Deriving Conditional State Probabilities From Transition Matrices

From a Landmark State to a Later State

At a landmark time, a prediction question can be stated precisely: given the state occupied at the landmark, what is the probability of each state at a later time? When transition probabilities are supplied separately for successive intervals, the answer combines them. Each possible intermediate state must be included, because a person can reach the final state by more than one path.

This tutorial develops that calculation using transition matrices. The emphasis is on conditioning at the landmark, the order of multiplication, and the path-by-path justification. The calculations are useful only when the transition probabilities describe compatible intervals and the state at each interval boundary contains the information needed to determine the next transition probabilities.

Use row vectors for state distributions. If the state at time \(u\) has distribution \(\boldsymbol{\pi}(u)\), then the distribution at \(v\) is \(\boldsymbol{\pi}(u)P(u,v)\), provided the transition probabilities apply to the conditioning information in question. If it is known that the state at landmark time \(L\) is \(r\), the initial row vector is the unit vector \(\boldsymbol{e}_r\): it has a 1 in position \(r\) and 0 elsewhere.

The matrices are multiplied in chronological order from left to right under this row-vector convention. The first matrix carries the distribution from the landmark to the first interval boundary; the next matrix carries that result forward again. These need not be identical matrices: transition probabilities may change with calendar time or with the length and conditions of the interval they describe.

Why Multiplication Includes Every Intermediate State

For two intervals, write \(P^{(1)}=P(t_0,t_1)\) and \(P^{(2)}=P(t_1,t_2)\). Given state \(i\) at \(t_0\), reaching state \(j\) at \(t_2\) can occur through any state \(k\) at \(t_1\). The law of total probability gives

Under the Markov property at the interval boundaries, \(P\{X(t_2)=j\mid X(t_1)=k,X(t_0)=i\}=P\{X(t_2)=j\mid X(t_1)=k\}\). Thus, the summand is \(p^{(1)}_{ik}p^{(2)}_{kj}\). Summing over \(k\) is exactly the definition of matrix multiplication:

For more than two intervals, apply the same argument repeatedly. Each product term corresponds to a sequence of intermediate states. The product is therefore not a shortcut that discards possible paths; it adds their probabilities while accounting for the probability of each step along each path.

The required assumption is often expressed as the Markov property at the interval boundaries: conditional on the state at an interval's start, the distribution of the state at its end does not also depend on the earlier path. The matrices must use the same state definitions and compatible conditioning throughout. If relevant earlier history affects transitions beyond the current state, state-only matrices may not yield the desired conditional probabilities.

Three States and Multiple Paths

With more states, the same calculation applies, but it becomes especially important to keep track of which intermediate states contribute to each final state. The following example shows the calculation entry by entry rather than treating a matrix product as a black box.

Conditioning on a Different Landmark State

The transition matrices can be unchanged while the answer changes because the landmark state changes. Selecting the appropriate unit vector is the operation that imposes the conditioning. It is not generally correct to average rows unless the landmark population actually has a distribution over states rather than a known state.

Checks, Alternatives, and Common Failure Modes

A sound calculation has several straightforward checks. First, confirm that each interval matrix has nonnegative entries and rows summing to 1. Second, check that the starting vector matches the stated landmark condition. Third, verify the order and dimensions of the matrices. Finally, the resulting row must have nonnegative entries summing to 1, apart from small rounding error. These checks can expose arithmetic mistakes, but they cannot establish that the transition probabilities or their assumptions are appropriate for a particular application.

If a transition matrix directly from \(t_0\) to \(t_m\) is already available under the same conditioning assumptions, multiplying interval matrices should give the same endpoint probabilities as using that matrix. This is the Chapman–Kolmogorov consistency relationship. Directly enumerating paths is another useful check in small examples: it makes omitted or double-counted routes visible. For many states or intervals, matrix multiplication is more systematic than listing every path.

• Reversing the order. Under the row-vector convention, use \(P(t_0,t_1)P(t_1,t_2)\), not the reversed product. The earlier interval acts first.
• Omitting a path. A state at the intermediate time contributes to the endpoint probability even if it is not the landmark state. Include every possible intermediate state in the sum.
• Using the wrong row. The row vector must represent the state known at the landmark. A different row answers a different conditional question.
• Assuming the answer is valid because it sums to one. A row that sums to one can still be wrong if the matrices use incompatible time intervals, state definitions, or conditioning assumptions.
• Multiplying entries element by element. Matrix multiplication requires summing products over intermediate states; multiplying corresponding entries does not combine paths.

A key limitation is that the product is only as appropriate as the transition matrices and their assumptions. If transitions over a later interval depend on information about the earlier path that is not represented by the current state, the state-only product need not equal the desired conditional probability. Also, probabilities estimated with substantial uncertainty remain uncertain after multiplication; the arithmetic alone does not provide an uncertainty interval.