entropy rate

A process $X$ with a countable index gives rise to the sequence of its joint entropies $H\_n(X\_1,\; X\_2,\; \backslash dots\; X\_n)$. If the limit exists, the entropy rate is defined as

:$H(X)\; :=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash tfrac\{1\}\{n\}\; H\_n.$

Note that given any sequence $(a\_n)\_n$ with $a\_0=0$ and letting $\backslash Delta\; a\_k\; :=\; a\_k\; -\; a\_\{k-1\}$, by telescoping one has $a\_n=\{\backslash textstyle\; \backslash sum\_\{k=1\}^n\}\backslash Delta\; a\_k$. The entropy rate thus computes the mean of the first $n$ such entropy changes, with $n$ going to infinity.

The behaviour of joint entropies from one index to the next is also explicitly subject in some characterizations of entropy.

While $X$ may be understood as a sequence of random variables, the entropy rate $H(X)$ represents the average entropy change per one random variable, in the long term.

It can be thought of as a general property of stochastic sources - this is the subject of the asymptotic equipartition property.

A stochastic process also gives rise to a sequence of conditional entropies, comprising more and more random variables.

For strongly stationary stochastic processes, the entropy rate equals the limit of that sequence

:$H(X)\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; H(X\_n|X\_\{n-1\},\; X\_\{n-2\},\; \backslash dots\; X\_1)$

The quantity given by the limit on the right is also denoted $H\text{'}(X)$, which is motivated to the extent that here this is then again a rate associated with the process, in the above sense.

Since a stochastic process defined by a Markov chain that is irreducible and aperiodic has a stationary distribution, the entropy rate is independent of the initial distribution{{Cite book |last1=Cover |first1=Thomas M. |title=Elements of information theory |last2=Thomas |first2=Joy A. |date=2006 |publisher=Wiley-Interscience |isbn=978-0-471-24195-9 |edition=2nd |location=Hoboken, N.J |page=78}}.

For example, consider a Markov chain defined on a countable number of states. Given its right stochastic transition matrix $P\_\{ij\}$ and an entropy

:$h\_i\; :=\; -\backslash sum\_\{j\}\; P\_\{ij\}\; \backslash log\; P\_\{ij\}$

associated with each state, one finds

:$\backslash displaystyle\; H(X)\; =\; \backslash sum\_\{i\}\; \backslash mu\_i\; h\_i,$

where $\backslash mu\_i$ is the asymptotic distribution of the chain.

In particular, it follows that the entropy rate of an i.i.d. stochastic process is the same as the entropy of any individual member in the process.

The entropy rate of hidden Markov models (HMM) has no known closed-form solution. However, it has known upper and lower bounds. Let the underlying Markov chain $X\_\{1:\backslash infty\}$ be stationary, and let $Y\_\{1:\backslash infty\}$ be the observable states, then we have$$H(Y\_n|X\_1\; ,\; Y\_\{1:n-1\}\; )\; \backslash leq\; H(Y)\; \backslash leq\; H(Y\_n|Y\_\{1:n-1\}\; )$$and at the limit of $n\; \backslash to\; \backslash infty$, both sides converge to the middle.{{Cite book |last1=Cover |first1=Thomas M. |title=Elements of information theory |last2=Thomas |first2=Joy A. |date=2006 |publisher=Wiley-Interscience |isbn=978-0-471-24195-9 |edition=2nd |location=Hoboken, N.J |chapter=4.5. Functions of Markov chains}}

The entropy rate may be used to estimate the complexity of stochastic processes. It is used in diverse applications ranging from characterizing the complexity of languages, blind source separation, through to optimizing quantizers and data compression algorithms. For example, a maximum entropy rate criterion may be used for feature selection in machine learning.{{cite journal |last1=Einicke |first1=G. A. |title=Maximum-Entropy Rate Selection of Features for Classifying Changes in Knee and Ankle Dynamics During Running |journal=IEEE Journal of Biomedical and Health Informatics |volume=28 |issue=4 |pages=1097–1103 |year=2018 |doi= 10.1109/JBHI.2017.2711487 |pmid=29969403 |s2cid=49555941 |hdl=10810/68978 |hdl-access=free }}

• Information source (mathematics)
• Markov information source
• Asymptotic equipartition property
• Maximal entropy random walk - chosen to maximize entropy rate

{{Reflist}}

• [http://staff.ustc.edu.cn/~cgong821/Wiley.Interscience.Elements.of.Information.Theory.Jul.2006.eBook-DDU.pdf Cover, T. and Thomas, J. Elements of Information Theory. John Wiley and Sons, Inc. Second Edition, 2006.]

Category:Information theory

Category:Entropy

Category:Markov models

Category:Temporal rates