Bayes Rule Formula

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[1] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.

Independently of Bayes, Pierre-Simon Laplace in 1774, and later in his 1812 Thorie analytique des probabilits, used conditional probability to formulate the relation of an updated posterior probability from a prior probability, given evidence. He reproduced and extended Bayes's results in 1774, apparently unaware of Bayes's work.[note 1][9] The Bayesian interpretation of probability was developed mainly by Laplace.[10]

Bayes Rule Formula

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About 200 years later, Sir Harold Jeffreys put Bayes's algorithm and Laplace's formulation on an axiomatic basis, writing in a 1973 book that Bayes' theorem "is to the theory of probability what the Pythagorean theorem is to geometry".[2]

Bayes' rule and computing conditional probabilities provide a solution method for a number of popular puzzles, such as the Three Prisoners problem, the Monty Hall problem, the Two Child problem and the Two Envelopes problem.

Once again, the answer can be reached without using the formula by applying the conditions to a hypothetical number of cases. For example, if the factory produces 1,000 items, 200 will be produced by Machine A, 300 by Machine B, and 500 by Machine C. Machine A will produce 5% 200 = 10 defective items, Machine B 3% 300 = 9, and Machine C 1% 500 = 5, for a total of 24. Thus, the likelihood that a randomly selected defective item was produced by machine C is 5/24 (~20.83%).

The interpretation of Bayes' rule depends on the interpretation of probability ascribed to the terms. The two predominant interpretations are described below. Figure 2 shows a geometric visualization.

The corresponding formula in terms of probability calculus is Bayes' theorem, which in its expanded form involving the prior probability/base rate a {\displaystyle a} of only A {\displaystyle A} , is expressed as:[23]

The Bayes theorem is a mathematical formula for calculating conditional probability in probability and statistics. In other words, it's used to figure out how likely an event is based on its proximity to another. Bayes law or Bayes rule are other names for the theorem.

Bayes theorem is a theorem in probability and statistics, named after the Reverend Thomas Bayes, that helps in determining the probability of an event that is based on some event that has already occurred. Bayes rule has many applications such as Bayesian interference, in the healthcare sector - to determine the chances of developing health problems with an increase in age and many others.

The Bayes theorem is based on finding P(A | B) when P(B | A) is given. Here, we will aim at understanding the use of the Bayes rule in determining the probability of events, its statement, formula, and derivation with the help of examples.

Bayes Law is a method to determine the probability of an event based on the occurrences of prior events. It is used to calculate conditional probability. Bayes theorem calculates the probability based on the hypothesis. Now, let us state and prove Bayes Theorem. Bayes rule states that the conditional probability of an event A, given the occurrence of another event B, is equal to the product of the likelihood of B, given A and the probability of A divided by the probability of B. It is given as:

Bayes formula exists for events and random variables. Bayes theorem formulas are derived from the definition of conditional probability. It can be derived for events A and B, as well as continuous random variables X and Y. Let us first see the formula for events.

Bayes theorem is a statistical formula to determine the conditional probability of an event. It describes the probability of an event based on prior knowledge of events that have already happened. Bayes rule is named after the Reverend Thomas Bayes and Bayesian probability formula for random events is $P(A|B) = \dfrac{P(B|A)P(A)}{P(B)}$, where

To determine the probability of an event A given that the related event B has already occurred, that is, P(A|B) using the Bayes Theorem, we calculate the probability of the event B, that is, P(B); the probability of event B given that event A has occurred, that is, P(B|A); and the probability of the event A individually, that is, P(A). Then, we substitute these values into the Bayes formula $P(A|B) = \dfrac{P(B|A)P(A)}{P(B)}$ to determine the probability.

Bayes theorem provides a method to determine the probability of a hypothesis based on its prior probability, the probabilities of observing various data given the hypothesis, and the observed data itself. It helps immensely in getting a more accurate result. Hence, whenever there is a conditional probability problem, the Bayes rule in Machine Learning is used.

Bayes' theorem is a mathematical identity which we canderive ourselves. Start with the definition of conditional probability and then expand the $\and$ term using the chain rule:$$\begin{align}\p(F|E) &= \frac{\p(F \and E)}{\p(E)} && \text{Def of }\href{ ../../part1/cond_prob/}{\text{conditional probability}} \\&= \frac{\p(E | F) \cdot \p(F)}{\p(E)} && \text{Substitute the }\href{ ../../part1/cond_prob/#chain_rule}{\text{chain rule}} \text{ for $\p(F \and E)$}\end{align}$$This theorem makes no assumptions about $E$ or $F$ so it will apply for any two events. Bayes' theorem is exceptionally useful because it turns out to be the ubiquitous way to answer the question: "how can I update a belief about something, which is not directly observable, given evidence." This is for good reason. For many "noisy" measurements it is straightforward to estimate the probability of the noisy observation given the true state of the world. However, what you would really like to know is the conditional probability the other way around: what is the probability of the true state of the world given evidence. There are countless real world situations that fit this situation:

There are names for the different terms in the Bayes' Rule formula. The term $\p(B|E)$ is often called the"posterior": it is your updated belief of $B$ after you take into account evidence $E$. The term $\p(B)$ is often called the "prior": it was your belief before seeing any evidence. The term $\p(E|B)$ is called the update and $\p(E)$ isoften called the normalization constant.

There are several techniques for handling the case where the denominator is not known. One technique is to use the law of total probability to expand out the term, resulting in another formula, called Bayes' Theorem with Law of Total Probability:$$\p(B|E) = \frac{\p(E | B) \cdot \p(B)}{\p(E|B)\cdot \p(B) + \p(E|B\c) \cdot \p(B\c)} $$

The numbers in this example are from the Mammogram test for breast cancer. The seriousness of cancer underscores the potential for bayesian probability to be applied to important contexts. The natural occurrence of breast cancer is 8%. The mammogram test returns a positive result 95% of the time for patients who have breast cancer. The test resturns a positive result 7% of the time for people who do not have breast cancer. In this demo you can enter different input numbers and it will recalculate.

We can also generalize the likelihood ratio by settingLR(H, H*; E) =PH(E)/PH*(E).This compares E's predictability on the basis of Hwith its predictability on the basis of H*. We can use thesetwo quantities to formulate an even more general form of Bayes'Theorem.

A variety of arguments for conditioning (simple or Jeffrey-style) canbe found in the literature, but we cannot consider them here.[16] There is, however, one sort of justification in which Bayes' Theoremfigures prominently. It exploits connections between belief revisionand the notion of incremental evidence to show that conditioning isthe only belief revision rule that allows learners tocorrectly proportion their posterior beliefs to the new evidence theyreceive.

This requires an agent to retain his views about the relativeprobability of two hypotheses when he acquires evidence that supportsthe more probable hypothesis more strongly. It rules out obviouslyirrational belief revisions such as this: George is more confidentthat the New York Yankees will win the American League Pennant than heis that the Boston Rex Sox will win it, but he reverses himself whenhe learns (only) that the Yankees beat the Red Sox in last night'sgame.

This article connects to our class discussion mainly through the utilization of Bayes Theorem to describe the probability of an uncertain event, based on probabilities of conditions related to said event. Though not explicitly explained in the above article, the calculation Price did is as follows and is based on the formula extensively discussed in class:

Price made broad assumptions in his calculation, which of course would not be acceptable in current uses of Bayes Theorem when it comes to data analysis and medicine and the examples we did in class. However, the lack of concrete related probabilities in his philosophical argument made it necessary to do so. His usage of the rule has thus stood the test of time as he was the first person ever to apply it.

Bayes' formula is an important method for computing conditional probabilities. It is often used to compute posterior probabilities (as opposed to priorior probabilities) given observations. For example, a patient is observed to have a certain symptom, and Bayes' formula can be used to compute the probability that a diagnosis is correct, given that observation. We illustrate this idea with details in the following example: 2351a5e196