History-Dependent Transition Intensities

Why History Can Matter After State and Duration Are Known

A patient may currently be in remission and have spent six months in that state. A model using only the current state and the six-month duration assigns the same transition rates to every patient with those facts. But two patients might have reached remission by different routes, such as after one treatment or after several relapses and treatments. If that observed path carries information about what happens next, the transition rate may differ even when current state and duration agree.

A history-dependent transition intensity allows the instantaneous transition rate to depend on observed event history beyond the current state and time spent there. It is a conditional description of local transition risk, not by itself a claim that the history causes a transition. Its usefulness depends on whether the relevant history is observed, adequately represented, and informative for the prediction or scientific question.

Definition and Interpretation

Let $X(t)$ denote the state occupied at time $t$ in a process with a finite set of states. Let $\mathcal H_{t-}$ denote the observed history strictly before $t$: the sequence and times of transitions up to $t$, together with any other observed information included in the model. The left-limit notation excludes a transition at $t$ itself. When the current state is $i$, let $D(t)$ be the time elapsed since entry into $i$.

Definition: For two distinct states $i$ and $j$, the transition intensity $\lambda_{ij}(t\mid\mathcal H_{t-})$ is the conditional instantaneous rate of a transition from $i$ to $j$, given the observed history just before $t$. In precise local form, when $X(t-)=i$, $$ P\{X(t+\Delta)=j\mid\mathcal H_{t-}\} =\lambda_{ij}(t\mid\mathcal H_{t-})\Delta+o(\Delta), \qquad \Delta\downarrow 0. $$ The intensity has units of inverse time. It is a rate, not the probability of transition over a fixed, non-infinitesimal interval.

This local description is used under regularity conditions: the intensities are nonnegative, measurable, and locally integrable; the process has no simultaneous jumps; and it does not make infinitely many transitions in a finite interval. These conditions make the rates usable as a model for transition probabilities. A rate can vary with calendar time, duration, the observed path, or a combination of them. History dependence in this tutorial means dependence on information in $\mathcal H_{t-}$ beyond current state and duration.

A transition intensity of zero at one time or for one history means that there is no first-order instantaneous risk of that transition at that point. It does not automatically mean that the transition is impossible. The transition is ruled out by the model structure only if its intensity is zero almost everywhere in time along every relevant, attainable history, so its cumulative hazard is zero.

How the Specification Differs From State-Only and Duration-Only Models

A state-only transition-intensity model specifies rates using the current state, and may also allow dependence on calendar time. A semi-Markov specification allows rates to depend on the current state and the duration in that state. A history-dependent specification can distinguish histories that agree on both current state and duration.

Specification	Information used for the transition rate	What it treats as equivalent
State-only	Current state, possibly with calendar time	Paths with the same current state and calendar time
State and duration	Current state and time since entry, possibly with calendar time	Paths with the same current state, duration, and calendar time
History-dependent	Current state and duration, plus additional observed history	Only histories that agree on the information used by the model

The table describes possible specifications rather than a claim that every model uses all information listed in its row. In particular, a state-and-duration model may also include calendar time, and a history-dependent model may use a selected summary of past observations rather than the entire record. The defining distinction is whether the rate can differ for two histories that have the same current state and duration.

Worked Example: Same State and Duration, Different Observed Paths

Suppose a hypothetical model has states “active disease” and “remission.” Consider two patients who are both in remission and have been there for six months. Patient A entered remission after the initial treatment; Patient B entered remission after a relapse and a second treatment. Suppose the model assigns a constant relapse intensity over the next month of $0.04$ per month for A’s history and $0.12$ per month for B’s history. Assume there are no other exits from remission during this calculation and that each stated rate remains constant until relapse.

For a constant single-exit rate $\lambda$ over one month, the probability of at least one exit is $1-\exp(-\lambda\times 1)$. Thus, for Patient A, $$ P(\text{relapse within 1 month}\mid \text{history A}) =1-\exp(-0.04\times 1) =1-\exp(-0.04) \approx 0.03921. $$

For Patient B, $$ P(\text{relapse within 1 month}\mid \text{history B}) =1-\exp(-0.12\times 1) \approx 0.1131. $$

The probabilities differ because the modeled histories differ, even though current state and duration are identical. These are conditional model-based probabilities for the stated hypothetical rates; they do not establish that the different treatment paths caused the difference.

From Intensities to First-Exit Probabilities

A current state may have several possible destinations. The intensity for each destination is cause-specific: it describes the local rate of that particular transition, conditional on still occupying the current state. The total rate of leaving state $i$ is the sum of its outgoing intensities. Under the assumptions below, the destination-specific first-exit probability follows by combining that total rate with the rate for the destination of interest.

Result: Conditional on a starting history and on the history path that would be observed if no exit from state $i$ occurred, write $\lambda_{ik}(u)$ for the intensity to destination $k$ at time $u$ after the start. If the rates are integrable and there are no simultaneous transitions, the probability of remaining in $i$ through time $v$ is $$ S_i(v)=\exp\left\{-\int_0^v\sum_{k\ne i}\lambda_{ik}(u)\,du\right\}. $$ The probability that the first exit occurs by time $v$ and is to $j$ is $$ P(\text{first exit by }v\text{ is to }j) =\int_0^v S_i(u)\lambda_{ij}(u)\,du. $$

The formulas condition on the starting information. If observed covariates or other included history evolve while the process remains in $i$, the rates inside the integrals are evaluated along their no-exit path, conditional on the information used in the calculation. The result is not a universal unconditional probability; it must be interpreted for the conditioning information and rate model that define it.

A short justification comes from a small interval of length $\Delta$. Given survival in $i$ to time $u$, the probability of exiting during $(u,u+\Delta]$ is $\{\sum_{k\ne i}\lambda_{ik}(u)\}\Delta+o(\Delta)$. Therefore the survival function satisfies the differential equation $S_i'(u)=-S_i(u)\sum_{k\ne i}\lambda_{ik}(u)$, with $S_i(0)=1$. Solving this equation gives the displayed exponential. The probability of a first exit to $j$ in a small interval is the probability of surviving to its start, multiplied by the local rate to $j$ and the interval length. Integrating those contributions gives the destination-specific formula.

Worked Example: Competing Destinations Under Two Histories

Suppose a hypothetical patient is in a stable disease state. From that state, the possible first exits are relapse and death. Conditional on one observed history, take the constant intensities to be $0.07$ per month for relapse and $0.03$ per month for death over the next month. Assume these intensities remain constant until the first exit, with no other exits.

The total exit intensity is $0.07+0.03=0.10$ per month. The probability that the first exit is relapse within one month is $$ \frac{0.07}{0.10}\{1-\exp(-0.10\times1)\} =0.70\{1-\exp(-0.10)\} \approx 0.06661. $$

For a second history, suppose the relapse intensity is $0.02$ per month and the death intensity remains $0.03$ per month. Then the total intensity is $0.05$ per month, and $$ P(\text{relapse is the first exit within 1 month}) =\frac{0.02}{0.05}\{1-\exp(-0.05)\} =0.40\{1-\exp(-0.05)\} \approx 0.01951. $$

In each calculation, the fraction is the proportion of the total instantaneous rate assigned to relapse, while the braces give the probability of any exit over the month. The rates and probabilities are conditional on the stated history. The calculation does not treat death as ordinary censoring: death is a competing destination that contributes to the total exit rate.

Pointwise Zeros, Structural Zeros, and Changing Rates

History-dependent rates can be zero for some histories or at some times without excluding a transition altogether. For example, a transition may not occur instantaneously during an initial period in a state but may become possible later. The probability accumulated over an interval depends on the integral of the rate, not on the rate at a single instant.

Worked Example: A Temporary Zero Intensity

Consider a hypothetical single-exit model in which the intensity depends on duration $d$ in the current state: $$ \lambda(d)= \begin{cases} 0, & 0\le d<1\text{ month},\\ 0.05\text{ per month}, & d\ge 1\text{ month}. \end{cases} $$ Assume the person enters the state at duration zero and the rate follows this specification until exit.

The cumulative intensity over the first two months is $$ \int_0^2\lambda(d)\,dd =\int_0^1 0\,dd+\int_1^2 0.05\,dd =0.05. $$ Thus the probability of exit by two months is $$ 1-\exp(-0.05)\approx 0.04877. $$

The rate is zero at durations below one month, but positive afterward, so the transition is not structurally ruled out. If the intensity were zero throughout every relevant duration and history, the model would instead exclude that transition.

Assumptions, Diagnostics, and Limitations

Interpreting a fitted history-dependent intensity requires more than a numerical rate estimate. The model should specify what counts as observed history, which transitions are possible, the time scale, and how histories are conditioned on. A useful interpretation also assumes that the observed process and its event times are recorded accurately enough to distinguish relevant histories. If histories are missing or measured coarsely, the fitted rate may not represent the conditional rate the model names.

Potential warning signs include very different estimated rates based on only a few people or transitions, estimates that change sharply under small modeling choices, or predictions that fail to match observed transitions in groups with similar modeled histories. A practical conceptual check is to ask whether the model assigns the same rate to histories that appear meaningfully different, or different rates where the data provide little information to support the distinction. These checks do not prove a model correct; they help identify poor fit, sparse support, and excessive complexity.

Compared with a state-only or state-and-duration model, a history-dependent specification can represent more conditional differences and may improve predictions when observed path information is genuinely useful. Its cost is additional data demands and more complex estimation. Many possible histories may occur rarely, leaving rates weakly determined; choices about which parts of the record matter can also be scientifically consequential. A detailed history-dependent model is not automatically preferable to a simpler specification.

The intensity remains a conditional association unless a separate causal framework and appropriate assumptions support causal interpretation. Also, a zero estimated intensity should not be read as proof that a transition can never occur: it may reflect sparse data, model restrictions, or a rate that is zero only for the observed time and history. These distinctions matter when translating a model into clinical or operational predictions.

Common Mistakes

Reading an intensity as a probability. A rate of $0.05$ per month is not a 5% probability over every interval. Under a constant single-exit rate, the one-month probability is $1-\exp(-0.05)$, approximately $0.04877$.
Ignoring competing exits. A destination-specific probability depends on the other outgoing intensities because they can remove individuals from the current state first.
Calling every zero rate an impossible transition. A pointwise zero means no instantaneous first-order risk at that time and history. Only an intensity that is zero almost everywhere in time along every relevant, attainable history, so its cumulative hazard is zero, rules out the transition under the model structure.
Claiming a causal effect from a conditional rate difference. Different rates across observed histories describe modeled conditional associations; they do not alone show that changing a person's history would change the outcome.

Key takeaway: A history-dependent transition intensity permits histories with the same current state and duration to have different local transition rates. Interpret those rates conditionally, account for all competing exits when calculating probabilities, and distinguish temporary zeros from structural exclusions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A patient has spent four months in state A. What additional model feature makes an intensity history-dependent rather than dependent only on state and duration?
A single-exit intensity is constant at $0.08$ per week for three weeks. Calculate the probability of exit by the end of that interval.
From state A, constant intensities to B and C are $0.04$ and $0.06$ per month. Calculate the probability that the first exit is to B within one month.
An intensity is zero for the first two days in a state and positive afterward. Does the zero alone rule out the transition? Explain.

Solutions. (1) The rate must be allowed to depend on additional observed information in the path before the current time, so two histories agreeing on current state and duration can have different rates. (2) Under the constant single-exit assumption, $P(\text{exit by 3 weeks})=1-\exp(-0.08\times3)=1-\exp(-0.24)\approx0.2134$. (3) The total rate is $0.04+0.06=0.10$ per month, so the probability is $\frac{0.04}{0.10}\{1-\exp(-0.10)\}=0.40\{1-\exp(-0.10)\}\approx0.03807$. (4) No. The rate is zero only during part of the duration range; if it becomes positive later, the model permits the transition after that period.