Why These Quantities Must Be Kept Separate
In a multiple-state model, it is useful to ask both “What is the chance of being in a given state later?” and “How quickly can a transition occur now?” These questions are related, but they do not ask for the same quantity. A transition probability concerns an interval of time; a transition intensity describes an instantaneous rate at a particular time.
Confusing the two can lead to large errors. For example, an intensity of \(0.12\) per year is not a 12% probability of transition in every period. Over a short interval, multiplying the intensity by the interval length gives a useful first-order approximation to a transition probability, but the interval length must be in the matching time unit, and the approximation is local.
Transition intensity: For \(i\ne j\), the transition intensity \(\mu_{ij}(t)\) is the instantaneous rate of transition from state \(i\) to state \(j\) at time \(t\), conditional on being in state \(i\) at time \(t\). When the limit exists, $$ \mu_{ij}(t)=\lim_{h\downarrow 0}\frac{P(\text{a transition from }i\text{ to }j\text{ during }[t,t+h]\mid X(t)=i)}{h}. $$ If time is measured in years, the intensity is measured per year. Unlike a probability, an intensity is not restricted to be at most 1.
What the Time Interval Changes
The probability \(p_{ij}(s,t)\) refers to the state at the end of an interval. The process may have moved through other states in between, depending on the model. By contrast, \(\mu_{ij}(t)\) describes the local rate of a direct transition from \(i\) to \(j\), conditional on being in \(i\) at that instant. It is not a probability of ending the interval in \(j\).
An intensity also needs a time unit. An intensity stated as \(0.08\) per year is a rate per year; it is not a probability of \(0.08\) over a year. To use it over a short interval, convert the interval into years. For example, 10 days is \(10/365\) years when a 365-day year is stipulated. Multiplying a rate per year by a duration in days without converting units is invalid.
The defining limit gives the local connection between the quantities. For a small positive interval \(h\), the conditional probability of the direct transition is approximately \(\mu_{ij}(t)h\). This is a short-interval approximation, not a general formula for an arbitrary finite-time transition probability.
Why the Short-Interval Approximation Holds
The intensity is defined as a limit of a probability divided by interval length. Rearranging that definition explains the approximation directly. Let \(r(h)\) be the conditional probability of the direct transition over the interval of length \(h\). If the defining limit equals \(\mu_{ij}(t)\), then
By the definition of this limit, \(r(h)/h\) approaches \(\mu_{ij}(t)\) as \(h\) becomes small. Equivalently, \(r(h)=\mu_{ij}(t)h+o(h)\). The product has no units: intensity has units of inverse time and \(h\) has units of time. That is consistent with a probability being dimensionless.
The approximation is most informative over a sufficiently short interval when the intensity does not change much during that interval. Over a longer interval, the process has more time to evolve, the intensity may vary, and intermediate transitions may affect the state at the endpoint. One intensity value at the start of the interval does not generally determine the finite-time transition probability.
Worked Example: Converting an Annual Intensity to a Short-Interval Probability
A hypothetical income-protection model has an active-to-disabled transition intensity of \(0.12\) per year at the start of a 15-day period. Assume this local intensity is representative over the short period. Estimate the probability of a direct transition from active to disabled during the period, conditional on being active at its start.
Method. Use the short-interval approximation \(\mu h\), after expressing the interval in years. The stated intensity is \(0.12\) per year, and the stipulated year has 365 days.
Step 1: Convert the duration. $$ h=\frac{15}{365}=0.0410959\text{ years}. $$
Step 2: Multiply by the intensity. $$ \mu h=0.12\left(\frac{15}{365}\right)=0.0049315. $$
Step 3: Interpret the result. The estimated probability is \(0.0049315\), or \(0.49315\%\). To four decimal places as a probability, it is \(0.0049\); expressed as a percentage to four decimal places, it is \(0.4932\%\).
The intensity \(0.12\) per year is a rate, while \(0.0049315\) is the approximate probability for this 15-day interval. The calculation is a local approximation, not an exact finite-time probability formula.
Time-Varying Intensities and Point-in-Time Values
When an intensity depends on time, the value at the beginning of a short interval gives the local rate for that moment. A value at one time should not be mistaken for a rate that necessarily applies throughout a long interval. Even when two intensities have the same units, their values may differ at different times.
A time-dependent intensity is written here as \(\mu_{ij}(t)\). Its argument indicates when the local rate is being evaluated. In a short-interval calculation, the intensity must be evaluated at the start time specified in the question, unless the model explicitly gives another convention. The duration still needs to be converted into the intensity's time unit.
Worked Example: Evaluating a Time-Dependent Intensity at the Correct Time
In a hypothetical long-term care model, the active-to-care intensity at time \(t\), measured in years from the valuation date, is $$ \mu_{AC}(t)=0.015+0.005t $$ per year. A member is active at \(t=4\). Estimate the probability of a direct transition to care during the next two months, using a 12-month year and the local short-interval approximation.
Method. Evaluate the intensity at the start time \(t=4\), convert two months to years, and multiply. This avoids using either the initial intensity at \(t=0\) or a rate with an inconsistent time unit.
Step 1: Find the intensity at the specified time. $$ \mu_{AC}(4)=0.015+0.005(4)=0.035\text{ per year}. $$
Step 2: Convert the duration. $$ h=\frac{2}{12}=\frac{1}{6}\text{ year}. $$
Step 3: Calculate the approximation. $$ \mu_{AC}(4)h=0.035\left(\frac{1}{6}\right)=0.0058333. $$
The estimated probability is \(0.0058333\), or \(0.58333\%\). To four decimal places as a probability, the result is \(0.0058\); as a percentage to four decimal places, it is \(0.5833\%\). This is an approximation over two months using the intensity at the start of the interval; it does not assert that the intensity stays at \(0.035\) per year throughout a longer period.
Finite-Time Probabilities Do Not Identify an Intensity by Themselves
A finite-time transition probability gives an endpoint probability over a specified interval. It does not, without additional model assumptions, reveal the instantaneous intensity at one time within that interval. The endpoint event may depend on how the process evolves throughout the interval, including changes in intensities and transitions through other states.
The reverse caution matters too: knowing an intensity at one instant does not usually determine the probability of being in a destination state at the end of a substantial interval. An intensity is conditional on the process currently being in the originating state. If the process leaves that state, the same direct-transition intensity is no longer describing its current situation.
Worked Example: Comparing an Endpoint Probability With a Local Rate
A hypothetical study reports that the probability of being in state \(B\) one year from now, given state \(A\) today, is \(0.08\). Separately, the model gives a current direct-transition intensity from \(A\) to \(B\) of \(0.06\) per year. Estimate the probability of a direct transition from \(A\) to \(B\) in the next seven days, conditional on being in \(A\) now. Use a 365-day year and the local approximation. Explain whether the one-year probability determines this estimate.
Method. Use the current intensity for the short seven-day estimate. Treat the one-year endpoint probability as a distinct quantity; do not divide it by one year and call that result the current intensity.
Step 1: Convert seven days to years. $$ h=\frac{7}{365}=0.0191781\text{ years}. $$
Step 2: Apply the local approximation. $$ 0.06\left(\frac{7}{365}\right)=0.0011507. $$
The estimated short-interval probability is \(0.0011507\), or \(0.11507\%\). To four decimal places as a probability, this is \(0.0012\); as a percentage to four decimal places, it is \(0.1151\%\).
The one-year endpoint probability \(0.08\) does not determine this short-interval estimate. It describes being in state \(B\) at the end of a year, whereas \(0.06\) per year is the current local rate of a direct transition from \(A\) to \(B\). Additional assumptions about the process over the year would be needed to relate them.
Common Mistake: Treating a Rate as a Probability
Suppose an intensity is \(0.12\) per year and the interval is 15 days. The incorrect calculation \(0.12=12\%\) treats the annual rate as the probability for the 15-day interval. It ignores the interval length. A second incorrect calculation is \(0.12\times15=1.8\), which multiplies a rate per year by a duration in days and so has inconsistent units.
The correction is to convert the duration to years and then use the short-interval approximation: $$ 0.12\text{ per year}\times\frac{15}{365}\text{ years}\approx0.0049. $$ This is approximately \(0.49\%\), not \(12\%\) or \(180\%\). For a longer interval, even a correctly converted product of intensity and duration need not equal the exact finite-time transition probability. The product is justified here as a short-interval approximation.
Check Your Understanding
For each item, distinguish the requested finite-interval probability from the local intensity and state any approximation being used.
- A hypothetical transition intensity is \(0.09\) per year. Using a 365-day year, estimate the direct-transition probability over 20 days by the short-interval approximation. (Answer: \(0.0049\), or \(0.4932\%\).)
- A time-dependent intensity is \(\mu(t)=0.01+0.004t\) per year, with \(t\) in years. At \(t=5\), estimate the direct-transition probability over the next month using a 12-month year. (Answer: \(\mu(5)=0.030\) per year and probability approximately \(0.0025\), or \(0.2500\%\).)
- A model gives \(p_{AB}(0,2)=0.14\). Is \(0.07\) per year necessarily the instantaneous intensity from \(A\) to \(B\) at time zero? (Answer: No. \(0.14/2=0.07\) is an average endpoint probability per year, not necessarily the instantaneous intensity; more model information is required.)
- A direct-transition intensity is \(1.8\) per year. Using a 365-day year, estimate the probability of that direct transition over one day by the short-interval approximation. Does the intensity itself represent a probability of 180%? (Answer: The approximation is \(1.8/365=0.0049\), or \(0.4932\%\); no, \(1.8\) per year is a rate, not a probability.)
- A local intensity is \(0.04\) per month, and a month is defined as 30 days. Estimate the direct-transition probability during the next 10 days by the short-interval approximation. (Answer: \(0.04(10/30)=0.0133\), or \(1.3333\%\).)