The Central Limit Theorem, a hands-on introduction

The central limit theorem can be informally summarized in few words: The sum of x₁, x₂, ... x_n samples from the same distribution is normally distributed, provided that n is big enough and that the distribution has a finite variance. to show this in an experimental way, let's define a function that sums n samples from the same distrubution for 100000 times:

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

def sum_random_variables(*kwarg, sp_distribution, n):
    # returns the sum of n random samples
    # drawn from sp_distribution
    v = [sp_distribution.rvs(*kwarg, size=100000) for _ in range(n)]
    return np.sum(v, axis=0)

This function takes in input the parameters of the distrubution, the function that implements the distrubution and n. It returns an array of 100000 elements, where each element is the sum of n samples. Given the Central Limit Theorem, we expect that the values in output are normally distributed if n is big enough. To verify this, let's consider a beta distribution with parameters alpha=1 and beta=2, run our function increasing n and plot the histogram of the values in output:

plt.figure(figsize=(9, 3))
N = 5
for n in range(1, N):
    plt.subplot(1, N-1, n)
    s = sum_random_variables(1, 2, sp_distribution=sps.beta, n=n)
    plt.hist(s, density=True)
plt.tight_layout()

On the far left we have the histogram with n=1 , the one with n=2 right next to it, and so on until n=4. With n=1 we have the original distribution, which is heavily skewed. With n=2 we have a distribution which is less skewed. When we reach n=4 we see that the distribution is almost symmetrical, resembling a normal distribution.

Let's do the same experiment using a uniform distribution:

plt.figure(figsize=(9, 3))
for n in range(1, N):
    plt.subplot(1, N-1, n)
    s = sum_random_variables(1, 1, sp_distribution=sps.beta, n=n)
    plt.hist(s, density=True)
plt.tight_layout()

Here we have that for n=2 the distribution is already symmetrical, resembling a triangle, and increasing n further we get closer to the shape of a Gaussian.

The same behaviour can be shown for discrete distributions. Here's what happens if we use the Bernoulli distribution:

plt.figure(figsize=(9, 3))
for n in range(1, N):
    plt.subplot(1, N-1, n)
    s = sum_random_variables(.5, sp_distribution=sps.bernoulli, n=n)
    plt.hist(s, bins=n+1, density=True, rwidth=.7)
plt.tight_layout()

We see again that for n=2 the distribution starts to be symmetrical and that the shape of a Gaussian is almost clear for n=4.

A Simple model that earned a Silver medal in predicting the results of the NCAAW tournament

This year I decided to join the March Machine Learning Mania 2021 - NCAAW challenge on Kaggle. It proposes to predict the outcome of each game into the basketball NCAAW tournament, which is a tournament for women at college level. Participants can assign a probability to each outcome and they're ranked on the leaderboard according to the accuracy of their prediction. One of the most attractive elements of the challenge is that the leaderboard is updated after each game throughout the tournament.

Since I have limited knowledge of basketball I decided to use a minimalistic model:

• It uses three features that are easy to interpret: seed, percentage of victories, and the average score of each team.
• It is based on linear Linear Regression, and it's tuned to predict extreme probability values only for games that are easy to predict.

The following visualizations give insight into how the model estimates the winning probability in a game between two teams:
Surprisingly, this model ranked 46th out of 451 submissions, placing itself in the top 11% of the leaderboard and earning a silver medal!

The notebook with the solution and some more charts can be found here.