Discrete Probability Distributions for Machine Learning

Author: Jason Brownlee

The probability for a discrete random variable can be summarized with a discrete probability distribution.

Discrete probability distributions are used in machine learning, most notably in the modeling of binary and multi-class classification problems, but also in evaluating the performance for binary classification models, such as the calculation of confidence intervals, and in the modeling of the distribution of words in text for natural language processing.

Knowledge of discrete probability distributions is also required in the choice of activation functions in the output layer of deep learning neural networks for classification tasks and selecting an appropriate loss function.

Discrete probability distributions play an important role in applied machine learning and there are a few distributions that a practitioner must know about.

In this tutorial, you will discover discrete probability distributions used in machine learning.

After completing this tutorial, you will know:

The probability of outcomes for discrete random variables can be summarized using discrete probability distributions.
A single binary outcome has a Bernoulli distribution, and a sequence of binary outcomes has a Binomial distribution.
A single categorical outcome has a Multinoulli distribution, and a sequence of categorical outcomes has a Multinomial distribution.

Let’s get started.

Discrete Probability Distributions for Machine Learning
Photo by John Fowler, some rights reserved.

Tutorial Overview

This tutorial is divided into five parts; they are:

Discrete Probability Distributions
Bernoulli Distribution
Binomial Distribution
Multinoulli Distribution
Multinomial Distribution

Discrete Probability Distributions

A random variable is the quantity produced by a random process.

A discrete random variable is a random variable that can have one of a finite set of specific outcomes. The two types of discrete random variables most commonly used in machine learning are binary and categorical.

Binary Random Variable: x in {0, 1}
Categorical Random Variable: x in {1, 2, …, K}.

A binary random variable is a discrete random variable where the finite set of outcomes is in {0, 1}. A categorical random variable is a discrete random variable where the finite set of outcomes is in {1, 2, …, K}, where K is the total number of unique outcomes.

Each outcome or event for a discrete random variable has a probability.

The relationship between the events for a discrete random variable and their probabilities is called the discrete probability distribution and is summarized by a probability mass function, or PMF for short.

For outcomes that can be ordered, the probability of an event equal to or less than a given value is defined by the cumulative distribution function, or CDF for short. The inverse of the CDF is called the percentage-point function and will give the discrete outcome that is less than or equal to a probability.

PMF: Probability Mass Function, returns the probability of a given outcome.
CDF: Cumulative Distribution Function, returns the probability of a value less than or equal to a given outcome.
PPF: Percent-Point Function, returns a discrete value that is less than or equal to the given probability.

There are many common discrete probability distributions.

The most common are the Bernoulli and Multinoulli distributions for binary and categorical discrete random variables respectively, and the Binomial and Multinomial distributions that generalize each to multiple independent trials.

Binary Random Variable: Bernoulli Distribution
Sequence of a Binary Random Variable: Binomial Distribution
Categorical Random Variable: Multinoulli Distribution
Sequence of a Categorical Random Variable: Multinomial Distribution

In the following sections, we will take a closer look at each of these distributions in turn.

There are additional discrete probability distributions that you may want to explore, including the Poisson Distribution and the Discrete Uniform Distribution.

Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution that covers a case where an event will have a binary outcome as either a 0 or 1.

x in {0, 1}

A “Bernoulli trial” is an experiment or case where the outcome follows a Bernoulli distribution. The distribution and the trial are named after the Swiss mathematician Jacob Bernoulli.

Some common examples of Bernoulli trials include:

The single flip of a coin that may have a heads (0) or a tails (1) outcome.
A single birth of either a boy (0) or a girl (1).

A common example of a Bernoulli trial in machine learning might be a binary classification of a single example as the first class (0) or the second class (1).

The distribution can be summarized by a single variable p that defines the probability of an outcome 1. Given this parameter, the probability for each event can be calculated as follows:

P(x=1) = p
P(x=0) = 1 – p

In the case of flipping a fair coin, the value of p would be 0.5, giving a 50% probability of each outcome.

Binomial Distribution

The repetition of multiple independent Bernoulli trials is called a Bernoulli process.

The outcomes of a Bernoulli process will follow a Binomial distribution. As such, the Bernoulli distribution would be a Binomial distribution with a single trial.

Some common examples of Bernoulli processes include:

A sequence of independent coin flips.
A sequence of independent births.

The performance of a machine learning algorithm on a binary classification problem can be analyzed as a Bernoulli process, where the prediction by the model on an example from a test set is a Bernoulli trial (correct or incorrect).

The Binomial distribution summarizes the number of successes k in a given number of Bernoulli trials n, with a given probability of success for each trial p.

We can demonstrate this with a Bernoulli process where the probability of success is 30% or P(x=1) = 0.3 and the total number of trials is 100 (k=100).

We can simulate the Bernoulli process with randomly generated cases and count the number of successes over the given number of trials. This can be achieved via the binomial() NumPy function. This function takes the total number of trials and probability of success as arguments and returns the number of successful outcomes across the trials for one simulation.

# example of simulating a binomial process and counting success
from numpy.random import binomial
# define the parameters of the distribution
p = 0.3
k = 100
# run a single simulation
success = binomial(k, p)
print('Total Success: %d' % success)

We would expect that 30 cases out of 100 would be successful given the chosen parameters (k * p or 100 * 0.3).

A different random sequence of 100 trials will result each time the code is run, so your specific results will differ. Try running the example a few times.

In this case, we can see that we get slightly less than the expected 30 successful trials.

Total Success: 28

We can calculate the moments of this distribution, specifically the expected value or mean and the variance using the binom.stats() SciPy function.

# calculate moments of a binomial distribution
from scipy.stats import binom
# define the parameters of the distribution
p = 0.3
k = 100
# calculate moments
mean, var, _, _ = binom.stats(k, p, moments='mvsk')
print('Mean=%.3f, Variance=%.3f' % (mean, var))

Running the example reports the expected value of the distribution, which is 30, as we would expect, as well as the variance of 21, which if we calculate the square root, gives us the standard deviation of about 4.5.

Mean=30.000, Variance=21.000

We can use the probability mass function to calculate the likelihood of different numbers of successful outcomes for a sequence of trials, such as 10, 20, 30, to 100.

We would expect 30 successful outcomes to have the highest probability.

# example of using the pmf for the binomial distribution
from scipy.stats import binom
# define the parameters of the distribution
p = 0.3
k = 100
# define the distribution
dist = binom(k, p)
# calculate the probability of n successes
for n in range(10, 110, 10):
	print('P of %d success: %.3f%%' % (n, dist.pmf(n)*100))

Running the example defines the binomial distribution and calculates the probability for each number of successful outcomes in [10, 100] in groups of 10.

The probabilities are multiplied by 100 to give percentages, and we can see that 30 successful outcomes has the highest probability at about 8.6%.

P of 10 success: 0.000%
P of 20 success: 0.758%
P of 30 success: 8.678%
P of 40 success: 0.849%
P of 50 success: 0.001%
P of 60 success: 0.000%
P of 70 success: 0.000%
P of 80 success: 0.000%
P of 90 success: 0.000%
P of 100 success: 0.000%

Given the probability of success is 30% for one trial, we would expect that a probability of 50 or fewer successes out of 100 trials to be close to 100%. We can calculate this with the cumulative distribution function, demonstrated below.

# example of using the cdf for the binomial distribution
from scipy.stats import binom
# define the parameters of the distribution
p = 0.3
k = 100
# define the distribution
dist = binom(k, p)
# calculate the probability of <=n successes
for n in range(10, 110, 10):
	print('P of %d success: %.3f%%' % (n, dist.cdf(n)*100))

Running the example prints each number of successes in [10, 100] in groups of 10 and the probability of achieving that many success or less over 100 trials.

As expected, after 50 successes or less covers 99.999% of the successes expected to happen in this distribution.

P of 10 success: 0.000%
P of 20 success: 1.646%
P of 30 success: 54.912%
P of 40 success: 98.750%
P of 50 success: 99.999%
P of 60 success: 100.000%
P of 70 success: 100.000%
P of 80 success: 100.000%
P of 90 success: 100.000%
P of 100 success: 100.000%

Multinoulli Distribution

The Multinoulli distribution, also called the categorical distribution, covers the case where an event will have one of K possible outcomes.

x in {1, 2, 3, …, K}

It is a generalization of the Bernoulli distribution from a binary variable to a categorical variable, where the number of cases K for the Bernoulli distribution is set to 2, K=2.

A common example that follows a Multinoulli distribution is:

A single roll of a die that will have an outcome in {1, 2, 3, 4, 5, 6}, e.g. K=6.

A common example of a Multinoulli distribution in machine learning might be a multi-class classification of a single example into one of K classes, e.g. one of three different species of the iris flower.

The distribution can be summarized with p variables from p1 to pK, each defining the probability of a given categorical outcome from 1 to K, and where all probabilities sum to 1.0.

P(x=1) = p1
P(x=2) = p1
P(x=3) = p3
…
P(x=K) = pK

In the case of a single roll of a die, the probabilities for each value would be 1/6, or about 0.166 or about 16.6%.

Multinomial Distribution

The repetition of multiple independent Multinoulli trials will follow a multinomial distribution.

The multinomial distribution is a generalization of the binomial distribution for a discrete variable with K outcomes.

An example of a multinomial process includes a sequence of independent dice rolls.

A common example of the multinomial distribution is the occurrence counts of words in a text document, from the field of natural language processing.

A multinomial distribution is summarized by a discrete random variable with K outcomes, a probability for each outcome from p1 to pK, and n successive trials.

We can demonstrate this with a small example with 3 categories (K=3) with equal probability (p=33.33%) and 100 trials.

Firstly, we can use the multinomial() NumPy function to simulate 100 independent trials and summarize the number of times that the event resulted in each of the given categories. The function takes both the number of trials and the probabilities for each category as a list.

The complete example is listed below.

# example of simulating a multinomial process
from numpy.random import multinomial
# define the parameters of the distribution
p = [1.0/3.0, 1.0/3.0, 1.0/3.0]
k = 100
# run a single simulation
cases = multinomial(k, p)
# summarize cases
for i in range(len(cases)):
	print('Case %d: %d' % (i+1, cases[i]))

We would expect each category to have about 33 events.

Running the example reports each case and the number of events.

A different random sequence of 100 trials will result each time the code is run, so your specific results will differ. Try running the example a few times.

In this case, we see a spread of cases as high as 37 and as low as 30.

Case 1: 37
Case 2: 33
Case 3: 30

We might expect the idealized case of 100 trials to result in 33, 33, and 34 cases for events 1, 2 and 3 respectively.

We can calculate the probability of this specific combination occurring in practice using the probability mass function or multinomial.pmf() SciPy function.

The complete example is listed below.

# calculate the probability for a given number of events of each type
from scipy.stats import multinomial
# define the parameters of the distribution
p = [1.0/3.0, 1.0/3.0, 1.0/3.0]
k = 100
# define the distribution
dist = multinomial(k, p)
# define a specific number of outcomes from 100 trials
cases = [33, 33, 34]
# calculate the probability for the case
pr = dist.pmf(cases)
# print as a percentage
print('Case=%s, Probability: %.3f%%' % (cases, pr*100))

Running the example reports the probability of less than 1% for the idealized number of cases of [33, 33, 34] for each event type.

Case=[33, 33, 34], Probability: 0.813%

Summary

In this tutorial, you discovered discrete probability distributions used in machine learning.

Specifically, you learned:

The probability of outcomes for discrete random variables can be summarized using discrete probability distributions.
A single binary outcome has a Bernoulli distribution, and a sequence of binary outcomes has a Binomial distribution.
A single categorical outcome has a Multinoulli distribution, and a sequence of categorical outcomes has a Multinomial distribution.

Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.

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