advanced-algorithmic-trading

Chapter 3
Bayesian Inference of a Binomial
Proportion
In the previous chapter we examined Bayes’ rule and considered how it allowed us to rationally
update beliefs about uncertainty as new evidence came to light. We mentioned briefly that
such techniques are becoming extremely important in the fields of data science and quantitative
finance.
In this chapter we are going to expand on the coin-flip example that we studied in the previous
chapter by discussing the notion of Bernoulli trials, the beta distribution and conjugate priors.
Our goal in this chapter is to allow us to carry out what is known as "inference on a binomial
proportion". That is, we will be studying probabilistic situations with two outcomes (e.g. a
coin-flip) and trying to estimate the proportion of a repeated set of events that come up heads
or tails.
Our goal is to estimate how fair a coin is. We will use that estimate to make
predictions about how many times it will come up heads when we flip it in the future.
While this may sound like a rather academic example, it is actually substantially more applicable
to real-world applications than may first appear. Consider the following scenarios:
• Engineering: Estimating the proportion of aircraft turbine blades that possess a structural
defect after fabrication
• Social Science: Estimating the proportion of individuals who would respond "yes" on a
census question
• Medical Science: Estimating the proportion of patients who make a full recovery after
taking an experimental drug to cure a disease
• Corporate Finance: Estimating the proportion of transactions in error when carrying
out financial audits
• Data Science: Estimating the proportion of individuals who click on an ad when visiting
a website
As can be seen, inference on a binomial proportion is an extremely important statistical
technique and will form the basis of many of the chapters on Bayesian statistics that follow.
23