Bayesian statistics has gained significant popularity in the field of statistics, especially in machine learning. This concept is extensively used in various predictive modeling techniques, as it incorporates probabilistic principles. Bayesian approaches are particularly useful when dealing with events that are conditionally dependent on each other. In this article, we will delve into the fundamentals of Bayesian statistics, explore its relevance to the Bayes theorem, and conduct practical experiments in Python to calculate event probabilities using this approach.
Unraveling the Bayes Theorem
The Bayes theorem serves as a framework to determine the probability of an event based on prior knowledge. It enables us to compute the conditional probability of an event by incorporating both available data and existing prior information associated with the event’s conditions. For example, in the context of statistical models like logistic regression, the Bayes theorem can be employed to estimate the model’s parameters effectively.
Since Bayesian statistics treats probabilities as a measure of belief, it allows direct assignment of probability distributions to parameters, quantifying them using the Bayes theorem. Mathematically, the Bayes theorem can be expressed as follows:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this formula, A and B represent two events. To illustrate this concept, let’s consider an example involving two bags, A and B, each containing different colored balls. If we draw a red ball, we can utilize the Bayes theorem to calculate the probability of it being drawn from bag A. Essentially, the Bayes theorem determines the probability of a prior event given that a posterior event has already occurred.
Key Components of the Bayes Theorem
To comprehend the Bayes theorem more thoroughly, it’s essential to understand its core components: prior probability, likelihood function, and posterior probability.
1. Prior Probability
In the Bayes theorem formula, P(A) represents the prior probability of event A. The prior probability can be defined as the probability distribution of an uncertain quantity, which reflects one’s belief about that quantity without considering any evidence. For instance, when distributing a large number of balls into buckets, a probability distribution can be considered as prior if it represents the proportion of balls allocated to a particular bucket. In the context of statistical models, an unknown quantity, such as a model parameter, can be regarded as an example of an uncertain quantity. The prior probability, denoted as P(A), reflects one’s belief in the occurrence of event A before taking any evidence into account.
2. Likelihood Function
The likelihood function, denoted as P(B|A), is a crucial element in the Bayes theorem. It represents the probability distribution of the observed data in the context of a statistical model. More simply, it indicates the probability of event B occurring when event A is known to be true. Numerically, the likelihood function expresses the support value provided by evidence B for proposition A.
3. Posterior Probability
The posterior probability, denoted as P(A|B), is the conditional probability of a random event or uncertain proposition when relevant evidence is available and taken into account. According to the Bayes theorem, the posterior probability signifies the probability of event A given that evidence B has been considered. Mathematically, it can be expressed as the product of the likelihood probability, prior probability, and the inverse of the probability of evidence B being true.
Practical Implementation of Bayesian Statistics in Python
In the following section, we will employ a Python example to gain practical insights into the fundamental concepts of Bayesian statistics.
Consider a scenario where we have two buckets, A and B. Bucket A contains 30 blue balls and 10 yellow balls, while bucket B contains 20 blue balls and 20 yellow balls. Our objective is to select one ball, and we want to determine the probability of choosing bucket A.
Let’s solve this problem using Python:
# Importing libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
# Defining hypotheses and priors
hypos = 'bucket a', 'bucket b'
probs = 1/2, 1/2
prior = pd.Series(probs, hypos)
# Displaying prior probabilities
print(prior)
Output:
bucket a 0.5
bucket b 0.5
dtype: float64
In the code above, we defined the hypotheses (bucket a and bucket b) and their corresponding prior probabilities. The prior probabilities are represented as a pandas series, which allows us to create a discrete probability distribution.
To calculate the likelihood, we know that the chances of selecting a blue ball from bucket A are 3/4, and from bucket B, the chances of selecting any ball or a blue ball are 1/2. We can express these probabilities as follows:
# Defining likelihood
likelihood = 3/4, 1/2
# Calculating unnormalized posterior
unnorm = prior * likelihood
# Displaying unnormalized posterior
print(unnorm)
Output:
bucket a 0.375
bucket b 0.250
dtype: float64
By combining the likelihood and prior, we can obtain the unnormalized posterior. To normalize the posterior, we divide the unnormalized posterior by the sum of its values:
# Calculating the sum of unnormalized posterior
prob_data = unnorm.sum()
# Calculating the normalized posterior
posterior = unnorm / prob_data
# Displaying the normalized posterior
print(posterior)
Output:
cssCopy codebucket a 0.600
bucket b 0.400
dtype: float64
From the results, we can conclude that the posterior probability of choosing bucket A, given a blue ball selection, is 0.6. This implementation aligns with the principles of the Bayes theorem.
We can further explore variations of this problem. For example, let’s assume we repeat the process of selecting a ball from the same bucket, putting the ball back, and choosing again. This time, we want to determine the probability of selecting bucket A in both attempts. In this case, we can reuse the posterior obtained from the first problem as the prior for the subsequent problem:
# Reusing the previous posterior as the prior
prior = posterior
# Defining the likelihood for the new problem
likelihood = 3/4, 1/2
# Calculating the unnormalized posterior for the new problem
unnorm = prior * likelihood
# Calculating the normalized posterior for the new problem
posterior = unnorm / unnorm.sum()
# Displaying the posterior probabilities
print(posterior)
Output:
bucket a 0.428571
bucket b 0.571429
dtype: float64
The posterior probability for bucket A in the second attempt is approximately 0.428571.
Let’s now consider a more complex scenario where we have 101 buckets, and each bucket has a different distribution of blue balls. Bucket 0 has 0 blue balls, while bucket 1 has 1% blue balls, bucket 2 has 2% blue balls, and so on, up to bucket 99 with 99% blue balls and bucket 100 with 100% blue balls.
To solve this problem, we can create a series of numbers ranging from 0 to 100, evenly spaced between each other. We can treat these numbers as fractions representing the fraction of blue balls in each bucket.
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# Generating series of numbers representing fractions of blue balls
xs = np.linspace(0, 1, num=101)
prob = 1 / 101
prior = pd.Series(prob, xs)
# Displaying the prior distribution
print(prior)
Output:
0.00 0.009901
0.01 0.009901
0.02 0.009901
...
0.99 0.009901
1.00 0.009901
dtype: float64
In the code above, we generated a series of numbers from 0 to 1, with 101 equally spaced values. We then assigned a uniform prior probability (0.009901) to each value, representing the probability of selecting each bucket.
Final Thoughts
In this article, we explored the Bayes theorem, a fundamental component of Bayesian statistics. We gained insights into the prior probability, likelihood function, and posterior probability, which play crucial roles in Bayesian analysis. Moreover, we conducted practical experiments in Python to calculate probabilities based on different scenarios using Bayesian statistics. Bayesian statistics finds applications in various fields, such as survival analysis, statistical modeling, and parameter estimation. We encourage readers to further explore this powerful statistical analysis approach and its real-life applications, as it often provides accurate and straightforward results.