Mutual information: consider random variables x and y,...

Mutual information: consider random variables $x$ and $y$, distributed according to a joint probability $p(x, y)$. The mutual information between the two variables is defined by $$ M(x, y) \equiv \sum_{x, y} p(x, y) \ln \left(\frac{p(x, y)}{p_x(x) p_y(y)}\right), $$ where $p_s$ and $p_y$ denote the unconditional probabilities for $x$ and $y$. (a) Relate $M(x, y)$ to the entropies $S(x, y), S(x)$, and $S(y)$ obtained from the corresponding probabilities. (b) Calculate the mutual information for the joint Gaussian form $$ p(x, y) \propto \exp \left(-\frac{a x^2}{2}-\frac{b y^2}{2}-c x y\right) $$

Mutual information: consider random variables $x$ and $y$, distributed according to a joint probability $p(x, y)$. The mutual information between the two variables is defined by
$$
M(x, y) \equiv \sum_{x, y} p(x, y) \ln \left(\frac{p(x, y)}{p_x(x) p_y(y)}\right),
$$
where $p_s$ and $p_y$ denote the unconditional probabilities for $x$ and $y$.
(a) Relate $M(x, y)$ to the entropies $S(x, y), S(x)$, and $S(y)$ obtained from the corresponding probabilities.
(b) Calculate the mutual information for the joint Gaussian form
$$
p(x, y) \propto \exp \left(-\frac{a x^2}{2}-\frac{b y^2}{2}-c x y\right)
$$

Key Concepts

Bivariate Gaussian Distribution

A Bivariate Gaussian (or joint Gaussian) distribution is the extension of the normal distribution to two dimensions, characterized by its means, variances, and covariance (or correlation coefficient). In problems involving mutual information, this distribution serves as an important example where the mutual information can be explicitly computed in terms of parameters that capture the dependency between the two variables.

Joint and Marginal Distributions

A joint probability distribution describes the likelihood of two random variables occurring simultaneously, while marginal distributions provide the probabilities of each variable independently, obtained by summing or integrating the joint distribution over the other variable. These distributions are crucial for calculating mutual information as they allow comparison of the joint behavior of variables with their independent behaviors.

Mutual Information

Mutual Information is a measure from information theory that quantifies the amount by which the uncertainty of one random variable is reduced when knowing another variable. It is computed as the difference between the sum of the individual entropies and the joint entropy, thereby capturing the dependence between the two variables.

Entropy

Entropy, specifically Shannon entropy in the context of information theory, quantifies the uncertainty or randomness inherent in a probability distribution. It provides a measure of the information content associated with a random variable, serving as a fundamental building block for defining quantities like mutual information.

Please give Ace some feedback