TY - GEN

T1 - On complexity and randomness of Markov-chain prediction

AU - Ratsaby, J.

N1 - Publisher Copyright:
© 2015 IEEE.

PY - 2015/6/24

Y1 - 2015/6/24

N2 - Let {Xt : t ∈ Z} be a sequence of binary random variables generated by a stationary Markov source of order k∗. Let β be the probability of the event "Xt = 1". Consider a learner based on a Markov model of order k, where k may be different from k∗, who trains on a sample sequence X(m) which is randomly drawn by the source. Test the learner's performance by asking it to predict the bits of a test sequence X(n) (generated by the source). An error occurs at time t if the prediction Yt differs from the true bit value Xt, 1 ≤ t ≤ n. Denote by Ξ(n) the sequence of errors where the error bit Ξt at time t equals 1 or 0 according to whether an error occurred or not, respectively. Consider the subsequence Ξ(ν) of Ξ(n) which corresponds to the errors made when predicting a 0, that is, Ξ(ν) consists of those bits Ξt at times t where Yt = 0: In this paper we compute an upper bound on the absolute deviation between the frequency of 1 in Ξ(ν) and β. The bound has an explicit dependence on k, k∗, m, ν, n. It shows that the larger k, or the larger the difference k - k∗, the less random that Ξ(ν) can become.

AB - Let {Xt : t ∈ Z} be a sequence of binary random variables generated by a stationary Markov source of order k∗. Let β be the probability of the event "Xt = 1". Consider a learner based on a Markov model of order k, where k may be different from k∗, who trains on a sample sequence X(m) which is randomly drawn by the source. Test the learner's performance by asking it to predict the bits of a test sequence X(n) (generated by the source). An error occurs at time t if the prediction Yt differs from the true bit value Xt, 1 ≤ t ≤ n. Denote by Ξ(n) the sequence of errors where the error bit Ξt at time t equals 1 or 0 according to whether an error occurred or not, respectively. Consider the subsequence Ξ(ν) of Ξ(n) which corresponds to the errors made when predicting a 0, that is, Ξ(ν) consists of those bits Ξt at times t where Yt = 0: In this paper we compute an upper bound on the absolute deviation between the frequency of 1 in Ξ(ν) and β. The bound has an explicit dependence on k, k∗, m, ν, n. It shows that the larger k, or the larger the difference k - k∗, the less random that Ξ(ν) can become.

UR - http://www.scopus.com/inward/record.url?scp=84938930694&partnerID=8YFLogxK

U2 - 10.1109/ITW.2015.7133078

DO - 10.1109/ITW.2015.7133078

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AN - SCOPUS:84938930694

T3 - 2015 IEEE Information Theory Workshop, ITW 2015

BT - 2015 IEEE Information Theory Workshop, ITW 2015

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2015 IEEE Information Theory Workshop, ITW 2015

Y2 - 26 April 2015 through 1 May 2015

ER -