Representing an Uncertain Starting State

Why the Starting State Needs Its Own Description

A multiple-state model describes how a person or policy can occupy different states and move between them over time. The transition structure alone does not say which state is occupied at the starting instant. That requires a separate description whenever the initial state is uncertain.

For example, a hypothetical long-term care portfolio might include people who are independent, people receiving moderate care, people receiving intensive care, and people who have died. If the model begins with a group whose members are in different states, the starting composition is described by probabilities. These probabilities answer a different question from an intensity matrix: they describe where the model starts, not how quickly transitions occur.

Definition: Let the possible states at time zero be $1,2,\ldots,m$, and let $Z(0)$ denote the state occupied at time zero. The initial-state distribution is the ordered collection of probabilities $$ \pi_i(0)=P(Z(0)=i), \qquad i=1,2,\ldots,m. $$ The entries are probabilities for the listed states in the stated order. They must be nonnegative and sum to one. The ordered entries may be displayed as $\pi(0)=(\pi_1(0),\ldots,\pi_m(0))$; whether this is written as a row or column is a notation convention.

The index $i$ identifies a state, and the argument $0$ identifies the time at which the distribution applies. The distribution is about the state at that instant. It is not a set of transition rates, nor does it by itself describe later state probabilities.

Validity Conditions and Their Meaning

The possible states at time zero must form an exhaustive and mutually exclusive set for the population being modelled. Every member is in exactly one of them at that instant. Consequently, each probability is at least zero, and the probabilities across all states add to one.

\pi_i(0)\geq 0\quad\text{for every }i, \qquad \sum_{i=1}^{m}\pi_i(0)=1.

These conditions are not just formal checks. If an entry is negative, it cannot represent the probability of occupying a state. If the entries sum to less than one, some probability mass has not been assigned to any state; if they sum to more than one, the states or probabilities overlap or have been counted inconsistently. A missing state, such as an absorbing state that is part of the model, can also cause an incomplete distribution.

List the states in order.
Use the same state definitions and order throughout the model description.

Assign a probability to each state at time zero.
Each entry describes the chance that a randomly selected model member starts in that state.

Check every entry and the total.
Verify that no entry is negative and that the sum is exactly one, allowing only for explicitly stated rounding.

A probability distribution may have zero entries. A zero means that the state is included in the model but no member of the specified starting group occupies it at time zero. It does not mean that the state is forbidden as a future destination. Similarly, a positive initial probability for a state does not say that members in that state will remain there.

Worked Example: Checking a Long-Term Care Starting Distribution

A hypothetical model has four states, in this order: independent $I$, moderate care $M$, intensive care $C$, and dead $D$. At time zero, the assigned probabilities are $0.72, 0.18, 0.07,$ and $0.03$, respectively. Check the distribution and calculate the expected immediate payment if the contract pays $0 in states $I$ and $D$, $4,000 in state $M$, and $12,000 in state $C$.

Method. Check the entries against the probability conditions, then calculate the expected payment by weighting each state-specific payment by its initial probability.

Step 1: Check nonnegativity. The four entries $0.72, 0.18, 0.07,$ and $0.03$ are all at least zero.

Step 2: Check the total.

$$ 0.72+0.18+0.07+0.03=0.90+0.07+0.03=0.97+0.03=1.00. $$

The initial distribution is valid.

Step 3: Calculate the expected immediate payment. The payment is $0 with probability $0.72$, $4,000 with probability $0.18$, $12,000 with probability $0.07$, and $0 with probability $0.03$. Therefore,

E[\text{payment at time }0] =0.72(0)+0.18(4{,}000)+0.07(12{,}000)+0.03(0) =0+720+840+0 =\$1{,}560.

The result is checked by noting that only the moderate-care and intensive-care states contribute a positive payment, and their weighted amounts total $1,560. The common mistake this method avoids is treating each state’s payment as equally likely rather than weighting it by the given initial probability.

Combining Starting Distributions Across Groups

Sometimes the starting population is assembled from subgroups with different state compositions. The overall initial distribution is then a weighted average of the subgroup distributions, where the weights are the proportions of the total population in each subgroup. The subgroup proportions must themselves add to one.

\pi_i(0)=\sum_{g=1}^{G} w_g\,\pi_{i\mid g}(0), \qquad \sum_{g=1}^{G}w_g=1.

Here $w_g$ is the proportion of the full starting population in subgroup $g$, and $\pi_{i\mid g}(0)$ is the probability that a member of that subgroup starts in state $i$. This is the law of total probability: a randomly selected member first belongs to one of the subgroups, and then occupies a state according to that subgroup’s distribution.

Worked Example: Aggregating Two Subgroup Distributions

A hypothetical care model has states $A$ (active), $B$ (receiving care), and $D$ (dead). One quarter of the starting population belongs to subgroup 1, with initial distribution $(0.80,0.20,0)$. The other three quarters belong to subgroup 2, with initial distribution $(0.40,0.50,0.10)$. Find and check the overall initial distribution.

Method. Multiply each subgroup’s state probabilities by that subgroup’s share of the population, then add the contributions state by state.

Step 1: Confirm the subgroup weights.

w_1=0.25,\qquad w_2=0.75,\qquad 0.25+0.75=1.

Each subgroup distribution is also valid: $0.80+0.20+0=1$, and $0.40+0.50+0.10=1$.

Step 2: Aggregate the probability of each state.

\begin{aligned} \pi_A(0)&=0.25(0.80)+0.75(0.40)=0.20+0.30=0.50,\\ \pi_B(0)&=0.25(0.20)+0.75(0.50)=0.05+0.375=0.425,\\ \pi_D(0)&=0.25(0)+0.75(0.10)=0+0.075=0.075. \end{aligned}

Thus the overall distribution, in the order $A,B,D$, is $(0.50,0.425,0.075)$.

Step 3: Check the result. All entries are nonnegative, and

$$ 0.50+0.425+0.075=0.925+0.075=1.000. $$

The aggregate is valid. The common mistake this method avoids is taking an unweighted average of the subgroup distributions, which would be wrong because the subgroups are not the same size.

Why the Validity Conditions Hold

The validity conditions follow directly from the definition of a probability distribution. At time zero, the events $\{Z(0)=1\},\ldots,\{Z(0)=m\}$ are mutually exclusive: the model member cannot occupy two different states at the same instant. They are also exhaustive: under the model, the member must occupy one of the listed states.

P(Z(0)=i)\geq 0 \quad\text{and}\quad \bigcup_{i=1}^{m}\{Z(0)=i\}=\Omega,

where $\Omega$ is the set of possible outcomes for the model member’s starting state. Because the events form a partition of $\Omega$, additivity of probability gives

\sum_{i=1}^{m}P(Z(0)=i) =P\left(\bigcup_{i=1}^{m}\{Z(0)=i\}\right) =P(\Omega)=1.

This also explains why a forgotten state matters. If the listed states do not cover every possible starting condition, their events do not form a partition of all outcomes, and their probabilities need not sum to one. The modeler must either include the omitted state or explicitly define a population for which that state is impossible.

Initial Distribution Is Not an Intensity Matrix

The initial distribution and the intensity matrix are both associated with a multiple-state model, but they encode different information. An initial distribution gives dimensionless probabilities at one specified instant. An intensity matrix gives instantaneous rates of transition between states, with units such as per year.

Quantity	What it describes	Typical entries	Units
Initial distribution	State occupied at time zero	Nonnegative probabilities summing to one	None
Intensity matrix	Instantaneous transition rates between states	Nonnegative off-diagonal rates and diagonal entries that balance each row	Per unit time

For instance, a hypothetical intensity matrix can have an off-diagonal entry of $0.06$ per year for a transition out of the active state and an entry of $0.04$ per year for a different transition from that state. These entries describe rates conditional on being in the relevant state; they do not say what proportion of the starting group is active. A separate initial distribution is needed to answer that question.

Conversely, knowing that 70% of a starting population is active does not determine the rate at which active members leave that state. The state composition and the transition mechanism are separate model inputs. The matrix’s diagonal entries are not probabilities of starting in a state either: they balance the outgoing rates and are typically negative.

Worked Example: Separating Starting Probabilities From Transition Rates

A hypothetical three-state model has states $A$ (active), $C$ (care), and $D$ (dead). Its constant intensity matrix, with rates measured per year, is

Q= \begin{pmatrix} -0.10&0.07&0.03\\ 0&-0.20&0.20\\ 0&0&0 \end{pmatrix}.

At time zero, an analyst states that the population is distributed in the ratio $2:3:5$ across $A,C,D$, respectively. Find the initial distribution, check the intensity matrix’s relevant row totals, and explain why the matrix alone would not provide this initial distribution.

Method. Convert the relative starting weights to probabilities by dividing by their total. Separately inspect the matrix as a set of rates, not as a probability distribution.

Step 1: Normalize the starting weights. The total relative weight is $2+3+5=10$. Thus,

\pi(0)=\left(\frac{2}{10},\frac{3}{10},\frac{5}{10}\right)=(0.20,0.30,0.50).

The entries are nonnegative and $0.20+0.30+0.50=1.00$, so this is a valid initial distribution.

Step 2: Check the matrix rates. In the active state, the outgoing rates are $0.07$ per year to care and $0.03$ per year to death. Their sum is $0.10$ per year, so the diagonal entry is $-0.10$ per year. In care, the outgoing rate is $0.20$ per year to death, balanced by the diagonal entry $-0.20$ per year. The dead state has no outgoing transitions, so its row is all zero.

Step 3: Keep the two objects distinct. The normalized ratio gives the starting probabilities $(0.20,0.30,0.50)$. The entries $0.07,0.03,$ and $0.20$ in $Q$ are rates per year; they do not add up across states to give the starting composition. Without the stated ratio or other starting-population information, $Q$ alone would not determine $\pi(0)$.

The common mistake this method avoids is normalizing the matrix’s rates as if they were probabilities. Rates describe transition behavior, whereas the normalized ratio describes the initial state.

Common Checks Before Using an Initial Distribution

Before using initial probabilities in a calculation, verify that the state order matches the model description and that the probabilities refer to the same starting population and time. Then check each entry and the total. If values are rounded, report that the sum differs from one only because of rounding; do not silently alter an entry to force a total of one.

Do not confuse weights with probabilities. Relative weights such as $2:3:5$ must be normalized before they are used as probabilities.
Do not normalize automatically. If stated probabilities sum to $1.05$, the data are inconsistent as given; normalization is justified only when the figures are explicitly relative weights or a rounding issue is established.
Do not read rates as starting shares. Intensities have a time unit and describe transition behavior; initial probabilities do not.
Include every possible initial state. If a state is omitted, confirm that it is impossible for the population being modelled to start there.

Key takeaway: An initial-state distribution is a probability description of the model’s state at time zero. Its entries are nonnegative and sum to one. It is separate from the intensity matrix, which describes instantaneous transition rates.

Check Your Understanding

For each problem, use the stated state order and distinguish probabilities from rates where relevant.

A hypothetical group starts in states $A,B,C$ with probabilities $0.55,0.25,0.20$. Check validity and find the probability of starting in either $B$ or $C$. (Answer: Valid; $0.45$)
A proposed distribution across three exhaustive states is $(0.40,0.35,0.20)$. Is it valid as stated, and how much probability is unassigned? (Answer: Invalid; the entries sum to $0.95$, so $0.05$ is unassigned)
One quarter of a population has distribution $(0.80,0.20)$ across states $A,B$, and three quarters has distribution $(0.40,0.60)$. Find the aggregate distribution. (Answer: $(0.50,0.50)$)
A model has an active-to-care intensity of $0.06$ per year and an active-to-death intensity of $0.04$ per year. Can these two rates determine the initial probability of being active? Explain. (Answer: No; they describe transition rates from the active state, not the starting-state distribution.)
At time zero, a hypothetical contract pays $0 in state $A$, $500 in state $B$, and $2,000 in state $C$. The initial probabilities are $(0.70,0.20,0.10)$. Find the expected time-zero payment and check the distribution. (Answer: Valid; $0.70(0)+0.20(500)+0.10(2{,}000)=\$300.00$)

What You'll Learn