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Bayesian Machine Learning for Optimization in Python

Advantages of Bayes’ Theorem over Deterministic Statistics

Explore the advantages of Bayes theorem over deterministic statistics. Learn how it incorporates prior knowledge, quantifies uncertainty, and allows continuous updates with new evidence. Gain insight into applying Bayesian methods to improve predictions and better handle uncertainty in data-driven problems.

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Bayes’ theorem is a statistical method used to update the probability of an event occurring after considering new evidence. Bayes’ theorem is used in various fields, including machine learning, natural language processing, and data analysis.

There are several advantages to using Bayes’ theorem over deterministic statistics:

Incorporation of prior knowledge

In Bayesian statistics, prior knowledge or beliefs about the probability of an event can be incorporated into the analysis. This is in contrast to deterministic statistics, which only consider the current data and do not consider any prior knowledge.

Example

Imagine that we are trying to predict the reliability of a particular software system. We have data on the reliability of similar systems, but we also have some prior knowledge about the distribution of reliability values based on our experience with comparable systems. In a Bayesian analysis, we could use this prior knowledge to inform our predictions about the system’s reliability.

Here is an example of how we can do this using Python:

Python 3.10.4
import scipy.stats as stats
import matplotlib.pyplot as plt
import numpy as np
# Define the prior distribution for the reliability of the system
prior = stats.beta(a=5, b=2)
noise=stats.beta(a=5,b=0.1)
# Plot the prior distribution
x = np.linspace(prior.ppf(0.01), prior.ppf(0.99), 100)
plt.plot(x, prior.pdf(x), 'r-', lw=2, alpha=0.6, label='prior')
# Generate some synthetic data
data = prior.rvs(size=100)+noise.rvs(size=100)
# Use Bayes theorem to update the prior distribution based on the data
posterior = stats.beta.fit(data)
# Convert the posterior distribution to a beta distribution object
posterior_dist = stats.beta(*posterior)
# Plot the posterior distribution
x = np.linspace(posterior_dist.ppf(0.01), posterior_dist.ppf(0.99), 100)
plt.plot(x, posterior_dist.pdf(x), 'b-', lw=1, alpha=0.6, label='posterior')
plt.legend()
plt.xlabel("Value of x")
plt.ylabel("Probability Distribution Function")
plt.title("Update of PDF on the basis of new information")
plt.show()
plt.savefig("output/prob.png")

In this example, the prior distribution represents the uncertainty about the system’s reliability based on our prior knowledge of similar systems as defined in line 5 with a=5 and b=1. Using Bayes’ theorem to update the initial ...