## Bayes' theorem

Bayes' theorem is a valuable instrument that allows for the computation of conditional probability. This formula centres around evaluating the mutual occurrence of two events, which is then divided by the likelihood of the second event's occurrence. This method of calculating conditional probability via Bayes' theorem is also effective for machine learning.

To utilize Bayes' theorem for computing conditional probability, adhere to these steps:

- Assume event A is valid, and deduce the probability of event B being true as well.
- Compute the probability of event A's occurrence.
- Multiply these two probabilities to get the desired outcome.
- Deduct the probability of event B from the total sum.

This process is particularly valuable when it is easier to calculate the conditional probability.

The Naive Bayes approach for classification and regression problem-solving in various fields is a common application of Bayes' theorem.

## Applications of Bayes' Theorem

In machine learning, the Naive Bayes method stands out as a frequently used application of the Bayes theorem. It is used often for natural language or Bayesian analysis tool processing.

Naive Bayes is based on the assumption that the evidence/attribute values – Bs in P(B1, B2, B3*A) – are independent. The assumption of independence simplifies computational complexity, even if it's not entirely accurate. Despite this, Naive Bayes' performance is satisfactory for classifications.

Several versions of the Naive Bayes classifier, such as the Multinomial, Bernoulli, and Gaussian, are widely employed in classification.

Multinomial Naive Bayes is a viable option for text classifications due to its ability in counting word frequency in documentation. The Bernoulli variant works similarly, but the outcomes are boolean, making it useful for determining whether particular words exist in a text. For continuous predictor/feature values, Gaussian Naive Bayes is used, where values are considered part of a Gaussian distribution.

## Bayes' Theorem in Real-World Use

Consider a scenario where you're participating in a game that requires you to distinguish who among the players is being truthful. In this case, Bayes' theorem can be used to predict behaviors denoted by categorical variables P1, P2, P3, and P4 and discern whether each participant is lying or telling the truth.

To determine if someone is untruthful, several indicators can help, just like in a poker game. In this context, lying can be denoted as L. Aim to predict the probability of a lie based on observed behavior where L indicates lying. Repeat this exercise for all behaviors observed and for all game participants.

New information intake effectively revises our probabilistic model, particularly the prior probabilities – a process commonly referred to as updating prior probabilities.