Why Exactness and Approximation Must Be Kept Separate
A duration-dependent model assigns different transition intensities to individuals who have spent different amounts of time in the same state. A Markov representation can retain that duration as part of the state description. In principle, this makes the representation exact: the expanded state keeps the information needed to predict future transitions. In practice, however, duration is often recorded or modeled using a finite set of bins. Binning replaces a continuum of possible durations with a small number of categories, and generally approximates the original process.
The distinction matters because the two constructions answer different questions. An exact expansion preserves the underlying transition law by retaining sufficient information. A finite-bin expansion is a computational or modeling approximation whose accuracy depends on the chosen bins and on how transition behavior is represented within them. Even with constant intensities within a bin, a calculation for the first exit from the current state is not automatically the probability of occupying a particular state at the interval endpoint: another transition might occur before that endpoint.
What the Exact Expansion Retains
Suppose the process is currently in transient state \(i\), has been there for duration \(a\), and can next move to one of several states \(j\). Write \(q_{ij}(a)\) for the transition intensity from \(i\) to \(j\) at duration \(a\), and define the total exit intensity by \(\lambda_i(a)=\sum_{j\ne i}q_{ij}(a)\). These intensities are conditional on being in state \(i\) at duration \(a\). The probability of remaining in \(i\) for at least a further \(u\) time units is
The probability that the first exit occurs within the next \(u\) time units and its destination is \(j\) is
Together, these expressions describe the first transition after the current time. Following a jump, the state changes and the duration clock resets to zero for the newly entered state, unless the model specifically defines a different clock rule. Retaining both current state and current duration therefore preserves the duration information needed to apply the transition law after each jump. If other history variables affect future intensities, they must also be retained for this exactness claim to hold.
This is an exact augmentation of the model state, not a claim that the process \(X(t)\) alone is Markov. Nor does it imply that a finite-state continuous-time Markov chain can represent every duration-dependent process without approximation. The augmented state includes a continuous duration coordinate, which increases with time between jumps and resets at a transition.
Worked Example: An Exact Duration-Augmented Description
Consider a hypothetical process in state \(A\) with two possible destinations, \(B\) and \(D\). Suppose the duration-specific intensities are \(q_{AB}(a)=0.12+0.04a\) and \(q_{AD}(a)=0.03\), measured per year when \(a\) is in years. At the current time, the process has been in \(A\) for \(a=2\) years. Find the probability of no exit over the next year and the probability that the first exit in that year is to \(B\).
At duration \(a+v=2+v\), the total intensity is \( \lambda_A(2+v)=0.12+0.04(2+v)+0.03=0.23+0.04v \) per year. Thus the integrated total intensity over the next year is \( \int_0^1(0.23+0.04v)\,dv=0.23+0.02=0.25 \).
The probability of no exit is therefore \( \exp(-0.25)=0.7788 \). The probability of a first exit to \(B\) is
The integral is evaluated numerically; the substituted intensity \(0.20+0.04v\) is \(q_{AB}(2+v)\). These are exact model probabilities for no exit and for a first exit to \(B\) under the stated duration-specific intensities. They are not, by themselves, probabilities of occupying \(A\) or \(B\) at the end of the year, since a first exit to \(B\) could be followed by another transition before then.
Replacing Duration With Bins
For a finite approximation, partition duration into intervals such as \( [b_k,b_{k+1}) \), with width \(h_k=b_{k+1}-b_k\). A model may represent a person in state \(i\) and bin \(k\), then assign a representative intensity to that bin. The representative value might be evaluated at the left endpoint, at the midpoint, or estimated as a bin-specific parameter. The choice is part of the approximation, not a consequence of state augmentation alone.
If the intensities are treated as constant within a bin, write them as \(q_{ij}\) and let \(\lambda_i=\sum_{j\ne i}q_{ij}\). Over an interval of length \(h\), the probability of no exit from \(i\) is \(e^{-\lambda_i h}\). The probability that the first exit occurs within the interval and is to \(j\) is
These formulas are exact for no exit and first-exit destination probabilities when the intensities truly are constant over the interval. They are not generally exact probabilities of the process's state at the interval endpoint in a multistate process that can make multiple jumps. For example, the path \(i\to j\to k\) may occur before the endpoint. To obtain endpoint-state probabilities, the calculation must account for such paths, using a suitable transition-matrix calculation when the model permits one, or a renewal or recursive calculation for the duration-dependent process.
Worked Example: Constant Bin Intensities and First-Exit Probabilities
In a hypothetical duration bin of width \(h=0.5\) years, suppose a person in state \(A\) has constant intensities \(q_{AB}=0.20\) and \(q_{AD}=0.10\) per year. Calculate the probability of no exit and the probabilities of first exit to each destination during the half-year.
The total intensity is \(\lambda_A=0.20+0.10=0.30\) per year. The no-exit probability is \(e^{-0.30(0.5)}=e^{-0.15}=0.8607\).
The first-exit probability to \(B\) is \( (0.20/0.30)(1-e^{-0.15})=0.0929 \). The first-exit probability to \(D\) is \( (0.10/0.30)(1-e^{-0.15})=0.0464 \). Their sum is \(0.1393\), which equals \(1-0.8607\), apart from rounding. This verifies that the first exit either occurs to \(B\) or to \(D\), or no exit occurs.
These calculations describe the first exit from \(A\). They do not say that \(0.0929\) is the probability of being in \(B\) at the half-year mark. If someone enters \(B\) early enough and can then leave \(B\), the endpoint probability of \(B\) must include paths that enter and remain there as well as account for paths that enter and leave.
How Bin Width Affects the Approximation
A simple way to see the effect of bin width is to approximate a changing total intensity by a constant within each bin. Consider a hypothetical state with total exit intensity \(\lambda(a)=0.10+0.04a\) per year, where duration \(a\) is in years. Starting at duration zero, the exact probability of no exit over two years is \(\exp\{-\int_0^2(0.10+0.04a)\,da\}=\exp(-0.28)=0.7558\). If each bin uses the intensity at its left endpoint, a wider bin gives a coarser approximation to the integrated intensity and hence to survival.
Worked Example: Comparing Two Duration-Bin Widths
For the intensity above, compare left-endpoint approximations using bins of width 1 year and 0.5 years over the first two years. Assume the only event is an exit from the current state, so that one minus survival is the probability of a first exit by two years.
With width 1 year, the two bin intensities are \(\lambda(0)=0.10\) and \(\lambda(1)=0.14\) per year. The approximated integrated intensity is \(0.10(1)+0.14(1)=0.24\). The approximate survival is \(e^{-0.24}=0.7866\), so the approximate first-exit probability is \(1-0.7866=0.2134\).
With width 0.5 years, the left-endpoint intensities are \(0.10, 0.12, 0.14,\) and \(0.16\) per year. The approximated integrated intensity is \(0.5(0.10+0.12+0.14+0.16)=0.26\). The approximate survival is \(e^{-0.26}=0.7711\), and the approximate first-exit probability is \(1-0.7711=0.2289\).
The exact first-exit probability is \(1-0.7558=0.2442\). In this example, narrowing the bins moves the approximation toward the exact result; the left-endpoint rule underestimates the increasing intensity within each bin. Smaller bins do not guarantee accuracy if the chosen representative values are inappropriate or if other parts of the model are misspecified.
Why Refining the Grid Helps—and What It Cannot Fix
For a smooth intensity, the integrated intensity is the quantity being approximated by a duration grid. The following bound gives a simple justification for refinement. Suppose \(\lambda(a)\) is Lipschitz continuous with constant \(L\), meaning \(|\lambda(a)-\lambda(b)|\le L|a-b|\), and use left-endpoint values on equal-width bins of width \(\Delta\) over a duration range of length \(T\). On one bin, the absolute integration error is at most \(\int_0^\Delta Lu\,du=L\Delta^2/2\). There are \(T/\Delta\) bins when \(T\) is an integer multiple of \(\Delta\). Therefore the total error in integrated intensity is at most \(LT\Delta/2\).
Since \(e^{-x}\) has derivative with absolute value at most 1 for \(x\ge 0\), an error of size \(\epsilon\) in a nonnegative integrated intensity changes the corresponding survival probability by at most \(\epsilon\). Thus, under these conditions, the survival approximation error also tends to zero as \(\Delta\) tends to zero. The bound concerns the integrated intensity approximation; it does not repair omitted predictors, incorrect destination-specific intensities, or an endpoint calculation that ignores multiple jumps.
Useful diagnostics include comparing predictions on successively finer grids, checking whether results depend strongly on bin boundaries, and inspecting whether estimated hazards change sharply within a bin. Large changes after refinement suggest that the coarse grid was inadequate or that the within-bin representation needs reconsideration. Sparse data in narrow bins can also make estimates unstable, so finer bins trade approximation bias against estimation variability.
Choosing a Representation
| Representation | What it retains | Main strength | Main limitation |
|---|---|---|---|
| Continuous duration augmentation | Current state and exact duration (and any other required history) | Preserves the specified duration-dependent transition law | Requires working with a continuous augmented state and evaluating duration-dependent probabilities |
| Finite duration bins | Current state and a duration category | Can simplify computation and interpretation | Approximates within-bin variation; results can depend on bin width and placement |
| Phase-type expansion | Current state and a finite latent phase | Can give a finite-state Markov representation for a phase-type holding-time model | Its holding-time law is constrained by the chosen phase-type representation; it is not the same as discretizing observed duration |
The continuous expansion is appropriate when the model requires exact duration information and computation can accommodate it. Binning can be useful for a manageable approximation, especially when the hazard is smooth and the bins are sufficiently fine for the intended precision. A phase-type representation is a distinct modeling strategy: it represents a holding-time distribution through latent phases rather than by assigning an observed duration category. The introductory material on phase-type expansions discusses that representation; it should not be treated as automatically equivalent to duration binning.
Common Mistakes and Practical Checks
- Calling a duration-bin model exact because the bin hazards are constant. Constant intensities make the no-exit and first-exit formulas exact for that interval. They do not make a one-transition-per-interval calculation an exact endpoint-state transition probability when multiple jumps are possible.
- Assuming a narrow grid guarantees a good model. Refinement controls some discretization error, but cannot correct an incorrect hazard specification, omitted history, or poor data support.
- Equating an exact expanded state with a finite-state Markov chain. Adding continuous duration can produce an exact Markov description on an expanded state space without producing a finite-state chain.
- Ignoring the clock rule. If duration is time since entry into the current state, it resets at state entry. If the model uses a different clock, the expanded coordinates must follow that rule instead.
Before interpreting an expanded-state calculation, state what the expanded coordinates are, whether they preserve the relevant history, how duration is represented between transitions, and whether the reported probability concerns the first exit or the state at a fixed endpoint. Those checks clarify both the model's exactness and the meaning of its numerical output.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A process has duration-dependent intensities, and the current state plus exact duration determine its future transition law. Is adding duration an exact expansion or a finite-bin approximation? What additional qualification is needed if another history variable also affects transitions?
- In a bin of width \(0.25\) years, a state has constant total exit intensity \(0.40\) per year. Calculate the probability of no exit in the interval.
- Why might the probability of a first exit to state \(B\) during an interval differ from the probability of occupying \(B\) at the interval endpoint?
- For a smooth increasing intensity, what generally happens to a left-endpoint approximation as bin width is reduced, and what does refinement fail to address?
- Give one important difference between a duration-bin expansion and a phase-type expansion.
Solutions. (1) It is an exact augmentation if the current state and exact duration capture all information needed to determine the future law. Any other history variable that affects future transitions must also be retained, or the claimed exactness does not follow. (2) The no-exit probability is \(e^{-0.40(0.25)}=e^{-0.10}=0.9048\). (3) A path can enter \(B\) and then leave \(B\) before the interval ends; first-exit and endpoint occupancy are different events. (4) Under regularity conditions, a finer grid reduces the integration error; it does not fix a misspecified intensity, omitted history, or sparse-data problems. (5) Duration bins categorize elapsed time, whereas a phase-type expansion represents holding time through latent phases and a phase-type distribution.