What is conditional entropy in information theory?

What is conditional entropy in information theory?

In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable given that the value of another random variable is known.

What is meant by joint entropy?

joint entropy is the amount of information in two (or more) random variables; conditional entropy is the amount of information in one random variable given we already know the other.

What is the entropy of a variable?

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.

How do you calculate HX?

H(X) = x p(x) log2 p(x) is both the marginal entropy of X, and its mutual informa- tion with itself. 3. H(X, Y ) = x y p(x, y) log2 p(x, y).

Why does conditioning reduce entropy?

Conditioning reduces entropy H(XY ) H(X) with equality of and only if X and Y are independent. Knowing another random variable Y reduces (on average) the uncertainty of variable X. X Y Z implies that Z Y X.

What is the chain rule of entropy?

With equality if and only if the Xi are independent. Proof: By the chain rule of entropies: … We know that the conditional entropy of a random variable X given another random variable Y is zero if and only if X is a function of Y. Hence we can estimate X from Y with zero probability of error if and only if H(XY) = 0.

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What is joint probability matrix?

Joint probability is a statistical measure that calculates the likelihood of two events occurring together and at the same point in time. Joint probability is the probability of event Y occurring at the same time that event X occurs.

What is entropy and mutual information?

The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected amount of information held in a random variable. … Mutual Information is also known as information gain.

How do you calculate entropy Hx?

The entropy H(X) is a measure of the uncertainty of the random variable X. It also quantifies how much information we gain on average when we learn the value of X. H ( X ) = p log p ( 1 p ) log ( 1 p ) : = H ( p ) . This is called binary entropy and we denote it by the symbol H(p).

Is entropy and enthalpy the same?

Enthalpy is the amount of internal energy contained in a compound whereas entropy is the amount of intrinsic disorder within the compound.

Can entropy be multiple?

Entropy is measured between 0 and 1. (Depending on the number of classes in your dataset, entropy can be greater than 1 but it means the same thing , a very high level of disorder.

What is Delta’s in chemistry?

S is the change in entropy (disorder) from reactants to products. R is the gas constant (always positive) T is the absolute temperature (Kelvin, always positive) What it means: If H is negative, this means that the reaction gives off heat from reactants to products.

What does Shannon entropy measure?

Shannon’s entropy quantifies the amount of information in a variable, thus providing the foundation for a theory around the notion of information. Storage and transmission of information can intuitively be expected to be tied to the amount of information involved.

What is entropy and how do you calculate it?

Entropy is a measure of probability and the molecular disorder of a macroscopic system. If each configuration is equally probable, then the entropy is the natural logarithm of the number of configurations, multiplied by Boltzmann’s constant: S = kB ln W.

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How is Shannon Entropy calculated in Python?

How to calculate Shannon Entropy in Python

  1. data = [1,2,2,3,3,3]
  2. pd_series = pd. Series(data)
  3. counts = pd_series. value_counts()
  4. entropy = entropy(counts)
  5. print(entropy)

Why is entropy non negative?

It is perhaps intuitive that the entropy should be non- negative because non-negativity implies that we always learn some number of bits upon learning random variable X (if we already know beforehand what the outcome of a random experiment will be, then we learn zero bits of information once we perform it).

Is conditional entropy always positive?

Unlike the classical conditional entropy, the conditional quantum entropy can be negative. This is true even though the (quantum) von Neumann entropy of single variable is never negative.

What is the relation between entropy and mutual information?

Thus, if we can show that the relative entropy is a non-negative quantity, we will have shown that the mutual information is also non-negative. = H(XZ) H(XY Z) = H(XZ) + H(Y Z) H(XY Z) H(Z). The conditional mutual information is a measure of how much uncertainty is shared by X and Y , but not by Z.

Is entropy concave or convex?

There exists a function S, called the thermodynamic entropy, with the following prop- erties: (i) In a simple thermodynamic system, the entropy is a differentiable, concave, extensive function of E,V,N; it is increasing in E and V .

Is mutual information convex?

Now, we prove that mutual information is convex in p(yx). More formally, we have the following. Let (X, Y ) have a joint probability distribution p(x, y) = p(x)p(yx).

Is relative entropy convex?

1.2 Properties of the relative entropy In general, D(p q) is now a strictly convex function of p on . This is verified just as we verified that the Shannon entropy is strictly concave.

What does XY mean in probability?

The notation P(xy) means P(x) given event y has occurred, this notation is used in conditional probability. There are two cases if x and y are dependent or if x and y are independent. Case 1) P(xy) = P(x&y)/P(y)

How do you solve joint probability?

Probabilities are combined using multiplication, therefore the joint probability of independent events is calculated as the probability of event A multiplied by the probability of event B. This can be stated formally as follows: Joint Probability: P(A and B)= P(A) * P(B)

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What is the distribution of X Y?

If X and Y are discrete random variables, the function given by f (x, y) = P(X = x, Y = y) for each pair of values (x, y) within the range of X is called the joint probability distribution of X and Y .

What does mutual information tell us?

Mutual information is one of many quantities that measures how much one random variables tells us about another. It is a dimensionless quantity with (generally) units of bits, and can be thought of as the reduction in uncertainty about one random variable given knowledge of another.

What does mutual information tell you?

Mutual information is a quantity that measures a relationship between two random variables that are sampled simultaneously. In particular, it measures how much information is communicated, on average, in one random variable about another. … That is, these variables share mutual information.

What is NMI score?

Normalized Mutual Information (NMI) is a normalization of the Mutual Information (MI) score to scale the results between 0 (no mutual information) and 1 (perfect correlation).

What is data entropy?

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. That is, the more certain or the more deterministic an event is, the less information it will contain.

What is entropy with example?

Entropy is a measure of the energy dispersal in the system. We see evidence that the universe tends toward highest entropy many places in our lives. A campfire is an example of entropy. … Ice melting, salt or sugar dissolving, making popcorn and boiling water for tea are processes with increasing entropy in your kitchen.

How is entropy calculated in digital communication?

If log is base 2 then the unit of entropy is bits. Entropy is a measure of uncertainty in a random variable and a measure of information it can reveal. A more basic explanation of entropy is provided in another module. If p=0 then H(X)=0, if p=1 then H(X)=0, if p=1/2 then H(X)=1 bits.