# Bayes Theory

We are given that Y has occurred, and we want to calculate the conditional probability of X C. Although machine C produces half of the total output, it produces a much smaller fraction of the defective items. The two main interpretations are described below.

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If the coin is flipped a number of times and the outcomes observed, that degree of belief may rise, fall or remain the same depending on the results. For more on the application of Bayes' theorem under the Bayesian interpretation of probability, see Bayesian inference. The two diagrams partition the same outcomes by A and B in opposite orders, to obtain the inverse probabilities.

Bayes' theorem serves as the link between these different partitionings. An entomologist spots what might be a rare subspecies of beetle , due to the pattern on its back. The rare subspecies accounts for only 0. In many applications, for instance in Bayesian inference , the event B is fixed in the discussion, and we wish to consider the impact of its having been observed on our belief in various possible events A. In such a situation the denominator of the last expression, the probability of the given evidence B , is fixed; what we want to vary is A. The posterior is proportional to the prior times the likelihood.

Denoting the constant of proportionality by c we have. For proposition A and evidence or background B , [4]. It is then useful to compute P B using the law of total probability :. In the special case where A is a binary variable :. However, terms become 0 at points where either variable has finite probability density.

## Bayes' Theorem -- from Wolfram MathWorld

A continuous event space is often conceptualized in terms of the numerator terms. It is then useful to eliminate the denominator using the law of total probability. For f Y y , this becomes an integral:. So the rule says that the posterior odds are the prior odds times the Bayes factor , or in other words, posterior is proportional to prior times likelihood. In short, posterior odds equals prior odds times likelihood ratio. The corresponding formula in terms of probability calculus is Bayes' theorem which in its expanded form is expressed as:.

Using the chain rule.

## Bayes' theorem

Price wrote an introduction to the paper which provides some of the philosophical basis of Bayesian statistics. In , he was elected a Fellow of the Royal Society in recognition of his work on the legacy of Bayes. The French mathematician Pierre-Simon Laplace reproduced and extended Bayes's results in , apparently unaware of Bayes's work. By modern standards, we should refer to the Bayes—Price rule.

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Loud arguments break out. Should you conduct another study? Or will that just waste more resources? How can we make sure our research has the right impact? The key part is how Bayesian reasoning deals with uncertainty. EX: Will a particular basketball team win a particular game? The calculation has two parts: an explicit statement of priors your assumptions about the likelihood of the event , and a procedure for updating that initial assumption as you learn new information. Exercise: Establishing and updating your priors Q1: Two teams are playing basketball.

What is the probability Team A wins the game? Q2: Two teams are playing basketball. Team A leads by 12 points at halftime. Q3: Two teams are playing basketball. Team A leads by 12 points with 2 minutes left in the game. So, what he asked his assistant to do was to throw another ball and tell him whether it landed to the left or to the right or in the front or behind of the first ball. This he would note down and then ask for more and more balls to be thrown on the table. What he realized was, that through this method he could keep updating his idea of where his first ball was.

But, of course he could never be completely certain but with each new piece of evidence he would get more and more accurate. I'm advising to watch the video below, it's cool anyway: Bayes theorem. The fundamental idea of Bayesian inference is to become "less wrong" with more data. The process is straightforward: we have an initial belief, known as a prior, which we update as we gain additional information. The conclusions drawn from the Bayes theorem are logical, but anti-intuitive. Almost always, people pay a lot of attention to the posterior probability, but they overlook the prior probability.

Bayes' Theorem - Hate it or Love it, can't ignore it!

Using this simple formula we already can construct some of the models but hold on to the flow.

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