Bayes Theorem - Revision history

Dirkjan at 19:53, 27 August 2020

2020-08-27T19:53:47Z

Dirkjan: Replaced content with "Bayes Theorem Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

2020-08-27T19:18:14Z

Replaced content with "Bayes Theorem Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

Dirkjan: Created page with "Bayes Theorem proposes to look at the end of a sequence of events, so we know the outcome, and we wish to find the probability of a certain event in the sequence (posterior pr..."

2020-08-25T07:57:29Z

Created page with "Bayes Theorem proposes to look at the end of a sequence of events, so we know the outcome, and we wish to find the probability of a certain event in the sequence (posterior pr..."

New page

Bayes Theorem proposes to look at the end of a sequence of events, so we know the outcome, and we wish to find the probability of a certain event in the sequence (posterior probability). As an example say a production line has 3 machines A, B and C.
{| class="wikitable"
!A
!B
!C
|-
|40%
|50%
|10%
|-
|}If we see defective items it is not certain from which machine they came. With Bayes theorem it is possible to identify the most likely machine and check that one first. Suppose the waste rates are:
{| class="wikitable"
!A
!B
!C
|-
|6%
|1%
|10%
|-
|}Then the probabilities per machine are:
*A 0.4
** 0.94 acceptable: 0.376
** 0.06 defective: 0.024
* B 0.5
** 0.99 acceptable: 0.495
** 0.01 defective: 0.005
* C 0.1
** 0.90 acceptable: 0.090
** 0.10 defective: 0.010
The overall probability of a defective item is:
{{{
P(defective) = 0.024 + 0.005 + 0.010 = 0.039
}}}
This can also be written as
{{{
P(defective) = P(defective|A or B or C)
}}}
To look at a particular outcome, for example for Machine A and a defective item, we could write the probability as
{{{
P(A and defective) = P(defective/defective from any machine)
}}}
Which is:
{{{
P(A and defective) = P(0.024 / 0.039) = 0.615
}}}

The general formulation of Bayes Theorem is:
<math>\huge {P(A_i|X)} =\frac</math>{{P}(X|A_i) P(A_i)}{\sum_{i=1}^{i=n}{P(X | A_i) P(A_i)}}<math></math>

Applied to the example <math>P(X|A_i)</math> represents the percentage of production that goes to machine <math>A_i</math>. <math>P(A_i)</math> represent the chance that this machine delivers a defective item. The multiplication of the two represent the overall probability that a specific machine delivered a defective item. <math>{\sum_{i=1}^{i=n}{P(X | A_i) P(A_i)}}</math> represents the total overal chance of getting a defective item in the full population (0.024 + 0.005 + 0.010 = 0.039)

Part of: [[M7-Quantitative Decision Methods-S2 - Probability and decision trees|M7-Quantitative Decision Methods-S2 - Probability and decision trees]]
See also: http://en.wikipedia.org/wiki/Bayes%27_theorem
[[Category:mbi]]

[[Category:Decision]]

[[Category:M7]]

[[Category:Methods]]

[[Category:Quantitative]]

[[Category:Term]]

← Older revision		Revision as of 19:18, 27 August 2020
Line 1:		Line 1:
−	Bayes Theorem proposes to look at the end of a sequence of events, so we know the outcome, and we wish to find the probability of a certain event in the sequence (posterior probability). As an example say a production line has 3 machines A, B and C.	+	Bayes Theorem
−	~~{\| class="wikitable"~~
−	!A
−	!B
−	!C
−	\|-
−	~~\|40%~~
−	~~\|50%~~
−	~~\|10%~~
−	\|-
−	~~\|}If we see defective items it is not certain from which machine they came. With Bayes theorem it is possible to identify the most likely machine and check that one first. Suppose the waste rates are:~~
−	~~{\| class="wikitable"~~
−	!A
−	!B
−	!C
−	\|-
−	~~\|6%~~
−	~~\|1%~~
−	~~\|10%~~
−	\|-
−	~~\|}Then the probabilities per machine are:~~
−	*A 0.4
−	** 0.94 acceptable: 0.376
−	** 0.06 defective: 0.024
−	* B 0.5
−	** 0.99 acceptable: 0.495
−	** 0.01 defective: 0.005
−	* C 0.1
−	** 0.90 acceptable: 0.090
−	** 0.10 defective: 0.010
−	~~The overall probability of a defective item is:~~
−	~~{{{~~
−	~~P(defective) = 0.024 + 0.005 + 0.010 = 0.039~~
−	~~}}}~~
−	~~This can also be written as~~
−	~~{{{~~
−	~~P(defective) = P(defective\|A or B or C)~~
−	~~}}}~~
−	~~To look at a particular outcome, for example for Machine A and a defective item, we could write the probability as~~
−	~~{{{~~
−	~~P(A and defective) = P(defective/defective from any machine)~~
−	~~}}}~~
−	~~Which is:~~
−	~~{{{~~
−	~~P(A and defective) = P(0.024 / 0.039) = 0.615~~
−	~~}}}~~
−
−	~~The general formulation of Bayes Theorem is:~~
−	~~<math>\huge {P(A_i\|X)} =\frac</math>{{P}(X\|A_i) P(A_i)}{\sum_{i=1}^{i=n}{P(X \| A_i) P(A_i)}}<math></math>~~
−
−	Applied to the example <math>P(X\|A_i)</math> represents the percentage of production that goes to machine <math>A_i</math>. <math>P(A_i)</math> represent the chance that this machine delivers a defective item. The multiplication of the two represent the overall probability that a specific machine delivered a defective item. <math>{\sum_{i=1}^{i=n}{P(X \| A_i) P(A_i)}}</math> represents the total overal chance of getting a defective item in the full population (0.024 + 0.005 + 0.010 = 0.039)
−
−	~~Part of: [[M7-Quantitative Decision Methods-S2 - Probability and decision trees\|M7-Quantitative Decision Methods-S2 - Probability and decision trees]]~~
−	~~See also: http://en.wikipedia.org/wiki/Bayes%27_theorem~~
	[[Category:mbi]]		[[Category:mbi]]

← Older revision		Revision as of 19:53, 27 August 2020
Line 1:		Line 1:
−	Bayes Theorem	+	Bayes Theorem proposes to look at the end of a sequence of events, so we know the outcome, and we wish to find the probability of a certain event in the sequence (posterior probability). As an example say a production line has 3 machines A, B and C.
		+	{\| class="wikitable"
		+	!A
		+	!B
		+	!C
		+	\|-
		+	\|40%
		+	\|50%
		+	\|10%
		+	\|-
		+	\|}If we see defective items it is not certain from which machine they came. With Bayes theorem it is possible to identify the most likely machine and check that one first. Suppose the waste rates are:
		+	{\| class="wikitable"
		+	!A
		+	!B
		+	!C
		+	\|-
		+	\|6%
		+	\|1%
		+	\|10%
		+	\|-
		+	\|}Then the probabilities per machine are:
		+	*A 0.4
		+	** 0.94 acceptable: 0.376
		+	** 0.06 defective: 0.024
		+	* B 0.5
		+	** 0.99 acceptable: 0.495
		+	** 0.01 defective: 0.005
		+	* C 0.1
		+	** 0.90 acceptable: 0.090
		+	** 0.10 defective: 0.010
		+	The overall probability of a defective item is:
		+	{{{
		+	P(defective) = 0.024 + 0.005 + 0.010 = 0.039
		+	}}}
		+	This can also be written as
		+	{{{
		+	P(defective) = P(defective\|A or B or C)
		+	}}}
		+	To look at a particular outcome, for example for Machine A and a defective item, we could write the probability as
		+	{{{
		+	P(A and defective) = P(defective/defective from any machine)
		+	}}}
		+	Which is:
		+	{{{
		+	P(A and defective) = P(0.024 / 0.039) = 0.615
		+	}}}
		+
		+	The general formulation of Bayes Theorem is:
		+	<math>{P(A_i\|X)} =\frac</math>{{P}(X\|A_i) P(A_i)}{\sum_{i=1}^{i=n}{P(X \| A_i) P(A_i)}}<math></math>
		+
		+	Applied to the example <math>P(X\|A_i)</math> represents the percentage of production that goes to machine <math>A_i</math>. <math>P(A_i)</math> represent the chance that this machine delivers a defective item. The multiplication of the two represent the overall probability that a specific machine delivered a defective item. <math>{\sum_{i=1}^{i=n}{P(X \| A_i) P(A_i)}}</math> represents the total overal chance of getting a defective item in the full population (0.024 + 0.005 + 0.010 = 0.039)
		+
		+	Part of: [[M7-Quantitative Decision Methods-S2 - Probability and decision trees\|M7-Quantitative Decision Methods-S2 - Probability and decision trees]]
		+	See also: http://en.wikipedia.org/wiki/Bayes%27_theorem
	[[Category:mbi]]		[[Category:mbi]]