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# P1.T2.500. Bayes theorem

#### Nicole Seaman

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Learning outcome: Describe Bayes’ theorem and apply this theorem in the calculation of conditional probabilities. Compare the Bayesian approach to the frequentist approach.

Questions:

500.1. According to Miller, each of the following is true about Bayes' theorem and Bayesian analysis EXCEPT which is false?

a. Bayes' theorem is often described as a procedure for updating beliefs about the world when presented with new information
b. Bayes' theorem updates a prior probability with evidence (aka, likelihood) to generate a posterior probability
c. Risk management, performance analysis and stress testing are areas where we often have very little data, and where the data tends to be noisy, such the frequentist approach is superior to the Bayesian approach
d. Although the theorem itself is simple, Bayes' Theorem can be applied to a wide range of problems (e.g., it is used in everything from spam filters to machine translation and to the software that controls self-driving cars) and its application can often be quite complex

500.2. You have a portfolio of bonds, each with a 1.0% probability of default. An analyst develops a model for forecasting bond defaults, but the model is only 70.0% accurate. In other words, of the bonds that actually default, the model identifies only 70.0% of them; likewise, of the bonds that do not default, the model correctly predicts that 70% will not default. Given that the model predicts that a bond will default, what is the probability that it actually defaults? (note: this is a variation on Miller's question 6-2).

a. 1.00%
b. 1.43%
c. 2.30%
d. 7.00%

500.3. You have a model that classifies Federal Reserve statements as either bullish or bearish. When the Fed makes a bullish announcement, you expect the market to be up 80.0% of the time. The market has 60.0% probability of being up, and a 40.0% probability of being flat or down (the only two states are up, or not up). The Fed makes bullish announcements 60.0% of the time. What is the probability that the Fed made a bearish announcement, given that the market was up? (note: this is a variation on Miller's question 6-7)

a. 20.0%
b. 40.0%
c. 60.0%
d. 80.0%

Answers here: