知の物理学研究センター / Institute for Physics of Intelligence (iπ)

【日時/Date】
2022年1月11日（火）10時30分～12時00分

【講演者/Speaker】
坂田 綾香/Ayaka SAKATA（統計数理研究所/Institute of Statistical Mathematics）

【講演タイトル/Title】
"Statistical physics for the Bayesian statistical decision: an application to Group Testing"

【概要/Abstract】
    Group testing is an efficient way to identify the patients' states, positive or negative, with fewer tests than the number of patients. The tests are performed on pools of mixed specimens taken from patients, where the number of the pools is smaller than that of the patients. The identification of the patients’ states from fewer number of test results is an underdetermined problem, and in general the patients’ states are not determined uniquely. However, it is known that if the fraction of the positive patients in the population is sufficiently small, in principle, one can identify the positive patients from the tests using appropriate inference method. 
     In this talk, we introduce Bayesian inference to formulate the group testing problem in order to account for the inevitable errors in realistic testing. There exist several deterministic inference algorithms for group testing, but it is known that the test errors deteriorate their performances. From the knowledges in estimation problems related to the group testing, it is expected that Bayesian inference in group testing is robust to the test errors.
     The basis of Bayesian inference is the posterior distribution with continuous values. Meanwhile, the goal is the identification of the patients' states, which corresponds to discrete variables. Therefore, for the identification of the patients' states, a mapping from continuous values to discrete variables is necessary. We introduce Bayesian decision-theoretic considerations to design the mapping and show that a cutoff-based function for the posterior marginal probabilities is optimal as a mapping.
     In implementing the cutoff-based decisions, the value of the posterior marginal probability is necessary. The computation of the posterior marginal probability corresponds to that of the thermal average in statistical physics. We introduce an approximated computation method known as message passing for the posterior marginal probability, and mention the relationship between the replica method in spin-glass theory.

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