The Maple Syrup Problem: The Normal-Normal Conjugate

Abstract

In this chapter, Bayesian methods are used to estimate the two parameters that identify a normal distribution, μ‎ and σ‎. Many Bayesian analyses consider alternative parameter values as hypotheses. The prior distribution for an unknown parameter can be represented by a continuous probability density function when the number of hypotheses is infinite. In the “Maple Syrup Problem,” a normal distribution is used as the prior distribution of μ‎, the mean number of millions of gallons of maple syrup produced in Vermont in a year. The amount of syrup produced in multiple years is determined, and assumed to follow a normal distribution with known σ‎. The prior distribution is updated to the posterior distribution in light of this new information. In short, a normal prior distribution + normally distributed data → normal posterior distribution.