Multihypothesis sequential probability ratio tests .I. Asymptotic optimality

Abstract

The problem of sequential testing of multiple hypotheses is considered, and two candidate sequential test procedures are studied. Both tests are multihypothesis versions of the binary sequential probability ratio test (SPRT), and are referred to as MSPRT's. The first test is motivated by Bayesian optimality arguments, while the second corresponds to a generalized likelihood ratio test. It is shown that both MSPRT's are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution, when the error probabilities or, more generally, risks associated with incorrect decisions are small. The results are first derived for the discrete-time case of independent and identically distributed (i.i.d.) observations and simple hypotheses. They are then extended to general, possibly continuous-time, statistical models that may include correlated and nonhomogeneous observation processes. It also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters, where the composite hypotheses, due to nuisance parameters, can be reduced to simple ones by using the principle of invariance. These results provide a complete generalization of the results given in [36], where it was shown that the quasi-Bayesian MSPRT is asymptotically efficient with respect to the expected sample size for i.i.d. observations. In a companion paper [12], based on the nonlinear renewal theory we find higher order approximations, up to a vanishing term, for the expected sample size that take into account the overshoot over the boundaries of decision statistics.

Multialternative drift-diffusion model predicts the relationship between visual fixations and choice in value-based decisions

“…In the model there is only ever one positive RDV at a time, which represents the difference in evidence between the items with the most and next-most evidence. Therefore, the model falls into the class of "best vs. next" models that are known to implement the asymptotically optimal MSPRT in the absence of attention biases (5,33,35). We emphasize, however, that once visual attention biases are present, neither the two-item version of the DDM, nor the multi-item version presented here, is optimal.…”

“…Furthermore, the optimal statistical test associated with the case of multiple alternatives is unknown, and only approximations have been proposed (32,33). Out of the alternatives that have been proposed, the multihypothesis sequential probability ratio test (MSPRT) is a particularly appealing one, because it reduces to the SPRT in the case of two options, and is asymptotically optimal in the sense that for sufficiently low error rates it minimizes expected decision time (5,25,33,35).Second, the classic DDM and leaky integrator models have ignored the role that visual attention plays in the choice process. This gap is an important limitation because both casual observation and previous research suggest that visual fixations play a role in the decision-making process (18,36,37).…”

Bayesian Sequential Change Diagnosis

“…Both change detection and sequential multi-hypothesis testing have been studied extensively. For recent reviews of these areas, we refer the reader to Basseville and Nikiforov [3], Dragalin, Tartakovsky and Veeravalli [8,9], and Lai [14], and the references therein.…”

Multisource Bayesian sequential binary hypothesis testing problem

“…The non-Bayes formulation of the sequential hypothesis testing problem has been studied by many authors, both in discrete-and continuous-time, and can be found in the recent reviews and contributions made by Lai (2000Lai ( , 2001, Dragalin et al (1999Dragalin et al ( , 2000, Lorden (1977). In the Bayesian framework, sequential hypothesis testing problems were studied in discrete-time for the identification of the distribution of i.i.d.…”