In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left.
Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this article concentrates on the discrete-time discrete-state-space case.
A Markov chain is an absorbing chain if
In an absorbing Markov chain, a state that is not absorbing is called transient.
Let an absorbing Markov chain with transition matrix P have t transient states and r absorbing states. Unlike a typical transition matrix, the rows of P represent sources, while columns represent destinations. Then
P= \begin{bmatrix} Q&R\\ 0&Ir \end{bmatrix},
The probability of transitioning from i to j in exactly k steps is the (i,j)-entry of Pk, further computed below. When considering only transient states, the probability found in the upper left of Pk, the (i,j)-entry of Qk.
A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Qk for all k (from 0 to ∞). It can be proven that
N:=
| infty | |
| \sum | |
| k=0 |
Qk=(It-Q)-1,
| infty | |
| {style\sum} | |
| k=0 |
qk=\tfrac{1}{1-q}
With the matrix N in hand, also other properties of the Markov chain are easy to obtain.[1]
The expected number of steps before being absorbed in any absorbing state, when starting in transient state i can be computed via a sum over transient states. The value is given by the ith entry of the vector
t:=N1,
By induction,
Pk= \begin{bmatrix} Qk&(1-Qk)NR\\ 0&Ir \end{bmatrix}.
B:=NR
An approximation of those probabilities can also be obtained directly from the (i,j)-entry of
Pk
\left(\limk
| k\right) | |
| P | |
| i,t+j |
=Bi,j
The probability of visiting transient state j when starting at a transient state i is the (i,j)-entry of the matrix
H:=(N-It)(N\operatorname{dg
The variance on the number of visits to a transient state j with starting at a transient state i (before being absorbed) is the (i,j)-entry of the matrix
N2:=N(2N\operatorname{dg
The variance on the number of steps before being absorbed when starting in transient state i is the ith entry of the vector
(2N-It)t-t\operatorname{sq
Consider the process of repeatedly flipping a fair coin until the sequence (heads, tails, heads) appears. This process is modeled by an absorbing Markov chain with transition matrix
P= \begin{bmatrix} 1/2&1/2&0&0\\ 0&1/2&1/2&0\\ 1/2&0&0&1/2\\ 0&0&0&1 \end{bmatrix}.
For this absorbing Markov chain, the fundamental matrix is
\begin{align} N&=(I-Q)-1= \left(\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix} - \begin{bmatrix} 1/2&1/2&0\\ 0&1/2&1/2\\ 1/2&0&0 \end{bmatrix} \right)-1\\[4pt] &= \begin{bmatrix} 1/2&-1/2&0\\ 0&1/2&-1/2\\ -1/2&0&1 \end{bmatrix}-1= \begin{bmatrix} 4&4&2\\ 2&4&2\\ 2&2&2 \end{bmatrix}. \end{align}
The expected number of steps starting from each of the transient states is
t=N1= \begin{bmatrix} 4&4&2\\ 2&4&2\\ 2&2&2 \end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} = \begin{bmatrix} 10\\ 8\\ 6 \end{bmatrix}.
Games based entirely on chance can be modeled by an absorbing Markov chain. A classic example of this is the ancient Indian board game Snakes and Ladders. The graph on the left[2] plots the probability mass in the lone absorbing state that represents the final square as the transition matrix is raised to larger and larger powers. To determine the expected number of turns to complete the game, compute the vector t as described above and examine tstart, which is approximately 39.2.
Infectious disease testing, either of blood products or in medical clinics, is often taught as an example of an absorbing Markov chain.[3] The public U.S. Centers for Disease Control and Prevention (CDC) model for HIV and for hepatitis B, for example,[4] illustrates the property that absorbing Markov chains can lead to the detection of disease, versus the loss of detection through other means.
In the standard CDC model, the Markov chain has five states, a state in which the individual is uninfected, then a state with infected but undetectable virus, a state with detectable virus, and absorbing states of having quit/been lost from the clinic, or of having been detected (the goal). The typical rates of transition between the Markov states are the probability p per unit time of being infected with the virus, w for the rate of window period removal (time until virus is detectable), q for quit/loss rate from the system, and d for detection, assuming a typical rate
λ
It follows that we can "walk along" the Markov model to identify the overall probability of detection for a person starting as undetected, by multiplying the probabilities of transition to each next state of the model as:
| p | |
| (p+q) |
| w | |
| (w+q) |
| d | |
| (d+q) |
The subsequent total absolute number of false negative tests—the primary CDC concern—would then be the rate of tests, multiplied by the probability of reaching the infected but undetectable state, times the duration of staying in the infected undetectable state:
| p | |
| (p+q) |
| 1 | |
| (w+q) |
λ