DOCSLIB.ORG

Entropy Rate

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Entropy Rate Lecture 6: Entropy Rate • Entropy rate H(X) • Random walk on graph Dr. Yao Xie, ECE587, Information Theory, Duke University Coin tossing versus poker • Toss a fair coin and see and sequence Head, Tail, Tail, Head ··· −nH(X) (x1; x2;:::; xn) ≈ 2 • Play card games with friend and see a sequence A | K r Q q J ♠ 10 | ··· (x1; x2;:::; xn) ≈? Dr. Yao Xie, ECE587, Information Theory, Duke University 1 How to model dependence: Markov chain • A stochastic process X1; X2; ··· – State fX1;:::; Xng, each state Xi 2 X – Next step only depends on the previous state p(xn+1jxn;:::; x1) = p(xn+1jxn): – Transition probability pi; j : the transition probability of i ! j P = j – p(xn+1) xn p(xn)p(xn+1 xn) – p(x1; x2; ··· ; xn) = p(x1)p(x2jx1) ··· p(xnjxn−1) Dr. Yao Xie, ECE587, Information Theory, Duke University 2 Hidden Markov model (HMM) • Used extensively in speech recognition, handwriting recognition, machine learning. • Markov process X1; X2;:::; Xn, unobservable • Observe a random process Y1; Y2;:::; Yn, such that Yi ∼ p(yijxi) • We can build a probability model Yn−1 Yn n n p(x ; y ) = p(x1) p(xi+1jxi) p(yijxi) i=1 i=1 Dr. Yao Xie, ECE587, Information Theory, Duke University 3 Time invariance Markov chain • A Markov chain is time invariant if the conditional probability p(xnjxn−1) does not depend on n p(Xn+1 = bjXn = a) = p(X2 = bjX1 = a); for all a; b 2 X • For this kind of Markov chain, define transition matrix 2 3 6 ··· 7 6P11 P1n7 = 6 ··· 7 P 46 57 Pn1 ··· Pnn Dr. Yao Xie, ECE587, Information Theory, Duke University 4 Simple weather model • X = f Sunny: S, Rainy R g • p(SjS) = 1 − β, p(RjR) = 1 − α, p(RjS) = β, p(SjR) = α " # 1 − β β P = α 1 − α #" $% " $%#" " " Dr. Yao Xie, ECE587, Information Theory, Duke University 5 • Probability of seeing a sequence SSRR: p(SSRR) = p(S)p(SjS)p(RjS)p(RjR) = p(S)(1 − β)β(1 − α) What will this sequence behave, after many days of observations? • What sequences of observations are more typical? • What is the probability of seeing a typical sequence? Dr. Yao Xie, ECE587, Information Theory, Duke University 6 Stationary distribution • Stationary distribution: a distribution µ on the states such that the distribution at time n + 1 is the same as the distribution at time n. • Our weather example: α β – If µ(S) = α+β, µ(R) = α+β " # 1 − β β P = α 1 − α – Then p(Xn+1 = S) = p(SjS)µ(S) + p(SjR)µ(R) α β α = − β + α = = µ : (1 )α + β α + β α + β (S ) Dr. Yao Xie, ECE587, Information Theory, Duke University 7 • How to calculate stationary distribution – Stationary distribution µi, i = 1; ··· ; jXj satisfies X XjXj µi = µ j p ji; (µ = µP); and µi = 1: j i=1 – “Detailed balancing”: # " " $% "#%" $" "#"" Dr. Yao Xie, ECE587, Information Theory, Duke University 8 Stationary process • A stochastic process is stationary if the joint distribution of any subset is invariant to time-shift p(X1 = x1; ··· ; Xn = xn) = p(X2 = x1; ··· ; Xn+1 = xn): • Example: coin tossing p(X1 = head; X2 = tail) = p(X2 = head; X3 = tail) = p(1 − p): Dr. Yao Xie, ECE587, Information Theory, Duke University 9 Entropy rate P • n = ; ··· ; = n = When Xi are i.i.d., entropy H(X ) H(X1 Xn) i=1 H(Xi) nH(X) n • With dependent sequence Xi, how does H(X ) grow with n? Still linear? • Entropy rate characterizes the growth rate • Definition 1: average entropy per symbol H(Xn) H(X) = lim n!1 n • Definition 2: rate of information innovation 0 H (X) = lim H(XnjXn−1; ··· ; X1) n!1 Dr. Yao Xie, ECE587, Information Theory, Duke University 10 0 • H (X) exists, for Xi stationary H(XnjX1; ··· ; Xn−1) ≤ H(XnjX2; ··· ; Xn−1) (1) ≤ H(Xn−1jX1; ··· ; Xn−2) (2) – H(XnjX1; ··· ; Xn−1) decreases as n increases – H(X) ≥ 0 – The limit must exist Dr. Yao Xie, ECE587, Information Theory, Duke University 11 0 • H(X) = H (X), for Xi stationary Xn 1 1 H(X ; ··· ; X ) = H(X jX − ; ··· ; X ) n 1 n n i i 1 1 i=1 0 • Each H(XnjX1; ··· ; Xn−1) ! H (X) • Cesaro mean: P ! = 1 n ! ! If an a, bn n i=1 ai, bi a, then bn a. • So 1 H(X ; ··· ; X ) ! H0(X) n 1 n Dr. Yao Xie, ECE587, Information Theory, Duke University 12 AEP for stationary process 1 − log p(X ; ··· ; X ) ! H(X) n 1 n −nH(X) • p(X1; ··· ; Xn) ≈ 2 • Typical sequences in typical set of size 2−nH(X) • We can use nH(X) bits to represent typical sequence Dr. Yao Xie, ECE587, Information Theory, Duke University 13 Entropy rate for Markov chain • For Markov chain H(X) = lim H(XnjXn−1; ··· ; X1) = lim H(XnjXn−1) = H(X2jX1) • By definition p(X2 = jjX1 = i) = Pi j • Entropy rate of Markov chain X H(X) = − µiPi j log Pi j i j Dr. Yao Xie, ECE587, Information Theory, Duke University 14 Calculate entropy rate is fairly easy 1. Find stationary distribution µi 2. Use transition probability Pi j X H(X) = − µiPi j log Pi j i j Dr. Yao Xie, ECE587, Information Theory, Duke University 15 Entropy rate of weather model α β Stationary distribution µ(S) = α+β, µ(R) = α+β " # 1 − β β P = α 1 − α β X = α α + − α − α H( ) α + β[ log (1 ) log(1 )] α β = β + α α + βH( ) α + βH( ) ! αβ ( p ) ≤ ≤ αβ H 2α + β H Maximum when α = β = 1=2: degenerate to independent process Dr. Yao Xie, ECE587, Information Theory, Duke University 16 Random walk on graph • An undirected graph with m nodes f1;:::; mg • Edge i ! j has weight Wi j ≥ 0 (Wi j = W ji) • A particle walks randomly from node to node • Random walk X1; X2; ··· : a sequence of vertices • Given Xn = i, next step chosen from neighboring nodes with probability Wi j Pi j = P k Wik Dr. Yao Xie, ECE587, Information Theory, Duke University 17 Dr. Yao Xie, ECE587, Information Theory, Duke University 18 Entropy rate of random walk on graph • Let X X Wi = Wi j; W = Wi j j i; j:i> j • Stationary distribution is µ = Wi i 2W • Can verify this is a stationary distribution: µP = µ • Stationary distribution / weight of edges emanating from node i (locality) Dr. Yao Xie, ECE587, Information Theory, Duke University 19 Dr. Yao Xie, ECE587, Information Theory, Duke University 20 Summary • AEP Stationary process X1; X2; ··· , Xi 2 X: as n ! 1 −nH(X) p(x1; ··· ; xn) ≈ 2 • Entropy rate 1 H(X) = lim H(XnjXn−1;:::; X1) = lim H(X1;:::; Xn) n!1 n n!1 • Random walk on graph µ = Wi i 2W Dr. Yao Xie, ECE587, Information Theory, Duke University 21.
Load more

Recommended publications
  • Entropy Rate of Stochastic Processes
    Entropy Rate of Stochastic Processes Timo Mulder Jorn Peters [email protected] [email protected] February 8, 2015 The entropy rate of independent and identically distributed events can on average be encoded by H(X) bits per source symbol. However, in reality, series of events (or processes) are often randomly distributed and there can be arbitrary dependence between each event. Such processes with arbitrary dependence between variables are called stochastic processes. This report shows how to calculate the entropy rate for such processes, accompanied with some denitions, proofs and brief examples. 1 Introduction One may be interested in the uncertainty of an event or a series of events. Shannon entropy is a method to compute the uncertainty in an event. This can be used for single events or for several independent and identically distributed events. However, often events or series of events are not independent and identically distributed. In that case the events together form a stochastic process i.e. a process in which multiple outcomes are possible. These processes are omnipresent in all sorts of elds of study. As it turns out, Shannon entropy cannot directly be used to compute the uncertainty in a stochastic process. However, it is easy to extend such that it can be used. Also when a stochastic process satises certain properties, for example if the stochastic process is a Markov process, it is straightforward to compute the entropy of the process. The entropy rate of a stochastic process can be used in various ways. An example of this is given in Section 4.1.
    [Show full text]
  • Modeling Dependence in Data: Options Pricing and Random Walks
    UNIVERSITY OF CALIFORNIA, MERCED PH.D. DISSERTATION Modeling Dependence in Data: Options Pricing and Random Walks Nitesh Kumar A dissertation submitted in partial fulfillment of the requirements for the degree Doctor of Philosophy in Applied Mathematics March, 2013 UNIVERSITY OF CALIFORNIA, MERCED Graduate Division This is to certify that I have examined a copy of a dissertation by Nitesh Kumar and found it satisfactory in all respects, and that any and all revisions required by the examining committee have been made. Faculty Advisor: Harish S. Bhat Committee Members: Arnold D. Kim Roummel F. Marcia Applied Mathematics Graduate Studies Chair: Boaz Ilan Arnold D. Kim Date Contents 1 Introduction 2 1.1 Brief Review of the Option Pricing Problem and Models . ......... 2 2 Markov Tree: Discrete Model 6 2.1 Introduction.................................... 6 2.2 Motivation...................................... 7 2.3 PastWork....................................... 8 2.4 Order Estimation: Methodology . ...... 9 2.5 OrderEstimation:Results. ..... 13 2.6 MarkovTreeModel:Theory . 14 2.6.1 NoArbitrage.................................. 17 2.6.2 Implementation Notes. 18 2.7 TreeModel:Results................................ 18 2.7.1 Comparison of Model and Market Prices. 19 2.7.2 Comparison of Volatilities. 20 2.8 Conclusion ...................................... 21 3 Markov Tree: Continuous Model 25 3.1 Introduction.................................... 25 3.2 Markov Tree Generation and Computational Tractability . ............. 26 3.2.1 Persistentrandomwalk. 27 3.2.2 Number of states in a tree of fixed depth . ..... 28 3.2.3 Markov tree probability mass function . ....... 28 3.3 Continuous Approximation of the Markov Tree . ........ 30 3.3.1 Recursion................................... 30 3.3.2 Exact solution in Fourier space .
    [Show full text]
  • Entropy Rate Estimation for Markov Chains with Large State Space
    Entropy Rate Estimation for Markov Chains with Large State Space Yanjun Han Department of Electrical Engineering Stanford University Stanford, CA 94305 [email protected] Jiantao Jiao Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720 [email protected] Chuan-Zheng Lee, Tsachy Weissman Yihong Wu Department of Electrical Engineering Department of Statistics and Data Science Stanford University Yale University Stanford, CA 94305 New Haven, CT 06511 {czlee, tsachy}@stanford.edu [email protected] Tiancheng Yu Department of Electronic Engineering Tsinghua University Haidian, Beijing 100084 [email protected] Abstract Entropy estimation is one of the prototypicalproblems in distribution property test- ing. To consistently estimate the Shannon entropy of a distribution on S elements with independent samples, the optimal sample complexity scales sublinearly with arXiv:1802.07889v3 [cs.LG] 24 Sep 2018 S S as Θ( log S ) as shown by Valiant and Valiant [41]. Extendingthe theory and algo- rithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with S states from a sample path of n observations. We show that Provided the Markov chain mixes not too slowly, i.e., the relaxation time is • S S2 at most O( 3 ), consistent estimation is achievable when n . ln S ≫ log S Provided the Markov chain has some slight dependency, i.e., the relaxation • 2 time is at least 1+Ω( ln S ), consistent estimation is impossible when n . √S S2 log S . Under both assumptions, the optimal estimation accuracy is shown to be S2 2 Θ( n log S ).
    [Show full text]
  • A Study of Hidden Markov Model
    University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2004 A Study of Hidden Markov Model Yang Liu University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mathematics Commons Recommended Citation Liu, Yang, "A Study of Hidden Markov Model. " Master's Thesis, University of Tennessee, 2004. https://trace.tennessee.edu/utk_gradthes/2326 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Yang Liu entitled "A Study of Hidden Markov Model." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. Jan Rosinski, Major Professor We have read this thesis and recommend its acceptance: Xia Chen, Balram Rajput Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Yang Liu entitled “A Study of Hidden Markov Model.” I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics.
    [Show full text]
  • Entropy Power, Autoregressive Models, and Mutual Information
    entropy Article Entropy Power, Autoregressive Models, and Mutual Information Jerry Gibson Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106, USA; [email protected] Received: 7 August 2018; Accepted: 17 September 2018; Published: 30 September 2018 Abstract: Autoregressive processes play a major role in speech processing (linear prediction), seismic signal processing, biological signal processing, and many other applications. We consider the quantity defined by Shannon in 1948, the entropy rate power, and show that the log ratio of entropy powers equals the difference in the differential entropy of the two processes. Furthermore, we use the log ratio of entropy powers to analyze the change in mutual information as the model order is increased for autoregressive processes. We examine when we can substitute the minimum mean squared prediction error for the entropy power in the log ratio of entropy powers, thus greatly simplifying the calculations to obtain the differential entropy and the change in mutual information and therefore increasing the utility of the approach. Applications to speech processing and coding are given and potential applications to seismic signal processing, EEG classification, and ECG classification are described. Keywords: autoregressive models; entropy rate power; mutual information 1. Introduction In time series analysis, the autoregressive (AR) model, also called the linear prediction model, has received considerable attention, with a host of results on fitting AR models, AR model prediction performance, and decision-making using time series analysis based on AR models [1–3]. The linear prediction or AR model also plays a major role in many digital signal processing applications, such as linear prediction modeling of speech signals [4–6], geophysical exploration [7,8], electrocardiogram (ECG) classification [9], and electroencephalogram (EEG) classification [10,11].
    [Show full text]
  • Source Coding: Part I of Fundamentals of Source and Video Coding
    Foundations and Trends R in sample Vol. 1, No 1 (2011) 1–217 c 2011 Thomas Wiegand and Heiko Schwarz DOI: xxxxxx Source Coding: Part I of Fundamentals of Source and Video Coding Thomas Wiegand1 and Heiko Schwarz2 1 Berlin Institute of Technology and Fraunhofer Institute for Telecommunica- tions — Heinrich Hertz Institute, Germany, [email protected] 2 Fraunhofer Institute for Telecommunications — Heinrich Hertz Institute, Germany, [email protected] Abstract Digital media technologies have become an integral part of the way we create, communicate, and consume information. At the core of these technologies are source coding methods that are described in this text. Based on the fundamentals of information and rate distortion theory, the most relevant techniques used in source coding algorithms are de- scribed: entropy coding, quantization as well as predictive and trans- form coding. The emphasis is put onto algorithms that are also used in video coding, which will be described in the other text of this two-part monograph. To our families Contents 1 Introduction 1 1.1 The Communication Problem 3 1.2 Scope and Overview of the Text 4 1.3 The Source Coding Principle 5 2 Random Processes 7 2.1 Probability 8 2.2 Random Variables 9 2.2.1 Continuous Random Variables 10 2.2.2 Discrete Random Variables 11 2.2.3 Expectation 13 2.3 Random Processes 14 2.3.1 Markov Processes 16 2.3.2 Gaussian Processes 18 2.3.3 Gauss-Markov Processes 18 2.4 Summary of Random Processes 19 i ii Contents 3 Lossless Source Coding 20 3.1 Classification
    [Show full text]
  • Relative Entropy Rate Between a Markov Chain and Its Corresponding Hidden Markov Chain
    ⃝c Statistical Research and Training Center دورهی ٨، ﺷﻤﺎرهی ١، ﺑﻬﺎر و ﺗﺎﺑﺴﺘﺎن ١٣٩٠، ﺻﺺ ٩٧–١٠٩ J. Statist. Res. Iran 8 (2011): 97–109 Relative Entropy Rate between a Markov Chain and Its Corresponding Hidden Markov Chain GH. Yariy and Z. Nikooraveshz;∗ y Iran University of Science and Technology z Birjand University of Technology Abstract. In this paper we study the relative entropy rate between a homo- geneous Markov chain and a hidden Markov chain defined by observing the output of a discrete stochastic channel whose input is the finite state space homogeneous stationary Markov chain. For this purpose, we obtain the rela- tive entropy between two finite subsequences of above mentioned chains with the help of the definition of relative entropy between two random variables then we define the relative entropy rate between these stochastic processes and study the convergence of it. Keywords. Relative entropy rate; stochastic channel; Markov chain; hid- den Markov chain. MSC 2010: 60J10, 94A17. 1 Introduction Suppose fXngn2N is a homogeneous stationary Markov chain with finite state space S = f0; 1; 2; :::; N − 1g and fYngn2N is a hidden Markov chain (HMC) which is observed through a discrete stochastic channel where the input of channel is the Markov chain. The output state space of channel is characterized by channel’s statistical properties. From now on we study ∗ Corresponding author 98 Relative Entropy Rate between a Markov Chain and Its ::: the channels state spaces which have been equal to the state spaces of input chains. Let P = fpabg be the one-step transition probability matrix of the Markov chain such that pab = P rfXn = bjXn−1 = ag for a; b 2 S and Q = fqabg be the noisy matrix of channel where qab = P rfYn = bjXn = ag for a; b 2 S.
    [Show full text]
  • Entropy Rates & Markov Chains Information Theory Duke University, Fall 2020
    ECE 587 / STA 563: Lecture 4 { Entropy Rates & Markov Chains Information Theory Duke University, Fall 2020 Author: Galen Reeves Last Modified: September 1, 2020 Outline of lecture: 4.1 Entropy Rate........................................1 4.2 Markov Chains.......................................3 4.3 Card Shuffling and MCMC................................4 4.4 Entropy of the English Language [Optional].......................6 4.1 Entropy Rate • A stochastic process fXig is an indexed sequence of random variables. They need not be in- n dependent nor identically distributed. For each n the distribution on X = (X1;X2; ··· ;Xn) is characterized by the joint pmf n pXn (x ) = p(x1; x2; ··· ; xn) • A stochastic process provides a useful probabilistic model. Examples include: ◦ the state of a deck of cards after each shuffle ◦ location of a molecule each millisecond. ◦ temperature each day ◦ sequence of letters in a book ◦ the closing price of the stock market each day. • With dependent sequences, hows does the entropy H(Xn) grow with n? (1) Definition 1: Average entropy per symbol H(X ;X ; ··· ;X ) H(X ) = lim 1 2 n n!1 n (2) Definition 2: Rate of information innovation 0 H (X ) = lim H(XnjX1;X2; ··· ;Xn−1) n!1 • If fXig are iid ∼ p(x), then both limits exists and are equal to H(X). nH(X) H(X ) = lim = H(X) n!1 n 0 H (X ) = lim H(XnjX1;X2; ··· ;Xn−1) = H(X) n!1 But what if the symbols are not iid? 2 ECE 587 / STA 563: Lecture 4 • A stochastic process is stationary if the joint distribution of subsets is invariant to shifts in the time index, i.e.
    [Show full text]
  • Information Theory 1 Entropy 2 Mutual Information
    CS769 Spring 2010 Advanced Natural Language Processing Information Theory Lecturer: Xiaojin Zhu [email protected] In this lecture we will learn about entropy, mutual information, KL-divergence, etc., which are useful concepts for information processing systems. 1 Entropy Entropy of a discrete distribution p(x) over the event space X is X H(p) = − p(x) log p(x). (1) x∈X When the log has base 2, entropy has unit bits. Properties: H(p) ≥ 0, with equality only if p is deterministic (use the fact 0 log 0 = 0). Entropy is the average number of 0/1 questions needed to describe an outcome from p(x) (the Twenty Questions game). Entropy is a concave function of p. 1 1 1 1 7 For example, let X = {x1, x2, x3, x4} and p(x1) = 2 , p(x2) = 4 , p(x3) = 8 , p(x4) = 8 . H(p) = 4 bits. This definition naturally extends to joint distributions. Assuming (x, y) ∼ p(x, y), X X H(p) = − p(x, y) log p(x, y). (2) x∈X y∈Y We sometimes write H(X) instead of H(p) with the understanding that p is the underlying distribution. The conditional entropy H(Y |X) is the amount of information needed to determine Y , if the other party knows X. X X X H(Y |X) = p(x)H(Y |X = x) = − p(x, y) log p(y|x). (3) x∈X x∈X y∈Y From above, we can derive the chain rule for entropy: H(X1:n) = H(X1) + H(X2|X1) + ..
    [Show full text]
  • Arxiv:1711.03962V1 [Stat.ME] 10 Nov 2017 the Entropy Rate of an Individual’S Behavior
    Estimating the Entropy Rate of Finite Markov Chains with Application to Behavior Studies Brian Vegetabile1;∗ and Jenny Molet2 and Tallie Z. Baram2;3;4 and Hal Stern1 1Department of Statistics, University of California, Irvine, CA, U.S.A. 2Department of Anatomy and Neurobiology, University of California, Irvine, CA, U.S.A. 3Department of Pediatrics, University of California, Irvine, CA, U.S.A. 4Department of Neurology, University of California, Irvine, CA, U.S.A. *email: [email protected] Summary: Predictability of behavior has emerged an an important characteristic in many fields including biology, medicine, and marketing. Behavior can be recorded as a sequence of actions performed by an individual over a given time period. This sequence of actions can often be modeled as a stationary time-homogeneous Markov chain and the predictability of the individual's behavior can be quantified by the entropy rate of the process. This paper provides a comprehensive investigation of three estimators of the entropy rate of finite Markov processes and a bootstrap procedure for providing standard errors. The first two methods directly estimate the entropy rate through estimates of the transition matrix and stationary distribution of the process; the methods differ in the technique used to estimate the stationary distribution. The third method is related to the sliding-window Lempel-Ziv (SWLZ) compression algorithm. The first two methods achieve consistent estimates of the true entropy rate for reasonably short observed sequences, but are limited by requiring a priori specification of the order of the process. The method based on the SWLZ algorithm does not require specifying the order of the process and is optimal in the limit of an infinite sequence, but is biased for short sequences.
    [Show full text]
  • Markov Decision Process Example
    Markov Decision Process Example Clarifying Stanleigh vernalised consolingly while Matthus always converts his lev lynch belligerently, he crossbreeding so subcutaneously. andDemetris air-minded. missend Answerable his heisters and touch-types pilose Tuckie excessively flanged heror hortatively flagstaffs turfafter while Hannibal Orbadiah effulging scrap and some emendated augmenters ungraciously, symptomatically. elaborative The example when an occupancy grid example of a crucial role of dots stimulus consists of theories, if π is. Below illustrates the example based on the next section. Data structure to read? MDP Illustrated mdp schematic MDP Example S 11 12 13 21 23 31 32 33 41 42 43 A. Both models by comparison of teaching mdp example, a trial starts after responding, and examples published in this in this technique for finding approximate solutions. A simple GUI and algorithms to experiment with Markov Decision Process miromanninomarkov-decision-process-examples. Markov Decision Processes Princeton University Computer. Senior ai scientist passionate about getting it simply runs into the total catch games is a moving object and examples are. Markov decision processes c Vikram Krishnamurthy 2013 6 2 Application Examples 21 Finite state Markov Decision Processes MDP xk is a S state Markov. Real-life examples of Markov Decision Processes Cross. Due to solve in a system does not equalizing strategy? First condition this remarkable and continues to. Example achieving a state satisfying property P at minimal. Example 37 Recycling Robot MDP The recycling robot Example 33 can be turned into his simple example had an MDP by simplifying it and providing some more. How does Markov decision process work? In state values here are you can be modeled as if we first accumulator hits its solutions are.
    [Show full text]
  • Redundancy Rates of Slepian-Wolf Coding∗
    Redundancy Rates of Slepian-Wolf Coding∗ Dror Baron, Mohammad Ali Khojastepour, and Richard G. Baraniuk Dept. of Electrical and Computer Engineering, Rice University, Houston, TX 77005 Abstract The results of Shannon theory are asymptotic and do not reflect the finite codeword lengths used in practice. Recent progress in design of distributed source codes, along with the need for distributed compression in sensor networks, has made it crucial to understand how quickly distributed communication systems can approach their ultimate limits. Slepian and Wolf considered the distributed encoding of length-n sequences x and y. If y is available at the decoder, then as n increases x can be encoded losslessly at rates arbitrarily close to the conditional entropy H(X Y ). However, for any finite n there is a positive probability that x and y are | not jointly typical, and so x cannot be decoded correctly. We examine a Bernoulli setup, where x is generated by passing y through a binary symmetric correlation channel. We prove that the finite n requires us to increase the rate above the conditional entropy by K()/√n, where is the probability of error. We also study the cost of universality in Slepian-Wolf coding, and propose a universal variable 1 rate scheme wherein the encoder for x receives PY = n i yi. For PY < 0.5, our redundancy rate is K0()/√n above the empirical conditional entropy. When 1/6 1/6 P PY 0.5 = O(n− ), K0()=Ω(n ), and another scheme with redundancy rate | −1/3 | O(n− ) should be used. Our results indicate that the penalties for finite n and unknown statistics can be large, especially for P 0.5.
    [Show full text]