L10.9 Mixed Bayes Rule

TL;DR

Bayes rule can be applied to situations involving both discrete and continuous random variables.

Transcript

We have seen two versions of the Bayes rule-- one involving two discrete random variables, and another that involves two continuous random variables. But there are many situations in real life when one has to deal simultaneously with discrete and continuous random variables. For example, you may want to recover a discrete digital signal that was se... Read More

Key Insights

• 📏 Bayes rule can be extended to situations involving both discrete and continuous random variables.
• 😑 The probability of two events happening can be expressed using the PMF and PDF functions.
• 👻 The Bayes rule for discrete and continuous random variables allows for making inferences about one variable given observations of the other.
• ❓ Inferences about discrete variables require knowledge of the unconditional distribution of the variable and the model of the noisy observation.
• ❓ Inferences about continuous variables require knowledge of the conditional densities of the variable under different scenarios for the related observation.
• 🍉 The total probability theorem is used to evaluate the denominator term in the Bayes rule equation.
• 🫡 The integral of the conditional density with respect to the variable must be equal to 1, ensuring the validity of the total probability theorem.

Questions & Answers

Q: How can Bayes rule be extended to situations involving both discrete and continuous random variables?

Bayes rule can be applied to situations with a discrete variable K and a continuous variable Y by using the multiplication rule and equating two expressions involving the PMF and PDF functions. The resulting Bayes rule provides a conditional probability of K given a specific value of Y.

Q: What information is necessary to apply the Bayes rule to situations with discrete and continuous random variables?

To apply the Bayes rule, one needs to know the unconditional distribution of the discrete variable K and have a model of the noisy observation Y under each possible conditional universe. This requires knowledge of the probability distributions of both K and Y.

Q: How can the Bayes rule be used to make inferences about a continuous random variable Y?

By rearranging the terms in the Bayes rule equation, it is possible to obtain a version of the rule that allows for making inferences about Y given the value of a related observation K. This version of the rule is useful when trying to estimate Y based on known values of K.

Q: What is the role of the denominator term in the Bayes rule equation?

The denominator term in the Bayes rule equation represents the total probability and needs to be evaluated. It can be determined using a version of the total probability theorem, which involves weighing the conditional densities of Y based on the probabilities of different discrete scenarios for K.

Summary & Key Takeaways

• Bayes rule can be extended to situations where one variable is discrete and the other is continuous.

• The probability of two events happening can be expressed in terms of the probability mass function (PMF) and probability density function (PDF).

• The Bayes rule for discrete and continuous random variables allows for making inferences about one variable given observations of the other.