov assumption is still needed to interpret that product as the model’s combined transition matrix.

Interpreting Finer-Step Sensitivity

For a numerical method that approximates transitions over time-varying intervals, compare the result on a coarse partition with the result obtained after subdividing intervals more finely. The refined calculation uses the transition approximation for each smaller interval and combines them in chronological order. In a Markov model, this combination is a matrix product.

A smaller difference between successive refinements is evidence of numerical stability for the chosen calculation. It is not a universal guarantee of accuracy: the method may have a systematic error, the intensities or time units may be wrong, or the state space may omit a relevant state. Nor must every pair of successive estimates move monotonically toward the true value. Check several partitions when practical, use an appropriate tolerance, and verify the inputs and boundary conditions.

When assessing the difference, be explicit about the comparison. For two matrices \(A\) and \(B\) on the same states and interval, one simple measure is the largest absolute entrywise difference, \(\max_{i,j}|A_{ij}-B_{ij}|\). Compare corresponding entries, not unrelated row totals: both valid matrices have row totals of one, so row sums alone will not reveal whether their individual transition probabilities differ.

Common Exam Trap: A Product Without the Markov Condition

A common error is to multiply interval matrices automatically after conditioning on an intermediate state, without checking whether the later transition probabilities depend on the earlier history. Consider a process with states \(0,1,M\). Suppose \(X_0\) is equally likely to be \(0\) or \(1\), \(X_1=M\) in either case, and \(X_2=X_0\). The later outcome remembers the initial state even though the intermediate state is \(M\).

If one uses the marginal conditional probability given \(X_1=M\), then \(P(X_2=0\mid X_1=M)=0.5\) and \(P(X_2=1\mid X_1=M)=0.5\). Multiplying as if this were a Markov transition row would predict that a process starting in state \(0\) has probability \(0.5\) of ending in state \(0\). But the specified process has \(X_2=X_0\), so that probability is \(1\). Here \(P(X_2=0\mid X_1=M,X_0=0)=1\), not \(0.5\). The intermediate state does not contain all the information needed for the product formula.

Correction: Confirm the Markov assumption, or directly verify that the conditional distribution over the second interval, given the intermediate state, is independent of the earlier state and history. Only then use the standard matrix product as a transition calculation.

A Practical Audit Sequence

These checks are useful in applications where a numerical transition matrix feeds later actuarial calculations: an invalid probability structure can contaminate every subsequent result. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central distinction is between structural validity—the matrix obeys probability and boundary requirements—and numerical accuracy—its entries adequately approximate the intended model.