Bayesian Theory

Bayesian theory, also known as Bayesian inference or Bayesian statistics, is a mathematical framework for updating beliefs or probabilities based on new evidence or data. It provides a principled approach to reasoning and making decisions under uncertainty by combining prior knowledge with observed evidence.

The core idea of Bayesian theory is to represent uncertain quantities, such as hypotheses, parameters, or predictions, as probability distributions. These distributions encode the degree of belief or uncertainty associated with different values or outcomes. Bayesian inference involves updating these distributions using Bayes’ theorem, which calculates the posterior probability of a hypothesis or parameter given the observed data.

The key elements of Bayesian theory include:

1. Prior Probability: The prior probability represents the initial belief or knowledge about a hypothesis or parameter before observing any data. It is often based on prior experience, expert knowledge, or previous data. The prior distribution encapsulates this prior belief and serves as the starting point for the Bayesian inference.
2. Likelihood Function: The likelihood function quantifies the probability of observing the data given a specific hypothesis or parameter value. It represents the likelihood of the observed data under different assumptions. The likelihood is typically derived from a statistical model that relates the data to the hypothesis or parameter of interest.
3. Posterior Probability: The posterior probability is the updated probability of a hypothesis or parameter after considering the observed data. It is calculated by combining the prior probability and the likelihood function using Bayes’ theorem. The posterior distribution reflects the updated belief or knowledge given the evidence.
4. Bayes’ Theorem: Bayes’ theorem provides the mathematical formula for updating the prior probability to obtain the posterior probability. It states that the posterior probability is proportional to the product of the prior probability and the likelihood function, with a normalization factor. Mathematically, Bayes’ theorem can be expressed as:

P(H | D) = (P(D | H) * P(H)) / P(D)

where: P(H | D) represents the posterior probability of hypothesis H given data D, P(D | H) is the likelihood of the data D given hypothesis H, P(H) is the prior probability of hypothesis H, P(D) is the probability of the observed data D.

Bayesian theory provides a coherent framework for updating beliefs as new evidence becomes available. It allows for the incorporation of prior knowledge, the quantification of uncertainty, and the iterative refinement of beliefs based on observed data. Bayesian inference can be applied to a wide range of problems, including parameter estimation, hypothesis testing, prediction, decision-making, and machine learning tasks. Its flexibility and interpretability make it a powerful tool for reasoning and analysis in various domains.

Bayes’ theorem:

Bayes’ theorem is also known as Bayes’ rule, Bayes’ law, or Bayesian reasoning, which determines the probability of an event with uncertain knowledge.

In probability theory, it relates the conditional probability and marginal probabilities of two random events.

Bayes’ theorem was named after the British mathematician Thomas Bayes. The Bayesian inference is an application of Bayes’ theorem, which is fundamental to Bayesian statistics.

It is a way to calculate the value of P(B|A) with the knowledge of P(A|B).

Bayes’ theorem allows updating the probability prediction of an event by observing new information of the real world.

Example: If cancer corresponds to one’s age then by using Bayes’ theorem, we can determine the probability of cancer more accurately with the help of age.

Bayes’ theorem can be derived using product rule and conditional probability of event A with known event B:

As from product rule we can write:

1. P(A ⋀ B)= P(A|B) P(B) or  

Similarly, the probability of event B with known event A:

1. P(A ⋀ B)= P(B|A) P(A)  

Equating right hand side of both the equations, we will get:

The above equation (a) is called as Bayes’ rule or Bayes’ theorem. This equation is basic of most modern AI systems for probabilistic inference.

It shows the simple relationship between joint and conditional probabilities. Here,

P(A|B) is known as posterior, which we need to calculate, and it will be read as Probability of hypothesis A when we have occurred an evidence B.

P(B|A) is called the likelihood, in which we consider that hypothesis is true, then we calculate the probability of evidence.

P(A) is called the prior probability, probability of hypothesis before considering the evidence

P(B) is called marginal probability, pure probability of an evidence.

In the equation (a), in general, we can write P (B) = P(A)*P(B|Ai), hence the Bayes’ rule can be written as:

Where A₁, A₂, A₃,…….., A_n is a set of mutually exclusive and exhaustive events.

Applying Bayes’ rule:

Bayes’ rule allows us to compute the single term P(B|A) in terms of P(A|B), P(B), and P(A). This is very useful in cases where we have a good probability of these three terms and want to determine the fourth one. Suppose we want to perceive the effect of some unknown cause, and want to compute that cause, then the Bayes’ rule becomes:

Example-1:

Question: what is the probability that a patient has diseases meningitis with a stiff neck?

Given Data:

A doctor is aware that disease meningitis causes a patient to have a stiff neck, and it occurs 80% of the time. He is also aware of some more facts, which are given as follows:

• The Known probability that a patient has meningitis disease is 1/30,000.
• The Known probability that a patient has a stiff neck is 2%.

Let a be the proposition that patient has stiff neck and b be the proposition that patient has meningitis. , so we can calculate the following as:

P(a|b) = 0.8

P(b) = 1/30000

P(a)= .02

Hence, we can assume that 1 patient out of 750 patients has meningitis disease with a stiff neck.

Application of Bayes’ theorem in Artificial intelligence:

Following are some applications of Bayes’ theorem:

• It is used to calculate the next step of the robot when the already executed step is given.
• Bayes’ theorem is helpful in weather forecasting.
• It can solve the Monty Hall problem.

Books on Bayesian Theory