Think Bayes, 2nd Edition

Chapter 7. Minimum, Maximum, and Mixture

In the previous chapter we computed distributions of sums. In this chapter, we’ll compute distributions of minimums and maximums, and use them to solve both Forward and Inverse Problems.

Then we’ll look at distributions that are mixtures of other distributions, which will turn out to be particularly useful for making predictions.

But we’ll start with a powerful tool for working with distributions, the cumulative distribution function.

Cumulative Distribution Functions

So far we have been using probability mass functions to represent distributions. A useful alternative is the cumulative distribution function, or CDF.

As an example, I’ll use the posterior distribution from the Euro Problem, which we computed in “Bayesian Estimation”.

Here’s the uniform prior we started with:

import numpy as np
from empiricaldist import Pmf

hypos = np.linspace(0, 1, 101)
pmf = Pmf(1, hypos)
data = 140, 250

And here’s the update:

from scipy.stats import binom

def update_binomial(pmf, data):
    """Update pmf using the binomial distribution."""
    k, n = data
    xs = pmf.qs
    likelihood = binom.pmf(k, n, xs)
    pmf *= likelihood
    pmf.normalize()

update_binomial(pmf, data)

The CDF is the cumulative sum of the PMF, so we can compute it like this:

cumulative = pmf.cumsum()

Here’s what it looks like, along with the PMF:

The range of the CDF is always from 0 to 1, in contrast ...