On Sequential Detection for GaltonWatson Processes
Abstract
A sequential detection problem in which the observation sequence is a realization of a Galton
Watson process is considered. A Bayesian framework for this problem is introduced. Based on a
dynamic programming argument, an optimal decision rule is obtained. The optimal sequential
detector is shown to have a stationary structure analogous to the standard sequential probability
ratio test. A detailed analysis of this problem for various cost assignments is presented along
with some numerical examples.
1
Introduction
The sequential detection problem for independent and identically distributed (i.i.d.) observations
has been discussed extensively in the literature (see
e.g.
[7]). Pioneering work in the area is due
to Wald [9]. Various extensions of this problem have already been considered, many of which fall
under the class of optimal stopping problems [6].
This paper considers sequential detection for GaltonWatson processes, a popular model for
population growth and cell kinetics.
This problem is fundamentally different from the standard
sequential detection problem in which observations are i.i.d. random variables. Indeed, the iterative
structure of GaltonWatson processes introduces a strong time dependence among observations.
In general, sequential detection for GaltonWatson processes cannot be reduced to the simpler
detection problem where observations are i.i.d. random variables.
We consider the instance in which the detector has to choose between two hypotheses,
H
0
and
H
1
. We introduce a Bayesian framework for the sequential detection problem. The optimal
solution is obtained as a limiting case of the truncated sequential detection problem in which a
decision must be made by time
T
. The population size together with the probability of hypothesis
H
0
conditioned on past observations form a sufficient statistic for a dynamic programming solution
to the sequential detection problem. The optimal detector is shown to have a stationary structure
analogous to the standard sequential probability ratio test, albeit more complex.
The remainder of this paper is organized as follows: In Section 2, we review the definition of
GaltonWatson processes and introduce a performance metric against which sequential detectors
are evaluated. In Section 3, we show that the problem of selecting an optimal terminal decision
rule presents no particular difficulties. Indeed, the problem of optimal sequential detection lies with
the selection of an optimal stopping rule. We derive an optimal solution to the truncated sequen
tial detection problem and to the infinite horizon sequential detection problem in Section 4 & 5,
respectively. Finally, we include concluding remarks in Section 6. Some numerical examples can
be found in Appendix A; while Appendix C contains proofs for all the Propositions and Lemmas.
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 Fall '07
 Enjeti
 Dynamic Programming, Conditional Probability, Probability theory, Sequential probability ratio test, Optimal Detector

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