Bayes' theorem - Revision history

Aschlafly: Presbyterian minister

2021-03-31T01:36:03Z

Presbyterian minister

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2018-08-03T02:09:29Z

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2018-08-03T02:00:24Z

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Aschlafly at 01:45, August 3, 2018

2018-08-03T01:45:33Z

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2018-08-03T01:44:45Z

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Aschlafly: While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the Enigma code of the Nazis during World War II.

2018-08-03T01:43:52Z

While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the Enigma code of the Nazis during World War II.

DavidB4-bot: /* Applications and Examples */clean up & uniformity

2016-07-13T02:59:41Z

Applications and Examples: clean up & uniformity

Whizkid at 19:21, February 8, 2010

2010-02-08T19:21:21Z

TK at 11:40, February 8, 2010

2010-02-08T11:40:55Z

Aschlafly: Reverted edits by ConservaTroll (Talk) to last version by Foxtrot

2008-10-15T13:44:21Z

Reverted edits by ConservaTroll (Talk) to last version by Foxtrot

← Older revision		Revision as of 01:36, March 31, 2021
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−	'''Bayes' Theorem''', also known as Bayes' Rule, is used in [[statistics]] and [[probability]] theory to relate marginal probabilities and conditional probabilities. In the context of Bayesian probability theory, it is used to update degrees of belief (probabilities) given new information. The theorem was first developed by the Reverend Thomas Bayes (1702-1761), an English nonconformist minister, and later improved by the French mathematician Pierre Laplace.	+	'''Bayes' Theorem''', also known as Bayes' Rule, is used in [[statistics]] and [[probability]] theory to relate marginal probabilities and conditional probabilities. In the context of Bayesian probability theory, it is used to update degrees of belief (probabilities) given new information. The theorem was first developed by the Reverend Thomas Bayes (1702-1761), an English nonconformist [[Presbyterian]] minister, and later improved by the French mathematician Pierre Laplace.

	While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the [[Enigma]] code of the [[Nazi]]s during [[World War II]].<ref name="Bayes">https://www.bayestheorem.net/history/</ref>		While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the [[Enigma]] code of the [[Nazi]]s during [[World War II]].<ref name="Bayes">https://www.bayestheorem.net/history/</ref>

@@ Line 3: / Line 3: @@
 While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the [[Enigma]] code of the [[Nazi]]s during [[World War II]].<ref name="Bayes">https://www.bayestheorem.net/history/</ref>
-Rev. Bayes' interest this is summed as follows:
+Rev. Bayes' interest in this was follows:
 {{cquote|Apart from his [[faith]], Bayes had a deep love and interest in mathematics and was considered an amateur mathematician. In his later years, he became fascinated with probability, specifically inverse probability. No one knows why for sure, but one thing is clear: Bayes became consumed with figuring out ... the approximate probability of a future event he knew nothing about except its past, that is, the number of times it had occurred or failed to occur.}}
 == Theorem ==

@@ Line 1: / Line 1: @@
 '''Bayes' Theorem''', also known as Bayes' Rule, is used in [[statistics]] and [[probability]] theory to relate marginal probabilities and conditional probabilities.  In the context of Bayesian probability theory, it is used to update degrees of belief (probabilities) given new information. The theorem was first developed by the Reverend Thomas Bayes (1702-1761), an English nonconformist minister, and later improved by the French mathematician Pierre Laplace.
-While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the [[Enigma]] code of the [[Nazi]]s during [[World War II]].<ref>https://www.bayestheorem.net/history/</ref>
+While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the [[Enigma]] code of the [[Nazi]]s during [[World War II]].<ref name="Bayes">https://www.bayestheorem.net/history/</ref>
 == Theorem ==

@@ Line 20: / Line 20: @@
 In a scientific context, ''X'' may be a [[hypothesis]], and ''Y'' may be experimental [[data]].  The theorem can then be used to determine the degree of belief in the hypothesis by using the experimental data.
 == Applications and Examples ==
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                <math>p(\theta|y) = \frac{p(\theta)p(y|\theta)}{p(y)}</math>
 [[Category:Probability and Statistics]]

@@ Line 11: / Line 11: @@
 where X and Y are independent statements, I is available background information, and
-<math>P(X|Y,I)</math> is the [[posterior probability]] for ''X'' given ''Y'' and ''I'',
+<math>P(X|Y,I)</math> is the posterior probability for ''X'' given ''Y'' and ''I'',
-<math>P(Y|X,I)</math> is the [[likelihood]] for ''Y'' given ''X'' and ''I'',
+<math>P(Y|X,I)</math> is the likelihood for ''Y'' given ''X'' and ''I'',
-<math>P(X|I)</math> is the [[prior probability]] for ''X'' given only ''I'', and
+<math>P(X|I)</math> is the prior probability for ''X'' given only ''I'', and
-<math>P(Y|I)</math> is sometimes called the [[evidence (statistics)|evidence]] or probability for ''Y'' given only ''I''.
+<math>P(Y|I)</math> is sometimes called the evidence or probability for ''Y'' given only ''I''.
 In a scientific context, ''X'' may be a [[hypothesis]], and ''Y'' may be experimental [[data]].  The theorem can then be used to determine the degree of belief in the hypothesis by using the experimental data.

← Older revision		Revision as of 01:43, August 3, 2018
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−	'''Bayes' Theorem''', also known as Bayes' Rule, is used in [[statistics]] and [[probability]] theory to relate marginal probabilities and conditional probabilities. In the context of [[Bayesian probability theory]], it is used to update degrees of belief (probabilities) given new information. The theorem ~~is attributed to~~ the Reverend [[Thomas Bayes]] (1702-1761) an English nonconformist minister, ~~but it is likely to have been known to earlier mathematicians such as [[~~Pierre Laplace]].	+	'''Bayes' Theorem''', also known as Bayes' Rule, is used in [[statistics]] and [[probability]] theory to relate marginal probabilities and conditional probabilities. In the context of Bayesian probability theory, it is used to update degrees of belief (probabilities) given new information. The theorem was first developed by the Reverend Thomas Bayes (1702-1761), an English nonconformist minister, and later improved by the French mathematician Pierre Laplace.

		+	While held in disdain by some mathematicians for years, Bayes Theorem is credited with helping decipher the [[Enigma]] code of the [[Nazi]]s during [[World War II]].<ref>https://www.bayestheorem.net/history/</ref>

	== Theorem ==		== Theorem ==

← Older revision		Revision as of 02:59, July 13, 2016
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−	[[~~category~~:Probability and Statistics]]	+	[[Category:Probability and Statistics]]

@@ Line 1: / Line 1: @@
-'''Bayes' theorem''' is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous [[mathematics|mathematical]] way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].
+== Theorem ==
-Let A and B be two uncertain events. We denote the probability of A by p(A) and probability of B by p(B). We denote the event that A and B are both true by AB, and the probability of AB by p(AB). In general, the fact that A has already occurred will affect the chances that B will occur, and vice versa. We denote the probability that B will occur, given that A has occurred, by p(B|A). We call this the probability of B, conditional on A. The event AB can occur in two ways. A can occur first, then B occurs, or B occurs first and then A occurs. The rules of probability require that the probability of two independent events both occurring is the product of the probabilities. Thus
+Bayes' theorem is stated mathematically as
-                     p(AB) = p(A)p(B|A) = p(B)p(A|B)
+where X and Y are independent statements, I is available background information, and
-This can be rearranged to be
+<math>P(X|Y,I)</math> is the [[posterior probability]] for ''X'' given ''Y'' and ''I'',
-                             p(B|A) = p(B)p(A|B)/p(A)
+<math>P(Y|X,I)</math> is the [[likelihood]] for ''Y'' given ''X'' and ''I'',
-This is Bayes' theorem. The theorem is attributed to the Reverend [[Thomas Bayes]] (1702-1761) an English nonconformist minister, but it is likely to have been known to earlier mathematicians such as [[Pierre Laplace]].
+<math>P(X|I)</math> is the [[prior probability]] for ''X'' given only ''I'', and
-The theorem has many applications.
+<math>P(Y|I)</math> is sometimes called the [[evidence (statistics)|evidence]] or probability for ''Y'' given only ''I''.
-Example 1.
+In a scientific context, ''X'' may be a [[hypothesis]], and ''Y'' may be experimental [[data]].  The theorem can then be used to determine the degree of belief in the hypothesis by using the experimental data.
 Suppose a particular disease afflicts 1% of the population. Suppose that a test for the disease is 95% accurate. Suppose that someone tests positive for the disease but there is no other evidence that they have the disease. What is the probability that they have the disease?
-Let A be the event that the test result is positive. Let B be the event that the person actually has the disease.
+Let X be the event that the test result is positive. Let Y be the event that the person actually has the disease.
-Before the test result is known our probability for the person having the disease is p(B) = 1% = 0.01. The probability that the person has a positive test result, giving that they have the disease, is 95% = 0.95. The denominator term, p(A) is a little more complex since A can occur in two different ways. If the person has the disease, they will test positive with probability 0.95. If the person does not have the disease they will test positive with probability 5%. We denote the probability that B is not true by B'. The laws of probability require that B' = 1 - B. Thus
+Before the test result is known our probability for the person having the disease is p(Y) = 1% = 0.01. The probability that the person has a positive test result, giving that they have the disease, is 95% = 0.95. The denominator term, p(X) is a little more complex since X can occur in two different ways. If the person has the disease, they will test positive with probability 0.95. If the person does not have the disease they will test positive with probability 5%. We denote the probability that Y is not true by Y'. The laws of probability require that Y' = 1 - Y. Thus
+<math> P(X|Y,I)=\frac{P(Y|X,I)P(X|I)}{P(Y|I)} = (0.01)(0.95) + (0.99)(0.05) = 0.059 </math>
 Hence
-p(B|A) 0.01 x 0.95/0.059 = 0.16
+<math>\frac{P(X|Y)(0.01)(0.95)}{0.059} = 0.16</math>
 In other words, there is only a 16% chance that a person testing positive actually has the disease.
-An extended form of Bayes's theorem is obtained by noting that it applies to [[probability distribution]]s as well as to events. Let ''y'' be a (vector valued) observable quantity that we want to use to estimate some unknown, unobservable (vector valued) quantity <math>\theta</math>. Prior to seeing the data ''y'', we summarise our knowledge about <math>\theta</math> by a probability distribution <math>p(\theta)</math>. Assume that we have a model of the relationship between ''y'' and <math>\theta</math>. Call this <math>p(y|\theta)</math>. We can use Bayes' theorem to update our knowledge of <math>\theta</math> by incorporating the information contained in the observed data ''y''.
+An extended form of Bayes's theorem is obtained by noting that it applies to [[probability distribution]]s as well as to events. Let ''y'' be a (vector valued) observable quantity that we want to use to estimate some unknown, unobservable (vector valued) quantity <math>\theta</math>. Prior to seeing the data ''y'', we summarize our knowledge about <math>\theta</math> by a probability distribution <math>p(\theta)</math>. Assume that we have a model of the relationship between ''y'' and <math>\theta</math>. Call this <math>p(y|\theta)</math>. We can use Bayes' theorem to update our knowledge of <math>\theta</math> by incorporating the information contained in the observed data ''y''.
 We have
-               <math>p(\theta|y) = p(\theta)p(y|\theta)/p(y)</math>
+               <math>p(\theta|y) = \frac{p(\theta)p(y|\theta)}{p(y)}</math>
 [[category:Probability and Statistics]]

← Older revision		Revision as of 11:40, February 8, 2010
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−	~~{{merge\|Bayes theorem}}~~	+

	'''Bayes' theorem''' is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous [[mathematics\|mathematical]] way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].		'''Bayes' theorem''' is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous [[mathematics\|mathematical]] way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].

← Older revision		Revision as of 13:44, October 15, 2008
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−	==~~SERVER ERROR 404~~- ~~PAGE NOT FOUND~~==	+	{{merge\|Bayes theorem}}
		+
		+	'''Bayes' theorem''' is a result in [[probability theory]] that shows how to invert a [[conditional probability]]. An extended form of the theorem provides a rigorous [[mathematics\|mathematical]] way of learning from uncertain data. This latter form is the basis of [[Bayesian statistics]].
		+
		+	Let A and B be two uncertain events. We denote the probability of A by p(A) and probability of B by p(B). We denote the event that A and B are both true by AB, and the probability of AB by p(AB). In general, the fact that A has already occurred will affect the chances that B will occur, and vice versa. We denote the probability that B will occur, given that A has occurred, by p(B\|A). We call this the probability of B, conditional on A. The event AB can occur in two ways. A can occur first, then B occurs, or B occurs first and then A occurs. The rules of probability require that the probability of two independent events both occurring is the product of the probabilities. Thus
		+
		+
		+	p(AB) = p(A)p(B\|A) = p(B)p(A\|B)
		+
		+	This can be rearranged to be
		+
		+	p(B\|A) = p(B)p(A\|B)/p(A)
		+
		+	This is Bayes' theorem. The theorem is attributed to the Reverend [[Thomas Bayes]] (1702-1761) an English nonconformist minister, but it is likely to have been known to earlier mathematicians such as [[Pierre Laplace]].
		+
		+	The theorem has many applications.
		+
		+	Example 1.
		+
		+	Suppose a particular disease afflicts 1% of the population. Suppose that a test for the disease is 95% accurate. Suppose that someone tests positive for the disease but there is no other evidence that they have the disease. What is the probability that they have the disease?
		+
		+	Let A be the event that the test result is positive. Let B be the event that the person actually has the disease.
		+
		+	Before the test result is known our probability for the person having the disease is p(B) = 1% = 0.01. The probability that the person has a positive test result, giving that they have the disease, is 95% = 0.95. The denominator term, p(A) is a little more complex since A can occur in two different ways. If the person has the disease, they will test positive with probability 0.95. If the person does not have the disease they will test positive with probability 5%. We denote the probability that B is not true by B'. The laws of probability require that B' = 1 - B. Thus
		+
		+	p(A) = p(B)p(A\|B) + p(B')p(A\|B') = 0.01 x 0.95 + 0.99 x 0.05 = 0.059
		+
		+	Hence
		+
		+	p(B\|A) 0.01 x 0.95/0.059 = 0.16
		+
		+	In other words, there is only a 16% chance that a person testing positive actually has the disease.
		+
		+	An extended form of Bayes's theorem is obtained by noting that it applies to [[probability distribution]]s as well as to events. Let ''y'' be a (vector valued) observable quantity that we want to use to estimate some unknown, unobservable (vector valued) quantity <math>\theta</math>. Prior to seeing the data ''y'', we summarise our knowledge about <math>\theta</math> by a probability distribution <math>p(\theta)</math>. Assume that we have a model of the relationship between ''y'' and <math>\theta</math>. Call this <math>p(y\|\theta)</math>. We can use Bayes' theorem to update our knowledge of <math>\theta</math> by incorporating the information contained in the observed data ''y''.
		+
		+	We have
		+
		+	<math>p(\theta\|y) = p(\theta)p(y\|\theta)/p(y)</math>
		+	[[category:Probability and Statistics]]