Question

Mutual information by another name In the chapter, we introduced the concept of mutual information as the average decrease in the missing information associated with one variable when the value of another variable in known. In terms of probability distributions, this can be written mathematically as \\[I=\sum_{y} p(y)\left[-\sum_{x} p(x) \log _{2} p(x)+\sum_{x} p(x | y) \log _{2} p(x | y)\right]\\] where the expression in square brackets is the difference in missing information, \(S_{x}-S_{x} y,\) associated with probability of \(x, p(x),\) and with probabilify of \(x\) conditioned on \(y, p(x | y)\) Using the relation between the conditional probability \(p(x | y)\) and the joint probability \(p(x, y)\) \\[p(x | y)=\frac{p(x, y)}{p(y)}\\] show that the formula for mutual information given in Equation 21.77 can be used to derive the formula used in the chapter (Equation 21.17 ), namely \\[I=\sum_{x, y} p(x, y) \log _{2}\left[\frac{p(x, y)}{p(x) p(y)}\right]\\].

Short answer

\nThe desired formula for mutual information \(I=\sum_{x,y} p(x, y) \log _{2}\left[\frac{p(x, y)}{p(x) p(y)}\right]\) is derived.

Step-by-step solution

Expand mutual information formula

Starting with the mutual information formula: \(I=\sum_{y} p(y)\left[-\sum_{x} p(x) \log _{2} p(x)+\sum_{x} p(x | y) \log _{2} p(x | y)\right]\), we should expand the expression using the properties of the logarithm where the sum of the logs is the log of the product. Therefore, we rewrite it as: \(I=\sum_{y}p(y)\sum_{x}p(x|y)\log_{2}\left[ \frac{p(x)}{p(x|y)}\right]\)

Substitute Conditional Probability

Replace the conditional probability \(p(x|y)\) as \(\frac{p(x, y)}{p(y)}\) in the formula of mutual information, thus obtaining: \(I=\sum_{y}p(y)\sum_{x}\frac{p(x, y)}{p(y)}\log_{2}\left[ \frac{p(x)p(y)}{p(x, y)}\right]\). This simplifies to: \(I=\sum_{x,y}p(x, y)\log_{2}\left[ \frac{p(x)p(y)}{p(x, y)}\right]\)

Rearrange the Formula

Next, take the reciprocal of the fraction inside the logarithm and change the sign of the logarithm to obtain the desired formula for mutual information: \(I=\sum_{x,y}p(x, y)\log_{2}\left[ \frac{p(x, y)}{p(x)p(y)}\right]\) Hence, the formula for mutual information in Equation 21.77 can be used to derive the formula used in the chapter, which is Equation 21.17

Key concepts

Probability Distributions

Probability distributions are fundamental concepts in statistics and probability theory. They describe how the probabilities are distributed over the values of a random variable. Imagine a random variable as a way to assign values to the outcomes of a random event. A probability distribution tells us how likely each of these values is to occur.

The two basic types of probability distributions are discrete and continuous. Discrete probability distributions happen with countable events, like rolling dice. Continuous ones apply to events with an infinite number of possibilities, like the exact height of people. Each type of distribution has its own characteristics and equations.

• For discrete distributions, probabilities are described by a probability mass function (PMF), which gives the probability that a discrete random variable is exactly equal to some value.
• For continuous distributions, we use a probability density function (PDF), which is used to specify the probability of the random variable falling within a particular range of values.

Probability distributions help us make predictions and decisions. For example, understanding the distribution of one set of data can help anticipate future outcomes based on patterns.

Conditional Probability

Conditional probability is the likelihood of an event occurring, given that another event has already occurred. It's key to understanding relationships between events, and it's crucial in fields like statistics, machine learning, and decision-making. Conditional probability helps us refine our predictions based on new information.

The formula for calculating conditional probability is:
\[p(A | B) = \frac{p(A \cap B)}{p(B)}\]
where \(p(A \cap B)\) is the probability of both events \(A\) and \(B\) happening together, and \(p(B)\) is the probability of event \(B\) occurring. If \(p(B)\) is zero, the conditional probability is undefined since we can't condition on an event that never happens.

• Keystone concept: It helps us better understand and predict outcomes by considering the context or conditions under which they occur.
• Application: In real-world scenarios, it's used to fine-tune models and predictions.

Using the conditional probability in our exercise, we saw that we can replace \(p(x|y)\) by \(\frac{p(x,y)}{p(y)}\) to break down the mutual information formula.

Joint Probability

Joint probability relates to the probability of two events happening together. It's particularly useful when exploring the relationships between multiple random variables. Calculating joint probabilities allows us to analyze and understand the interconnectedness of different events.

The probability of two events \(A\) and \(B\) happening together is given by the joint probability \(p(A, B)\). This is computed by multiplying the probability of event \(A\) by the probability of event \(B\), provided that they are independent. If they're not independent, more information is required to calculate \(p(A, B)\).

• Calculation: For independent events, \(p(A, B) = p(A) \times p(B)\). For dependent events, \(p(A, B) = p(A|B) \times p(B)\).
• Importance: It helps to quantify the likelihood of multiple events occurring at the same time.

In our mutual information exercise, we used joint probability to derive the formula by expressing \(p(x, y)\) within conditional probability and ultimately refined it into the desired mutual information expression. This shows the power of joint probability in handling complex probabilistic calculations.