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# Entropy of a discrete random variable

### Entropy (information theory) - Wikipedi

• In information theory, the entropy of a random variable is the average level of information, surprise, or uncertainty inherent in the variable's possible outcomes. The concept of information entropy was introduced by Claude Shannon in his 1948 paper A Mathematical Theory of Communication
• 2 Entropy Deﬁnition The entropy of a discrete random variable X with pmf pX(x) is H(X) = − X x p(x)logp(x) = −E[ log(p(x)) ] (1) The entropy measures the expected uncertainty in X. We also say that H(X) is approximately equal to how much information we learn on average from one instance of the random variable X
• 2 Entropy Deﬁnition The entropy of a discrete random variable X with pmf p X(x) is H(X) = − X x p(x)logp(x) = −E[ log(p(x)) ] (1) The entropy measures the expected uncertainty in X. We also say that H(X) is approximately equal to how much information we learn on average from one instance of the random variable X
• You probably heard that the entropy of a discrete random variable with distinct outcomes is defined as where is the probability of the event. The RHS is sometimes denoted as. You might be wondering, how can we make sense of that entropy formula intuitively
• Abstract It is well known that the entropy H(X) of a discrete random variable Xis always greater than or equal to the entropy H(f(X)) of a function fof X, with equality if and only if fis one-to-one. In this paper, we give tight bounds on H(f(X)) when the function fis not one-to-one, and we illustrate a few scenarios where this matters
• 2.3 RELATIVE ENTROPY AND MUTUAL INFORMATION The entropy of a random variable is a measure of the uncertainty of the random variable; it is a measure of the amount of information required on the average to describe the random variable. In this section we introduce two related concepts: relative entropy and mutual information

@NetranjitBorgohain that's a different method, but again it expects a different set of parameters entropy_joint(X, base=2, fill_value=-1, estimator='ML', Alphabet_X=None, keep_dims=False) see documentation for details - nickthefreak Mar 28 '19 at 15:2 (d) Follows because the (conditional) entropy of a discrete random variable is nonnegative, i.e., H(X|g(X)) ≥ 0, with equality iﬀ g(X) is a one-to-one mapping. 2. A measure of correlation. Let X1 and X2 be identically distributed, but not necessarily independent. Let ρ = 1− H(X1|X2) H(X1). (a) Show ρ = I(X1;X2) H(X1). (b) Show 0 ≤ ρ ≤ 1 Further information: Entropy (information theory) If X is a discrete random variable with distribution given by. Pr ⁡ ( X = x k ) = p k for k = 1 , 2 , . {\displaystyle \operatorname {Pr} (X=x_ {k})=p_ {k}\quad {\mbox { for }}k=1,2,\ldots } then the entropy of X is defined as. H ( X ) = − ∑ k ≥ 1 p k log ⁡ p k 9.2 THE AEP FOR CONTINUOUS RANDOM VARIABLES One of the important roles of the entropy for discrete random variables is in the AEP, which states that for a sequence of i.i.d. random variables, PK,X2, * l l , X, ) is close to 2-nHU) with high probability. This enable

This was the main topic of the 1959 paper from Alfred Rényi intitled On the dimension and entropy of probability distributions: I am questioning the assumptions under which the discrete entropy is well-defined Entropy power inequality for a family of discrete random variables. In 2011 IEEE International Symposium on Information Theory Proceedings (ISIT) , pages 1945-1949. IEEE, 2011

In statistical mechanics, the entropy is proportional to the logarithm of the number of resolvable microstates associated with a macrostate. In classical mechanics, this quantity contains an arbitrary additive constant associated with the size of a microstate that is considered resolvable Minimum entropy of a discrete random variable - find the appropriate distributions. 1 Entropy of random variable which denotes the number of heads landed after 3 coin flip the entropy of a discrete random variable is the so plug-in estimator [1]. The Miller-Madow estimator [30] adds a bias correction to the plug-in estimator. This correction depends on the ratio between the number of symbols from the alphabet that appear at least once in the sample and the sample size

### Shannon's entropy of random variables and partitions Yoo Bo

1. Entropy provides a measure of the average amount of information needed to represent an event drawn from a probability distribution for a random variable. Kick-start your project with my new book Probability for Machine Learning, including step-by-step tutorials and the Python source code files for all examples
2. Definition; The entropy H(X) of a discrete random variable X is defined bY H(X) = - c p(d log pm - (2.1) We also write H(p) for the above quantity. The log is to the base 2 and entropy is expressed in bits. For example, the entropy of a fair coin toss is 1 bit
3. e what the
4. Entropy as well as entropy preservation is well-deﬁned only in the context of purely discrete or continuous random variables. However for a mixture of discrete and continuous random variables, which arise in many interesting situations, the notions of entropy and entropy preservation have not been well understood
5. Entropy: Let U a discrete random variable taking values in alphabet U. The entropy of U is given by: H(U) , E[S(U)] = E log 1 p(U) = E log(p(U)) = X u p(u)logp(u) (2) Where U represents all u values possible to the variable. The entropy is a property of the underlying distribution P U(u);u 2Uthat measures the amount of randomness or surprise in.
6. An Intuitive Explanation of the Information Entropy of a Random Variable Or: How to Play Twenty Questions. Daniel Shawcross Wilkerson 8 and 15 October 2006, 26 January 2009 Abstract. Information Entropy is not often explained well, yet it is quite straightforward to understand and that understanding can be quite helpful in everyday life

### python 3.x - Entropy estimation from discrete variable ..

1. Entropy Let X be a discrete random variable with alphabet Xand probability mass function p(x) p(x) = PrfX = xg; x 2X The entropy of the variable X is de ned by H(X) = X x2X p(x)log p(x) The logarithm can be in any base, but normally base 2 is used. The unit of the entropy is then called bits. If we use base b in the logarithm, we denote the.
2. The entropy of a random variable gives the information theory of the random variable (either discrete or continuous). It is also known as information..
3. Let X be a discrete random variable with support S and f: S → S ′ be a bijection. Then it is well-known that the entropy of X is the same as the entropy of f (X).This entropy preservation property has been well-utilized to establish non-trivial properties of discrete stochastic processes, e.g. queuing process cite{prg03}

Deﬁnition: The entropy H(X) of a discrete random variable X with pmf p X (x) is given by H(X)= x p X (x)log p X (x)=E p X (x)[log p X (X)] University of Illinois at Chicago ECE 534, Natasha Devroye Order these in terms of entropy. University of Illinois at Chicago ECE 534, Natasha Devroy Entropy 2015, 17 7121 Following Thomas and Cover, it worth noting that we can choose sequences of distributions on XX(0), (1),... such that the limit () 1 1 () k A X j H SHXj k = = does not exist [16]. The greatest value of HA()S X is equal to l bits per time-step. This value is obtained when {()}X k is a sequence of independent and uniformly distributed random variables 1 Entropy Lecture 2 Scribe. Definition: Entropy is a measure of uncertainty of a random variable. The entropy of a discrete random variable X with alphabet X is :cex When the base of the logarithm is 2, entropy is measured in bits Example: One can model the temperature in a city (e.g. Amherst) as a random variable, X. The Entropy of Discrete Random Variables Suppose that we are performing a random experiment with outcomes indicated by a discrete random variable X.The entropy of X measures the amount of surprise that we have at hearing the outcome: we're more surprised by unlikely events Determine the entropy of a continuos random variable with distributions The differential entropy of a discrete random variable can be consid- ered to be ---co. Note that 2- = 0, agreeing with the idea that the volume of the support set of a discrete random variable is zero. 9.4 JOINT AND CONDITIONAL DIFFERENTIAL ENTROPY As in the discrete case, we can extend the

SIAM J. on Discrete Mathematics. Browse SIDMA; SIAM J. on Financial Mathematics. Browse SIFIN; SIAM J. on Imaging Sciences. Browse SIIMS; SIAM J. on Mathematical Analysis. Browse SIMA; SIAM J. on Matrix Analysis and Applications. Browse SIMAX; SIAM J. on Numerical Analysis. Browse SINUM; SIAM J. on Optimization. Browse SIOPT; SIAM J. on. Solution for Let X be a discrete random variable taking values {x1, x2, . . . , xn} with probability {p1, p2, . . . , pn}. The entropy of the random variable i The entropy of the random variable is defined as H(X) = − p i log (p i). Find the probability mass function for the above discrete random variable that maximizes the entropy

### Maximum entropy probability distribution - Wikipedi

Conditional Entropy LetY be a discrete random variable with outcomes, {y1,...,ym}, which occur with probabilities, pY(yj).The avg. infor-mation you gain when told the outcome of Y is Unlike entropy that is only well-deﬁned for discrete random variables, in general we can deﬁne the mutual information between two real-valued random variables (no necessarily continuous or discrete) as follows. Deﬁnition 1 (Mutual information) The mutual information between two random variables X and Y is deﬁned as I(X;Y) = su 2 Entropy For information theory, the fundamental value we are interested in for a random variable X is the entropy of X. We'll consider X to be a discrete random variable. The entropy can be thought of as any of the following intuitive de nitions: 1. The amount of randomness in X (in bits) 2 In this paper, we propose a novel method for increasing the entropy of a sequence of independent, discrete random variables with arbitrary distributions. The method uses an auxiliary table and a novel theorem that concerns the entropy of a sequence in which the elements are a bitwise exclusive-or sum of independent discrete random variables

• The information source can be modeled as a discrete-time random process {X k,k 1}. • {X k,k 1} is an inﬁnite collection of random variables indexed by the set of positive integers. The index k is referred to as the time index. • Random variables X k are called letters. • Assume that H(X k) < 1 for all k 1. Entropy Consider a probability space (;F;P) and a discrete valued random variable X on . Because Xis discrete we can de ne a new variable Y = p(X) taking values in the open interval 0 <y 1, by Y = yfor all x2R such that X = xand P(X= x) = y. The information variable I(X) and the entropy H(X) are then given by the formula A normally distributed random variable is not discrete. Cover and Thomas (1991) define differential entropy h(X) for a continuous random variable X, analogous to H(Y) for a discrete Y. They warn in the beginning of chapter 9 that there are some important differences. I think this is one of the important differences. And

Entropy (joint entropy included), is a property of the distributionthat a random variable follows. The available sample (and hence the timing of observation) plays no role in it. Copying for Cover & Thomas, the joint entropy $H(X,Y)$ of two discrete random variables $X, Y,$ with joint distribution $p(x,y)$, is defined a Discrete Random Variables. Discrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers). Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability Recommend & Share. Recommend to Library. Email to a frien

i. sason: entropy bounds for discrete random v ariables via coupling 5 T o verify that it is included in the feasible set of (22), note that due to the constraints of this optimization proble a discrete random variable deﬁned over a ﬁnite or countably inﬁnite alphabet. This work reﬁnes bounds on the entropy difference of two discrete random variables via the use of maximal couplings, leading to sharpened bounds that depend on both the local and total variation distances. The reader is also referred to a recen We can also see that our joint entropy is minimized when both variables are deterministic. Finally, if we rotate our plot to face the p 1 p_1 p 1 or p 2 p_2 p 2 axis head on, we should see a familiar shape in the outline of our curv

This quantity is the called the entropy of X. De nition 2. Let X be a discrete random variable. The (Shannon) entropy of X, denoted H(X), equals the nonnegative number X x2range(X) p X(x)log 2(1=p X(x)): The associated unit of measurement is bits.1 Convention: In entropy calculations, we always de ne 0log 2(1=0) = 0. That way it is even oka $$w = f(x,y,z)$$ where w is also a discrete random variable (to which I also know the PMF). Let us imagine that $f(\cdot)$ is something simple like $$f(x,y,z) = x+y -z$$ Now, imagine we are interested in the Shannon entropy of this relation, and we apply it thus: $$H(w) = H(f(x,y,z)).$$ Is the following true One of the important roles of the entropy for discrete random variables is in the AEP, which states that for a sequence of i.i.d. random variables, nH(X) X ) is close to 2 with high probability. This enables X2,... us to define the typical set and characterize the behavior of typical sequences. We can do the same for a continuous random variable Entropy Bounds for Discrete Random Variables via Maximal Coupling Igal Sason, Senior Member, IEEE Abstract This paper derives new bounds on the difference of the entropies of two discrete random variables in terms of the local and total variation distances between their probability mass functions I am trying to explore a number of things regarding the entropy of random strings and am wondering how a character set of random size would affect the entropy of strings made from that set. Using the following formula, I need to take the log of a discrete random variable H = L\\log_2 N..

BasicsMatchingsHomomorphismsIndependent Sets DeﬁnitionProperties Entropy Deﬁnition The entropy of a discrete random variable X is H(X) = X x p(x)lo Entropy Definition 1 The entropy H X of a discrete random variable X with from TELE 9754 at University of New South Wale The entropy of a random variable is a function which attempts to characterize the \unpredictability of a random variable. Consider a random variable X rep- resenting the number that comes up on a roulette wheel and a random variable Y representing the number that comes up on a fair 6-sided die. The entropy of X is greater than the entropy of Y

### Discrete entropy of the integer part of a random variable

A discrete random variable X is completely deﬁned1by the set of values it can take, X, which we assume to be a ﬁnite set, and its probability distribution {pX(x)}x∈X. The value pX(x) is the probability that the random variable Xtakes the value x. The probability distribution pX: X → [0,1] must satisfy the normalization condition The formula for continuous entropy is a (seemingly) logical extension of the discrete case. In fact, we merely replace the summation with an integral. De nition (Continuous entropy). The continuous entropy h(X) of a continu-ous random variable Xwith density f(x) is de ned as: h(X) = R S f(x)log 1 f(x) dx where Sis the support set of the random.

### Entropy and Thinning of Discrete Random Variables

1. It is well known that the entropy H(X) of a discrete random variable X is always greater than or equal to the entropy H(f(X)) of a function f of X, with equality if and only if f is one-to-one. In this paper, we give tight bounds on H(f(X)), when the function f is not one-to-one, and we illustrate a few scenarios, where this matters
2. In entropy: Estimation of Entropy, Mutual Information and Related Quantities. Description Usage Arguments Details Value Author(s) See Also Examples. Description. discretize puts observations from a continuous random variable into bins and returns the corresponding vector of counts.. discretize2d puts observations from a pair of continuous random variables into bins and returns the.
3. The two quantities entropy rate and the previous one correspond to two different notions of entropy rate. The first is the per symbol entropy rate of the n random variables, and the second is the conditional entropy rate of the last random variable given the past. We now prove that for stationary processes both limits exist and are equal

### information theory - Estimating the Shanon entropy of a

The discrete random variable x is binomial distributed if, for example, it describes the probability of getting k heads in N tosses of a coin, 0 ≤ k ≤ N.Let p be the probability of getting a head and q = 1 - p be the probability of getting a tail. Then x takes discrete values according to the densit It is easy to calculate the entropy on discrete numbers or categorical data, which si equal to minus the summation of (countable) but infinite random variable Maximum di erential entropy: For any random variable X with pdf f(x) such that E[X2] = Z x2f(x)dx= P it holds that h(X) 1 2 log2ˇeP with equality i f(x) = N(0;P). Mikael Skoglund, Information Theory 5/26 Typical Sets for Continuous Variables A discrete-time continuous-amplitude i.i.d. process fX mg, with marginal pdf f(x) of support X. It.

### Neural Joint Entropy Estimatio

1. Entropy of a function of a random variable G. Kesidis CS&E and EE Depts The Pennsylvania State University University Park, PA, 16802 kesidis@engr.psu.edu Aug. 20, 2007 I. INTRODUCTION Regarding p. 24 of Chapter 1 of the text [1]: If X = g(Y ) for some (Borel measurable) function g (i.e., the σ-ﬁeld generated by X, σ(X) ⊂ σ(Y )) and X and.
2. Entropy is defined here as a functional mapping a distribution on χ to real numbers. In practice, we often deal with a pair of random variables (X, Y) defined on a joint space χ × γ. Then P is the joint distribution of (X, Y), and P X is its corresponding marginal distribution on X, P X (x) = Σ y P(x,y)
3. In this section we obtain some characterization results for the Shannon entropy of discrete doubly truncated random variable. Let X be a discrete random variable, which takes the values x 1, x 2, , x l with probabilities p 1, p 2, , p l, respectively, where l ∈ N. Here, N represents the set of natural numbers

Past couple of days the question troubling me was that if the the entropy of a random variable (r.v) is what is the entropy of the r.v . Some point of time it looked it shouldn't change, and some times it seemed as if it should. I have been able to find the answer, although not completely, but still to a certain extent, thanks to google An information-theoretical measure of the degree of indeterminacy of a random variable. If $\xi$ is a discrete random variable defined on a probability space $( \Omega , \mathfrak A , {\mathsf P} )$ and assuming values $x _ {1} , x _ {2} \dots$ with probability distribution $\{ {p _ {k} } : {1 , 2 ,\dots } \}$, $p _ {k} = {\mathsf P} \{ \xi = x _ {k} \}$, then the entropy is defined. 3.Discrete random variables 3. 1 Discrete random variables. For the current purposes it is sufficient to think of a discrete random variable as variable with a fixed but unknown integer value larger than or equal to zero. The random variables we will encounter later will be mostly secret keys, messages, and message tags One might wonder why entropy is de ned as H(P)=∑p ilog 1 and if there are other de nitions. pi Indeed, the information-theoretic de nition of entropy is related to entropy in statistical physics. Also, it arises as answers to speci c operational problems, e.g., the minimum average number of bits to describe a random variable as discussed above An entropy estimator that achieves least square error is obtained through Bayesian estimation of the occurrence probabilities of each value taken by the discrete random variable. This Bayesian entropy estimator requires large amount of calculation cost if the random variable takes numerous sorts of values

### A Gentle Introduction to Information Entropy

Relative Entropy of joint distributions as Mutual Information Mutual Information, which is a measure of the amount of information that one random variable contains about another random variable. It is the reduction in the uncertainty of one random variable due to the knowledge of the other. Unlike Relative Entropy, Mutual Information is symmetric variable, a pair of random variables with one discrete and the other continuous. Our extensions are consistent in that there exist natural injections from discrete or continuous random variables into mixed-pair random variables such that their entropy remains the same. This extension of entropy allows us to obtain suﬃcien Definition. Let X be a random variable with a probability density function f whose support is a set .The differential entropy h(X) or h(f) is defined as . As with its discrete analog, the units of differential entropy depend on the base of the logarithm, which is usually 2 (i.e., the units are bits). See logarithmic units for logarithms taken in different bases

### statistics - Minimum entropy of a discrete random variable

1. Discrete random variables Expected value •The expected value, mean or average of a random variable xis: E[x] = = X x2X xP(x) = Xm i=1 v iP(v i) •The expectation operator is linear: E[ x+ 0y] = E[x] + 0E[y] Variance •The variance of a random variable is the moment of inertia of its probability mass function: Var[x] = ˙2 = E[(x )2] = X x2X.
2. The definition of Entropy given in 2.1 is with respect to a single random variable. We now extend the definition to a pair of random variables (X, Y). The joint entropy H(X, Y) of a pair of discrete random variables (X, Y) with a joint distribution p(x, y) is defined a
3. 1. Maximum entropy2. Poincar e inequality3. Entropy monotonicity4. Entropy concavity Entropy and thinning of discrete random variables Joint work with Daly, Harremo es, Hillion, Kontoyiannis, Y
4. Deﬁnition 1.1: The entropy of a discrete random variable X is deﬁned as: H(X) = X x∈X p(x)log 1 p(x) which can interpreted as the expected value H(X) = E p[log 1 p(x)]. 2 Source Coding Deﬁnition 2.1: A (binary) source code C for a random variable X is a mapping from X to a (ﬁnite) binary string. Let C(x) be the codeword corresponding to x and let l(x
5. entropy membership function of continuous and discrete fuzzy variables. On the basis of their work£we promote their ideas to solve the problem for maximum Entropy functions of discrete fuzzy random variables. The organization of our work is as follows: In section 2, some basic concepts and results on fuzzy random variables are reviewed

information-theoretic notion of the entropy of a discrete random variable, derive its basic properties, and show how it can be used as a tool for estimating the size of combinatorially de ned sets. The entropy of a random variable is essentially a measure of its degree of randomness, and was introduced by Claude Shannon in 1948 Deﬁne the ǫ-entropy of a discrete random variable Y by Hǫ(Y) = inf H(F) | Pr(F 6= Y) 6ǫ, where the inﬁmum is taken over σ(Y )-measurable (hence discrete) random variables F. When Y takes its values in N, one can restrict the random variables F to those taking their values in N∪{∞}. Indeed, if F takes more than on

### An Intuitive Explanation of the Information Entropy of a

Entropy of Discrete Fuzzy Measures Jean-Luc Marichal⁄ Marc Roubens Institute of Mathematics, University of Liµege Grande Traverse 12 - B37, Sart Tilman, B-4000 Liµege, Belgium. fjl.marichal,m.roubensg[at]ulg.ac.be Revised version, September 4, 2000 Abstract The concept of entropy of a discrete fuzzy measure has been recently introduce Entropy is a measure of uncertainty regarding a discrete random variable. I am being lazy here, because the embedding layer maps all 128 ASCII, which might be system control symbols. Perhaps the easiest way to circumvent this problem is to wrap the dataset with numpy Its entropy is 1 bit. Example 1.4 Imagine throwing M fair coins: the number of all possible out-comes is 2 M. The entropy equals M bits. Example 1.5 A fair dice with M faces has entropy log 2 M. Example 1.6 Bernouilli process. A random variable X can take values 0, 1 with probabilities p (0) = q, p (1) = 1 − q scipy.stats.entropy calculates the differential entropy for a continuous random variable. By which estimation method, and which formula, exactly is it calculating differential entropy? (i.e. th

We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information theoretic approaches to Poisson approximation, the maximum entropy property of the Poisson distribution, discrete concentration (Poincaré and logarithmic Sobolev. Shannon entropy, which quantifies the expected value of the information % contained in a message, usually in units such as bits. In this context, a 'message' means a specific realization of the random variable. Shannon denoted the entropy H of a discrete random variable X with possible values {x1 xn} as, H(X) = E(I(X)) 8.3 RELATION OF DIFFERENTIAL ENTROPY TO DISCRETE ENTROPY Consider a random variable X with density f(x)illustrated in Figure 8.1. Suppose that we divide the range of X into bins of length !. Let us assume that the density is continuous within the bins. Then, by the mean value theorem, there exists a value x i within each bin such that f( Peng-Hua Wang, May 14, 2012 Information Theory, Chap. 8 - p. 2/24 Chapter Outline Chap. 8 Differential Entropy 8.1 Deﬁnitions 8.2 AEP for Continuous Random Variables 8.3 Relation of Differential Entropy to Discrete Entropy 8.4 Joint and Conditional Differential Entropy 8.5 Relative Entropy and Mutual Information 8.6 Properties of Differential Entropy and Related Amount It is well known that the entropy H(X) of a discrete random variable X is always greater than or equal to the entropy H(f(X)) of a function f of X, with equality if and only if f is one-to-one. In this paper, we give tight bounds on H(f(X)) when the function f is not one-to-one, and we illustrate a few scenarios where this matters

i is a discrete random variable. • Entropy (Shannon Entropy) H(X)=−! x p(x)log p(x) • Joint Entropy H(X,Y)=−! x,y p(x,y)log p(x,y) • In information theory, entropy is the measure of the uncertainty contained in a discrete random variable, justiﬁed by fundamental coding theorems The entropy H(X) of a discrete random variable is de ned by H(X) = X x2X p(x)log b p(x) where b is the base of the logarithm used to specify units. Common values of b are 2, Euler's number e, and 10, and the unit of entropy is bit for b = 2, nat for b = e, and digit for b = 10

Let Nd[X]=12πee2H[X] denote the entropy power of the discrete random variable X where H[X] denotes the discrete entropy of X. In this paper, we show that for two independent discrete random variables X and Y, the entropy power inequality holds and it can be tight. The basic idea behind the proof is to perturb the discrete random variables using suitably designed continuous random variables. and relations between, random variables using real numbers. Often these numbers can serve as objective functions or con-straints for algorithms and learning agents. The basic measures are (1) entropy, (2) mutual information, and (3) relative entropy or KL divergence. There are a few forms of each and there are impor 08/03/20 - Let X_1, , X_n be independent random variables taking values in the alphabet {0, 1, , r}, and S_n = ∑_i = 1^n X_i. The She..

Measuring information flows using transfer entropy. Let $$log$$ denote the logarithm to the base 2, then informational gain is measured in bits. Shannon entropy (Shannon 1948) states that for a discrete random variable $$J$$ with probability distribution $$p(j)$$, where $$j$$ stands for the different outcomes the random variable $$J$$ can take, the average number of bits required to optimally. discrete random variables be, subject to given constraints on the individual dis-tributions (say, no value may be taken by a variable with probability greater than p, for p < 1)? This question has been posed and partially answered in a re-cent work of Babai [Entropy versus pairwise independence (preliminary version)

Inference of Entropies of Discrete Random Variables with Unknown Cardinalities Ilya Nemenman Kavli Institute for Theoretical Physics University of California Santa Barbara, CA 93106 nemenman@kitp.ucsb.edu Abstract We examine the recently introduced NSB estimator of entropies of severely undersampled discrete variables and devise a procedure for. random variable X can be estimated using histogram. That is, we can use the normalized frequency of each histogram bin as the probability p(x). In a simple example, given a 3D volume dataset, we can model the entire dataset as a discrete random variable X where each voxel carries a scalar value. The entropy H(X) indicates how much information.

where is the probability ascribed to the value of that turns out to be correct. This makes a discrete random variable, but not necessarily with integer values.. Now it is clear that the Shannon entropy is the expectation value of where is the probability assigned to the measured value of the random variable. can be interpreted as the needed length, in bits, of a message communicating a. For a discrete random variable X (a random variable whose range X is countable) with probability mass function p(x) = P(X = x), we can define the (Shannon or discrete) entropy of the random variable as H[X] = − E[log2p(X)] = − ∑ x ∈ Xp(x)log2p(x). That is, the entropy of the random variable is the expected value of one over the probability mass. If X and Y are discrete random variables and f(x, y) is the value of their joint probability distribution of (x, y), then the joint entropy of X and Y is. H(X, Y)=-Σ x є X Σ y є Y f(x, y) log f(x, y) The joint entropy represents the amount of information needed on average to specify the value of two discrete random variables. Conditional.

### What is entropy of a random variable? How to calculate it

Solved Expert Answer to The entropy H ( X ) of a discrete random variable X measures the uncertainty about predicting the value of X (Cover & Thomas Get Best Price Guarantee + 30% Extra Discount support@crazyforstudy.co On the Discrete Cumulative Residual Entropy 209 We show below that D-CRE dominates the discrete Shannon entropy. Theorem 3.1 . Let X be a discrete random variable with probability mass function and survival functionp(i) and R(i); i = 0;1:::;b; b 1 ; respectively. Then, d (X ) C exp(H(X ) 1 p(0)); whereH(X ) is discrete Shannon entropy ofX and C = exp h If is a discrete random variable defined on a probability space and assuming values with probability distribution , , then the entropy is defined by the formula (1) (here it is assumed that ). The base of the logarithm can be any positive number,.

The joint Shannon entropy (in bits) of two discrete random variables and with images and is defined as: where and are particular values of and , respectively, is the joint probability of these values occurring together, and is defined to be 0 if 1. Joint Entropy ↩. URL copied to clipboard Infinite entropy . This problem shows that the entropy of a discrete random variable can be infinite. Let [It is easy to show that A is finite by bounding the infinite sum by the integral of (x log 2 x)−1.] Show that the integer-valued random.. considered entropy estimation in the case of unknown alphabet size (Wolpert and DeDeo, 2013). In that paper, the authors infer entropy under a ( nite) Dirichlet prior, but treat the alphabet size itself as a random variable that can be either inferred from the data or integrated out. Other recent work considers countably in nite alphabets

### On entropy for mixtures of discrete and continuous variables

arXiv:1112.0061v1 [cs.IT] 1 Dec 2011 1 On the Entropy Region of Gaussian Random Variables* Sormeh Shadbakht and Babak Hassibi Department of Electrical Engineering California Inst The entropy of discrete random variables and the differ-ential entropy of continuous random variables are basic and important quantities in information theory [1]. Applications include the calculation of channel capacity [1, Ch. 8] and of rate-distortion functions [1, Ch. 10]. Unfortunately, a tractabl This quantity is called the entropy of the random process, and this concept is currently of great interest in communication theory studies. The concept of entropy is discussed in several books (see e.g. [2, 3]). Usually its formulation is ﬁrst given for discrete random variables; that is, if X is a discrete random variable with values {x 1,

Figure 1: High and low entropy distributions for Q-values in RL; a_i represent actions [homemade.] The form a l definition of entropy in RL has been taken from Information Theory, where entropy is calculated as shown in equation (1), for a discrete random variable x with probability mass function P(X).In RL, the formula becomes equation (2) because we calculate the entropy of the policy π(a|s. H(Z) and H(X) H(Z). Thus the addition of independent random variables adds uncertainty. (b)Give an example (of necessarily dependent random variables) in which H(X) >H(Z) and H(Y) >H(Z). (c)Under what conditions does H(Z) = H(X) + H(Y)? 2. Entropy of a disjoint mixture. Let X 1 and X 2 be discrete random variables draw The key to many proofs of Shannon's Entropy Power Inequality is the behaviour of entropy on scaling of continuous random variables. We believe that Rényi's operation of thinning discrete random variables plays a similar role to scaling, and give a sharp bound on how the entropy of ultra log-concave random variables behaves on thinning The entropy of discrete random variables can be harder to bound and control than their continuous counterparts. In particular, a number of functional inequalities and arguments based on scaling do not pass over, meaning that results such as a discrete entropy power inequality are elusive.I will review results from a number of papers, including a maximum entropy result fo

The entropy H(X) of a discrete random variable X is de ned by H(X) = X x p(x)logp(x) (A.2) We will also sometimes use the notation H[p] to denote the entropy of a random variablethat has a probability distribution p. Given severalrandom variables we then de ne De nition A.1.2: Joint Entropy The basic idea behind the proof is to perturb the discrete random variables using suitably designed continuous random variables. Then, the continuous entropy power inequality is applied to the sum of the perturbed random variables and the resulting lower bound is optimized More generally, a random variable with high entropy is closer to being a uniform random variable whereas a random variable with low entropy is less uniform (i.e. there is a high probability associated with only a few of its outcomes). This is depicted in the schematic below: Entropy is the limit to how efficiently one can communicate the. measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the Savage representation. Examples of scoring rules for probabilistic forecasts in the form of predictive densities include the logarithmic, spherical, pseudospherical, and quadrati random variables—which are neither discrete nor continuous— has not been available so far. Here, we present such an extension for the practically relevant class of integer-dimensional singular random variables. The proposed entropy deﬁnition contains the entropy of discrete random variables and the differential entropy IEEE Xplore, delivering full text access to the world's highest quality technical literature in engineering and technology. | IEEE Xplor

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