Gaussian Mixture Models (GMMs)

Male and Female Weight

Let's suppose we collected some data from some University students. We asked every student their gender (Male = M, Female = F) and their weight. We will build a model that can predict if you are a male or a female based only on your weight. I know this is not the ideal case for this, but this example will help to demonstrate a statistical trick to discriminate between two populations that show different behaviors in the same dataset.

Let's check the data

let's add some libraries we will use in our example:

from matplotlib import pyplot as plt
import numpy as np
import scipy.stats as stats

We will use some fake data to test this out and we will use sklearn to train-test split the data in a 70/30 proportion.

weight = [52, 46, 52, 47, 47, 49, 50, 47, 53, 55, 44, 52, 47, 48, 51, 54, 51, 51, 53, 52, 47, 52, 47, 50, 47, 53, 52, 44, 54, 52, 55, 55, 53, 42, 53, 50, 49, 49, 46, 52, 56, 48, 50, 56, 49, 44, 53, 46, 50, 51, 42, 37, 43, 53, 57, 50, 51, 51, 46, 48, 50, 47, 51, 47, 49, 49, 47, 44, 53, 51, 56, 58, 47, 44, 55, 53, 46, 53, 52, 47, 49, 52, 36, 49, 55, 49, 55, 55, 42, 50, 49, 37, 53, 45, 52, 56, 51, 55, 48, 58, 49, 56, 54, 48, 56, 49, 48, 57, 55, 58, 52, 51, 39, 43, 47, 45, 50, 54, 50, 49, 56, 43, 44, 50, 47, 36, 53, 47, 43, 51, 50, 50, 52, 54, 46, 49, 52, 52, 56, 52, 54, 43, 51, 52, 56, 48, 49, 55, 51, 64, 55, 46, 45, 52, 54, 46, 47, 56, 58, 50, 48, 46, 42, 54, 50, 53, 42, 55, 45, 54, 50, 51, 56, 48, 49, 47, 47, 52, 43, 47, 48, 47, 46, 53, 53, 51, 49, 47, 50, 57, 55, 51, 48, 55, 45, 49, 47, 47, 49, 54, 46, 49, 48, 50, 47, 48, 50, 52, 53, 51, 54, 34, 49, 53, 63, 49, 47, 50, 56, 51, 54, 46, 45, 50, 58, 43, 52, 49, 53, 50, 56, 48, 54, 45, 54, 46, 43, 51, 40, 57, 51, 45, 53, 50, 53, 52, 48, 56, 60, 48, 50, 51, 46, 47, 53, 52, 45, 47, 54, 49, 52, 53, 60, 58, 56, 53, 54, 44, 51, 54, 53, 51, 45, 53, 52, 52, 51, 42, 55, 54, 53, 45, 52, 49, 53, 58, 44, 47, 47, 45, 55, 53, 45, 53, 48, 50, 58, 52, 50, 45, 49, 55, 55, 54, 42, 47, 41, 46, 50, 55, 52, 47, 49, 48, 52, 51, 53, 50, 48, 45, 51, 53, 51, 61, 51, 41, 55, 43, 56, 53, 47, 53, 51, 51, 52, 56, 42, 42, 47, 48, 56, 46, 43, 43, 48, 50, 50, 50, 49, 51, 51, 47, 41, 49, 47, 44, 50, 52, 49, 49, 49, 45, 54, 54, 54, 45, 47, 49, 52, 51, 45, 49, 47, 54, 53, 55, 46, 53, 50, 51, 51, 53, 49, 52, 55, 47, 53, 51, 53, 49, 58, 53, 58, 53, 51, 49, 51, 42, 51, 46, 45, 61, 50, 40, 44, 40, 58, 46, 47, 50, 51, 45, 54, 45, 58, 43, 51, 48, 48, 57, 48, 50, 61, 47, 53, 56, 49, 40, 46, 53, 49, 50, 51, 58, 56, 61, 46, 55, 58, 49, 49, 49, 45, 50, 47, 53, 46, 53, 45, 52, 53, 47, 54, 54, 52, 55, 50, 45, 53, 42, 52, 52, 52, 52, 51, 49, 44, 61, 49, 47, 56, 48, 56, 41, 47, 50, 57, 57, 47, 53, 47, 51, 50, 49, 50, 47, 55, 50, 54, 60, 42, 53, 46, 58, 52, 54, 53, 46, 46, 55, 52, 50, 43, 48, 54, 44, 43, 45, 53, 58, 49, 48, 51, 48, 48, 39, 55, 56, 58, 52, 50, 49, 49, 54, 49, 49, 48, 55, 52, 47, 54, 48, 53, 50, 51, 49, 59, 47, 64, 51, 53, 54, 52, 48, 54, 56, 46, 52, 48, 41, 51, 55, 56, 49, 52, 48, 45, 50, 56, 49, 46, 51, 55, 43, 53, 47, 56, 64, 49, 46, 56, 45, 53, 50, 53, 50, 51, 48, 49, 53, 41, 44, 50, 50, 50, 49, 51, 53, 48, 51, 50, 50, 46, 59, 50, 46, 47, 44, 45, 48, 44, 49, 52, 52, 50, 51, 51, 49, 52, 49, 56, 51, 56, 50, 46, 43, 53, 61, 53, 44, 52, 57, 51, 56, 50, 53, 54, 46, 57, 52, 64, 52, 55, 54, 52, 52, 49, 53, 51, 57, 48, 50, 50, 52, 41, 51, 52, 50, 54, 51, 47, 51, 40, 54, 54, 52, 51, 54, 51, 46, 48, 53, 48, 49, 46, 47, 61, 49, 46, 39, 51, 48, 49, 53, 42, 52, 50, 55, 49, 46, 46, 51, 51, 56, 51, 52, 52, 54, 44, 46, 42, 36, 48, 50, 50, 50, 43, 50, 47, 55, 49, 50, 47, 55, 49, 53, 37, 50, 46, 53, 47, 52, 56, 46, 58, 52, 51, 51, 55, 51, 46, 50, 54, 53, 53, 46, 48, 49, 53, 51, 38, 48, 51, 49, 47, 54, 49, 52, 51, 43, 47, 48, 48, 48, 43, 53, 64, 56, 46, 55, 55, 47, 47, 58, 52, 51, 44, 47, 52, 61, 51, 54, 61, 50, 45, 43, 51, 50, 48, 52, 53, 51, 48, 50, 48, 47, 52, 51, 50, 47, 51, 58, 58, 62, 53, 46, 51, 56, 44, 49, 54, 55, 47, 54, 44, 54, 51, 46, 48, 53, 54, 50, 55, 48, 50, 53, 50, 54, 47, 46, 44, 52, 50, 53, 43, 46, 51, 48, 49, 48, 45, 47, 41, 55, 52, 48, 53, 45, 62, 45, 47, 48, 52, 53, 53, 56, 38, 44, 48, 49, 47, 46, 52, 50, 49, 50, 47, 49, 55, 53, 46, 48, 47, 55, 52, 48, 47, 53, 53, 43, 46, 56, 48, 51, 40, 51, 57, 50, 57, 51, 53, 42, 45, 52, 51, 44, 50, 50, 53, 46, 47, 54, 50, 50, 56, 56, 53, 51, 49, 52, 46, 49, 47, 47, 50, 50, 41, 49, 52, 42, 47, 54, 47, 47, 51, 56, 47, 49, 51, 47, 48, 56, 44, 60, 41, 50, 52, 44, 48, 53, 48, 54, 54, 52, 44, 50, 46, 55, 52, 50, 51, 44, 56, 49, 50, 53, 52, 54, 52, 51, 54, 45, 48, 47, 45, 53, 45, 52, 52, 49, 52, 49, 45, 51, 44, 51, 49, 54, 47, 49, 53, 49, 46, 45, 52, 47, 47, 47, 43, 48, 51, 53, 54, 56, 46, 50, 55, 59, 54, 44, 48, 59, 47, 41, 48, 50, 45, 53, 53, 48, 49, 50, 53, 43, 59, 47, 44, 47, 51, 50, 56, 57, 55, 49, 56, 56, 54, 42, 49, 56, 51, 58, 44, 47, 56, 45, 53, 54, 47, 45, 50, 48, 47, 57, 53, 55, 47, 53, 49, 50, 46, 50, 48, 51, 46, 44, 51, 56, 55, 51, 47, 50, 49, 49, 47, 46, 51, 47, 51, 50, 48, 43, 49, 46, 46, 56, 49, 51, 51, 54, 50, 52, 59, 53, 46, 47, 50, 46, 47, 46, 49, 48, 49, 42, 47, 52, 48, 61, 49, 50, 54, 49, 49, 55, 52, 47, 51, 42, 55, 41, 54, 55, 50, 51, 51, 50, 53, 52, 46, 54, 52, 51, 56, 51, 53, 53, 44, 44, 53, 56, 54, 50, 46, 49, 50, 59, 48, 50, 54, 48, 46, 51, 44, 47, 54, 51, 55, 54, 51, 39, 52, 52, 49, 44, 47, 60, 51, 34, 46, 45, 52, 55, 57, 55, 42, 50, 57, 52, 53, 52, 45, 44, 42, 47, 46, 55, 47, 51, 54, 51, 47, 50, 57, 46, 49, 47, 47, 44, 44, 55, 52, 47, 48, 41, 48, 53, 43, 51, 45, 51, 55, 49, 54, 48, 52, 47, 47, 55, 43, 54, 48, 52, 51, 49, 48, 46, 49, 50, 47, 50, 49, 60, 51, 52, 53, 48, 51, 50, 42, 51, 52, 50, 52, 47, 51, 54, 47, 44, 53, 48, 45, 54, 49, 47, 46, 51, 52, 49, 47, 50, 48, 59, 54, 45, 61, 53, 43, 52, 54, 53, 61, 54, 49, 51, 50, 57, 43, 44, 51, 52, 47, 46, 48, 48, 51, 56, 43, 56, 46, 50, 53, 48, 45, 43, 50, 56, 48, 52, 48, 46, 48, 47, 42, 52, 49, 51, 49, 54, 48, 47, 54, 53, 42, 45, 46, 52, 55, 50, 42, 52, 45, 44, 44, 54, 48, 46, 48, 51, 55, 52, 50, 50, 48, 56, 52, 41, 56, 55, 50, 42, 53, 46, 47, 50, 46, 53, 56, 48, 50, 52, 53, 46, 57, 49, 51, 52, 50, 53, 53, 47, 47, 46, 55, 54, 48, 49, 49, 49, 51, 45, 45, 50, 48, 55, 52, 48, 48, 41, 50, 50, 50, 50, 61, 43, 52, 48, 53, 54, 49, 44, 47, 55, 58, 48, 58, 46, 44, 54, 55, 47, 47, 45, 47, 52, 50, 46, 52, 49, 48, 47, 50, 56, 50, 56, 54, 49, 54, 50, 50, 45, 56, 48, 56, 60, 50, 47, 54, 51, 53, 50, 50, 57, 56, 41, 53, 49, 54, 51, 57, 49, 50, 52, 47, 49, 53, 55, 49, 53, 47, 51, 50, 37, 61, 56, 50, 49, 67, 49, 52, 57, 55, 46, 53, 51, 57, 50, 50, 49, 49, 48, 56, 60, 42, 48, 49, 52, 56, 47, 57, 40, 60, 50, 52, 51, 54, 51, 47, 48, 59, 47, 53, 52, 49, 49, 49, 50, 39, 52, 48, 42, 50, 49, 50, 46, 52, 48, 51, 51, 51, 46, 49, 46, 48, 45, 55, 47, 47, 60, 57, 50, 51, 46, 49, 48, 47, 38, 52, 55, 51, 44, 48, 54, 51, 59, 51, 56, 46, 48, 51, 45, 56, 47, 51, 53, 50, 45, 53, 59, 50, 56, 53, 48, 42, 50, 54, 49, 50, 60, 50, 65, 58, 54, 47, 45, 55, 44, 51, 48, 49, 45, 46, 51, 48, 55, 53, 50, 46, 58, 59, 50, 49, 51, 54, 53, 61, 55, 52, 51, 45, 41, 52, 48, 50, 51, 52, 49, 51, 50, 50, 55, 44, 50, 47, 49, 55, 52, 58, 51, 50, 48, 45, 46, 55, 56, 43, 49, 50, 59, 53, 45, 47, 49, 40, 45, 55, 46, 50, 53, 50, 45, 52, 48, 53, 53, 51, 49, 52, 49, 52, 56, 36, 50, 45, 45, 49, 49, 56, 51, 50, 50, 48, 46, 50, 48, 48, 47, 52, 39, 44, 51, 55, 48, 55, 46, 53, 45, 54, 55, 57, 53, 49, 46, 53, 49, 49, 53, 52, 52, 51, 52, 51, 48, 53, 44, 39, 46, 48, 51, 52, 44, 54, 44, 49, 54, 55, 48, 51, 50, 54, 43, 53, 47, 47, 45, 54, 46, 52, 51, 55, 59, 48, 49, 55, 54, 51, 51, 44, 56, 50, 54, 47, 53, 58, 51, 54, 47, 43, 47, 55, 49, 75, 72, 59, 73, 74, 54, 62, 64, 68, 66, 82, 77, 79, 76, 63, 83, 76, 72, 75, 71, 66, 69, 67, 65, 67, 67, 69, 57, 52, 61, 72, 64, 74, 59, 72, 70, 68, 80, 70, 70, 72, 74, 78, 73, 76, 65, 68, 70, 75, 73, 68, 81, 76, 93, 73, 69, 66, 65, 61, 74, 74, 69, 76, 79, 73, 73, 82, 61, 62, 71, 83, 76, 61, 66, 77, 64, 72, 67, 68, 66, 67, 53, 79, 75, 61, 66, 71, 67, 71, 60, 72, 72, 72, 71, 69, 71, 78, 61, 71, 80, 69, 59, 64, 72, 75, 74, 69, 69, 70, 62, 61, 66, 77, 66, 67, 62, 64, 77, 69, 73, 78, 79, 63, 70, 72, 70, 70, 68, 70, 77, 88, 69, 62, 71, 74, 71, 58, 78, 68, 70, 74, 81, 68, 64, 67, 74, 64, 66, 65, 67, 65, 78, 78, 70, 64, 68, 72, 81, 68, 68, 64, 69, 64, 66, 65, 70, 80, 71, 59, 70, 69, 64, 78, 68, 71, 69, 69, 75, 62, 65, 59, 77, 62, 70, 72, 76, 84, 78, 60, 69, 74, 76, 76, 73, 73, 86, 65, 72, 72, 64, 82, 81, 73, 62, 65, 68, 74, 65, 73, 66, 80, 64, 65, 69, 65, 74, 74, 71, 65, 63, 66, 72, 72, 77, 77, 68, 75, 68, 72, 62, 68, 59, 68, 59, 73, 71, 82, 73, 71, 74, 73, 73, 71, 71, 75, 81, 72, 76, 75, 75, 84, 68, 65, 76, 71, 64, 67, 65, 65, 76, 80, 66, 80, 63, 72, 61, 76, 73, 61, 70, 72, 75, 71, 53, 73, 84, 74, 67, 68, 64, 70, 74, 68, 75, 63, 71, 73, 63, 72, 61, 68, 67, 62, 79, 65, 72, 72, 66, 69, 71, 68, 65, 81, 69, 70, 70, 61, 77, 73, 80, 73, 74, 60, 68, 71, 69, 81, 87, 66, 66, 71, 67, 65, 77, 67, 75, 66, 74, 77, 62, 72, 71, 80, 73, 64, 75, 69, 62, 67, 71, 70, 68, 64, 61, 80, 68, 83, 62, 70, 66, 75, 70, 61, 71, 64, 56, 55, 75, 78, 73, 62, 71, 67, 73, 65, 71, 67, 69, 66, 71, 69, 72, 77, 70, 68, 74, 70, 69, 68, 71, 80, 63, 70, 64, 70, 70, 68, 71, 65, 67, 63, 64, 73, 71, 62, 85, 73, 75, 73, 78, 69, 69, 77, 69, 70, 66, 61, 70, 71, 72, 68, 64, 60, 58, 68, 67, 71, 82, 60, 65, 73, 70, 74, 71, 69, 81, 60, 76, 70, 62, 67, 66, 67, 63, 72, 71, 62, 63, 65, 70, 66, 68, 79, 76, 79, 70, 68, 62, 66, 66, 75, 77, 68, 67, 73, 80, 58, 64, 78, 69, 65, 70, 70, 70, 75, 64, 59, 66, 74, 66, 84, 57, 79, 80, 63, 83, 84, 77, 74, 64, 67, 79, 64, 73, 70, 74, 75, 63, 68, 63, 70, 70, 66, 77, 70, 71, 79, 68, 70, 69, 66, 69, 69, 59, 73, 63, 73, 70, 63, 80, 78, 70, 77, 71, 68, 66, 78, 72, 71, 63, 77, 68, 71, 66, 73, 80, 77, 73, 70, 63, 61, 71, 67, 62, 80, 79, 62, 72, 61, 66, 69, 72, 75, 69, 76, 80, 76, 80, 76, 66, 77, 62, 72, 75, 62, 62, 61, 74, 72, 64, 70, 63, 69, 72, 73, 78, 63, 73, 63, 65, 69, 66, 72, 70, 85, 67, 63, 69, 72, 70, 70, 75, 76, 63, 66, 70, 67, 76, 74, 70, 70, 75, 67, 61, 61, 74, 77, 61, 68, 69, 78, 73, 76, 70, 63, 76, 62, 79, 73, 66, 79, 71, 64, 61, 69, 72, 72, 71, 75, 66, 77, 62, 63, 63, 64, 60, 69, 67, 72, 82, 75, 81, 76, 65, 66, 71, 82, 78, 59, 70, 72, 74, 73, 80, 75, 70, 78, 86, 77, 66, 68, 71, 68, 67, 78, 64, 71, 65, 64, 73, 73, 60, 72, 70, 76, 75, 68, 70, 60, 71, 57, 78, 74, 71, 70, 71, 59, 73, 76, 64, 76, 76, 70, 75, 65, 62, 70, 62, 64, 69, 68, 78, 81, 72, 67, 84, 67, 79, 65, 72, 62, 65, 71, 69, 61, 65, 58, 68, 68, 75, 60, 73, 63, 68, 87, 71, 69, 70, 61, 77, 75, 75, 72, 66, 75, 82, 71, 73, 61, 66, 81, 79, 71, 67, 66, 77, 64, 70, 70, 76, 69, 76, 74, 77, 71, 78, 70, 67, 64, 75, 69, 84, 64, 74, 65, 69, 73, 66, 77, 63, 74, 67, 61, 68, 59, 64, 59, 66, 65, 55, 66, 71, 75, 65, 58, 68, 75, 79, 69, 69, 67, 75, 71, 78, 75, 68, 54, 72, 75, 75, 62, 81, 67, 66, 78, 75, 59, 64, 74, 65, 67, 72, 67, 79, 66, 51, 72, 65, 64, 66, 61, 71, 67, 67, 65, 63, 68, 82, 62, 69, 87, 66, 72, 65, 58, 70, 83, 76, 70, 69, 75, 72, 63, 65, 66, 80, 72, 76, 68, 66, 78, 64, 73, 68, 69, 66, 80, 80, 68, 56, 67, 78, 59, 71, 70, 64, 81, 77, 78, 74, 61, 69, 70, 79, 61, 63, 71, 66, 74, 70, 69, 77, 76, 77, 75, 69, 77, 69, 68, 71, 77, 64, 65, 72, 69, 79, 68, 79, 68, 78, 73, 62, 58, 55, 76, 85, 75, 72, 77, 67, 68, 66, 83, 63, 88, 67, 77, 75, 80, 66, 86, 65, 77, 67, 63, 72, 75, 66, 68, 65, 71, 73, 72, 68, 80, 68, 81, 74, 73, 73, 77, 74, 68, 77, 70, 66, 72, 71, 77, 62, 65, 72, 78, 77, 69, 67, 78, 72, 66, 70, 64, 74, 75, 69, 66, 67, 72, 78, 65, 57, 74, 67, 61, 72, 66, 63, 66, 67, 72, 75, 65, 66, 75, 69, 62, 68, 61, 67, 67, 77, 70, 71, 65, 66, 66, 77, 68, 67, 74, 68, 70, 78, 60, 78, 69, 75, 73, 66, 76, 69, 66, 67, 65, 72, 74, 61, 65, 73, 77, 80, 75, 72, 66, 66, 70, 73, 77, 72, 67, 75, 71, 75, 68, 70, 77, 72, 64, 69, 66, 71, 61, 81, 74, 63, 65, 71, 77, 56, 81, 66, 64, 79, 72, 74, 64, 66, 74, 75, 71, 77, 74, 63, 86, 78, 85, 74, 68, 85, 63, 73, 61, 66, 68, 80, 69, 74, 65, 67, 57, 71, 70, 71, 68, 61, 62, 58, 65, 79, 77, 68, 64, 66, 72, 67, 74, 59, 73, 76, 69, 65, 74, 72, 66, 79, 79, 67, 72, 67, 68, 70, 60, 78, 78, 73, 74, 70, 50, 75, 68, 83, 71, 77, 66, 77, 80, 81, 69, 65, 70, 65, 63, 67, 65, 83, 69, 54, 74, 73, 71, 66, 70, 82, 54, 66, 69, 66, 62, 69, 67, 71, 61, 73, 72, 71, 65, 78, 70, 77, 70, 68, 66, 69, 71, 60, 68, 73, 69, 74, 67, 69, 73, 72, 73, 69, 65, 55, 74, 71, 67, 73, 62, 74, 75, 65, 70, 72, 69, 64, 70, 72, 73, 75, 70, 68, 66, 71, 81, 68, 80, 78, 72, 66, 65, 66, 64, 63, 74, 72, 67, 76, 76, 69, 61, 66, 75, 69, 66, 75, 58, 78, 67, 68, 73, 70, 77, 70, 80, 80, 73, 70, 66, 71, 72, 73, 63, 75, 66, 65, 87, 82, 75, 60, 65, 71, 67, 78, 77, 74, 65, 74, 68, 74, 76, 72, 70, 73, 62, 63, 71, 70, 70, 68, 73, 80, 78, 59, 69, 73, 63, 73, 61, 67, 81, 76, 69, 68, 58, 66, 63, 68, 59, 68, 65, 79, 72, 64, 68, 75, 67, 68, 67, 71, 70, 73, 72, 74, 67, 66, 74, 59, 72, 60, 72, 74, 81, 64, 72, 79, 67, 78]
sex = ['F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 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'M', 'M']

Let's split the data into 70/30 sets.

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(weight, sex, test_size=0.3, random_state=0)

Now let's plot the data for X_train and X_test colored by gender:

plt.hist([x for (x,y) in list(zip(X_train,y_train)) if y=='M'], color='blue', alpha=0.5, bins=15)
plt.hist([x for (x,y) in list(zip(X_train,y_train)) if y=='F'], color='red', alpha=0.5, bins=15)
plt.show()

Now its time for the histogram for the test set:

plt.hist([x for (x,y) in list(zip(X_test,y_test)) if y=='M'], color='blue', alpha=0.5, bins=15)
plt.hist([x for (x,y) in list(zip(X_test,y_test)) if y=='F'], color='red', alpha=0.5, bins=15)
plt.show()

As you can observe, it's a multimodal distribution. This is something very typical in many scenarios, such as when there are patients with certain diseases (cancer/no cancer) or maybe in the banking industry, to label if a customer will be labeled as declined for a loan (declined/approved).

We can use a Gaussian Mixture, a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters. SKlearn already has this implemented, and we only need to tell the Gaussian mixture class how many labels or classes exist in the dataset. In this case, we know there are only two genders (male and female), so we will set the parameter n_components=2. Sklearn will automatically identify both distributions and estimate each distribution's means. The trick is simple: if new data comes for a prediction, the Gaussian Mixture Model will predict with a value Xi is close to one mean or two the other and will assign the label accordingly:

import numpy as np
from sklearn.mixture import GaussianMixture

X = np.array(X_train).reshape(np.array(X_train).shape[0], 1)
gm = GaussianMixture(n_components=2, random_state=0).fit(X)

print(gm.means_)

This will print gm.means_[0] = 70.18 and gm.means_[1] = 50.12. We can plot the X_train and y_train distribution again with some vertical lines to check if this works correctly.

plt.hist([x for (x,y) in list(zip(X_train,y_train)) if y=='M'], color='blue', alpha=0.5, bins=15)
plt.hist([x for (x,y) in list(zip(X_train,y_train)) if y=='F'], color='red', alpha=0.5, bins=15)

plt.axvline(gm.means_[0], color='k', linestyle='dashed', linewidth=1)
plt.axvline(gm.means_[1], color='k', linestyle='dashed', linewidth=1)
plt.show()

The gm model was trained on the train set data, and the two means were added to the histogram. All looks good. Now, the only thing left is to predict the test set and get the accuracy. The following snippet will do the trick:

y_pred = gm.predict(np.array(X_test).reshape(np.array(X_test).shape[0], 1))
y_pred = ['M' if x == 0 else 'F' for x in y_pred]

from sklearn.metrics import accuracy_score
accuracy = accuracy_score(y_test, y_pred)

print("Accuracy:", accuracy)

Accuracy: 0.97

Summary

The Gaussian Mixture is a tool for segregating data with multiple distributions. In this case, two normal distributions represented males and females, which SKlearn effortlessly separated. In some cases, the data might be a little more complex with more dimensions; this might require another type of Mixture Model, such as a Variational Bayesian Mixture Model that can help cluster across multiple classes. You can check an example of such a scenario with sklearn mixtures.

Acknowledgments

This code was inspired by an old post from Luis Perez, PhD, about when we worked at the university building AI content for students.