When to look at a noisy Markov chain in sequential decision making if measurements are costly?

A decision maker records measurements of a finite-state Markov chain corrupted by noise. The goal is to decide when the Markov chain hits a specific target state. The decision maker can choose from a finite set of sampling intervals to pick the next time to look at the Markov chain. The aim is to optimize an objective comprising of false alarm, delay cost and cumulative measurement sampling cost. Taking more frequent measurements yields accurate estimates but incurs a higher measurement cost. Making an erroneous decision too soon incurs a false alarm penalty. Waiting too long to declare the target state incurs a delay penalty. What is the optimal sequential strategy for the decision maker? The paper shows that under reasonable conditions, the optimal strategy has the following intuitive structure: when the Bayesian estimate (posterior distribution) of the Markov chain is away from the target state, look less frequently; while if the posterior is close to the target state, look more frequently. Bounds are derived for the optimal strategy. Also the achievable optimal cost of the sequential detector as a function of transition dynamics and observation distribution is analyzed. The sensitivity of the optimal achievable cost to parameter variations is bounded in terms of the Kullback divergence. To prove the results in this paper, novel stochastic dominance results on the Bayesian filtering recursion are derived. The formulation in this paper generalizes quickest time change detection to consider optimal sampling and also yields useful results in sensor scheduling (active sensing)

Reinforcement Learning: Stochastic Approximation Algorithms for Markov Decision Processes

This article presents a short and concise description of stochastic approximation algorithms in reinforcement learning of Markov decision processes. The algorithms can also be used as a suboptimal method for partially observed Markov decision processes

Quickest Detection with Social Learning: Interaction of local and global decision makers

We consider how local and global decision policies interact in stopping time problems such as quickest time change detection. Individual agents make myopic local decisions via social learning, that is, each agent records a private observation of a noisy underlying state process, selfishly optimizes its local utility and then broadcasts its local decision. Given these local decisions, how can a global decision maker achieve quickest time change detection when the underlying state changes according to a phase-type distribution? The paper presents four results. First, using Blackwell dominance of measures, it is shown that the optimal cost incurred in social learning based quickest detection is always larger than that of classical quickest detection. Second, it is shown that in general the optimal decision policy for social learning based quickest detection is characterized by multiple thresholds within the space of Bayesian distributions. Third, using lattice programming and stochastic dominance, sufficient conditions are given for the optimal decision policy to consist of a single linear hyperplane, or, more generally, a threshold curve. Estimation of the optimal linear approximation to this threshold curve is formulated as a simulation-based stochastic optimization problem. Finally, the paper shows that in multi-agent sensor management with quickest detection, where each agent views the world according to its prior, the optimal policy has a similar structure to social learning

Partially Observed Markov Decision Processes. Problem Sets and Internet Supplement

This document is an internet supplement to my book "Partially Observed Markov Decision Processes - From Filtering to Controlled Sensing" published by Cambridge University Press in 2016. This internet supplement contains exercises, examples and case studies. The material appears in this internet supplement (instead of the book) so that it can be updated. This document will evolve over time and further discussion and examples will be added. This internet supplement document is work in progress and will be updated periodically. I welcome constructive comments from readers of the book and this internet supplement

Controlled Sequential Information Fusion with Social Sensors

A sequence of social sensors estimate an unknown parameter (modeled as a state of nature) by performing Bayesian Social Learning, and myopically optimize individual reward functions. The decisions of the social sensors contain quantized information about the underlying state. How should a fusion center dynamically incentivize the social sensors for acquiring information about the underlying state? This paper presents five results. First, sufficient conditions on the model parameters are provided under which the optimal policy for the fusion center has a threshold structure. The optimal policy is determined in closed form, and is such that it switches between two exactly specified incentive policies at the threshold. Second, it is shown that the optimal incentive sequence is a sub-martingale, i.e, the optimal incentives increase on average over time. Third, it is shown that it is possible for the fusion center to learn the true state asymptotically by employing a sub-optimal policy; in other words, controlled information fusion with social sensors can be consistent. Fourth, uniform bounds on the average additional cost incurred by the fusion center for employing a sub-optimal policy are provided. This characterizes the trade-off between the cost of information acquisition and consistency for the fusion center. Finally, when it is sufficient to estimate the state with a degree of confidence, uniform bounds on the budget saved by employing policies that guarantee state estimation in finite time are provided

Average-Consensus Algorithms in a Deterministic Framework

We consider the average-consensus problem in a multi-node network of finite size. Communication between nodes is modeled by a sequence of directed signals with arbitrary communication delays. Four distributed algorithms that achieve average-consensus are proposed. Necessary and sufficient communication conditions are given for each algorithm to achieve average-consensus. Resource costs for each algorithm are derived based on the number of scalar values that are required for communication and storage at each node. Numerical examples are provided to illustrate the empirical convergence rate of the four algorithms in comparison with a well-known "gossip" algorithm as well as a randomized information spreading algorithm when assuming a fully connected random graph with instantaneous communication.Comment: 53 pages, 2 figures, 1 table. Short version submitted to IEEE Trans. Signal Processin

Dependence Structure Analysis Of Meta-level Metrics in YouTube Videos: A Vine Copula Approach

This paper uses vine copula to analyze the multivariate statistical dependence in a massive YouTube dataset consisting of 6 million videos over 25 thousand channels. Specifically we study the statistical dependency of 7 YouTube meta-level metrics: view count, number of likes, number of comments, length of video title, number of subscribers, click rates, and average percentage watching. Dependency parameters such as the Kendall's tau and tail dependence coefficients are computed to evaluate the pair-wise dependence of these meta-level metrics. The vine copula model yields several interesting dependency structures. We show that view count and number of likes' are in the central position of the dependence structure. Conditioned on these two metrics, the other five meta-level metrics are virtually independent of each other. Also, Sports, Gaming, Fashion, Comedy videos have similar dependence structure to each other, while the News category exhibits a strong tail dependence. We also study Granger causality effects and upload dynamics and their impact on view count. Our findings provide a useful understanding of user engagement in YouTube

How to Calibrate your Adversary's Capabilities? Inverse Filtering for Counter-Autonomous Systems

We consider an adversarial Bayesian signal processing problem involving "us" and an "adversary". The adversary observes our state in noise; updates its posterior distribution of the state and then chooses an action based on this posterior. Given knowledge of "our" state and sequence of adversary's actions observed in noise, we consider three problems: (i) How can the adversary's posterior distribution be estimated? Estimating the posterior is an inverse filtering problem involving a random measure - we formulate and solve several versions of this problem in a Bayesian setting. (ii) How can the adversary's observation likelihood be estimated? This tells us how accurate the adversary's sensors are. We compute the maximum likelihood estimator for the adversary's observation likelihood given our measurements of the adversary's actions where the adversary's actions are in response to estimating our state. (iii) How can the state be chosen by us to minimize the covariance of the estimate of the adversary's observation likelihood? "Our" state can be viewed as a probe signal which causes the adversary to act; so choosing the optimal state sequence is an input design problem. The above questions are motivated by the design of counter-autonomous systems: given measurements of the actions of a sophisticated autonomous adversary, how can our counter-autonomous system estimate the underlying belief of the adversary, predict future actions and therefore guard against these actions

Adaptive Polling in Hierarchical Social Networks using Blackwell Dominance

Consider a population of individuals that observe an underlying state of nature that evolves over time. The population is classified into different levels depending on the hierarchical influence that dictates how the individuals at each level form an opinion on the state. The population is sampled sequentially by a pollster and the nodes (or individuals) respond to the questions asked by the pollster. This paper considers the following problem: How should the pollster poll the hierarchical social network to estimate the state while minimizing the polling cost (measurement cost and uncertainty in the Bayesian state estimate)? This paper proposes adaptive versions of the following polling methods: Intent Polling, Expectation Polling, and the recently proposed Neighbourhood Expectation Polling to account for the time varying state of nature and the hierarchical influence in social networks. The adaptive polling problem in a hierarchical social network is formulated as a partially observed Markov decision process (POMDP). Our main results exploit the structure of the polling problem, and determine novel conditions for Blackwell dominance to construct myopic policies that provably upper bound the optimal policy of the adaptive polling POMDP. The LeCam deficiency is used to determine approximate Blackwell dominance for general polling problems. These Blackwell dominance conditions also facilitate the comparison of Renyi Divergence and Shannon capacity of more general channel structures that arise in hierarchical social networks. Numerical examples are provided to illustrate the adaptive polling policies with parameters estimated from YouTube data

Sequential Detection of Market shocks using Risk-averse Agent Based Models

This paper considers a statistical signal processing problem involving agent based models of financial markets which at a micro-level are driven by socially aware and risk- averse trading agents. These agents trade (buy or sell) stocks by exploiting information about the decisions of previous agents (social learning) via an order book in addition to a private (noisy) signal they receive on the value of the stock. We are interested in the following: (1) Modelling the dynamics of these risk averse agents, (2) Sequential detection of a market shock based on the behaviour of these agents. Structural results which characterize social learning under a risk measure, CVaR (Conditional Value-at-risk), are presented and formulation of the Bayesian change point detection problem is provided. The structural results exhibit two interesting prop- erties: (i) Risk averse agents herd more often than risk neutral agents (ii) The stopping set in the sequential detection problem is non-convex. The framework is validated on data from the Yahoo! Tech Buzz game dataset