Editing Bayes' theorem (section)

====Alternative form====

{| class="wikitable floatright"
|+ [[Contingency table]]
! {{diagonal split header|<br />Proposition|&nbsp;&nbsp;Background}} !! B !! {{tmath|\lnot B}}<br />(not {{mvar|B}}) !! Total
|-
|-
! {{mvar|A}}
| |<math>P(B|A)\cdot P(A)</math><br /><math>= P(A|B)\cdot P(B)</math> || |<math>P(\neg B|A)\cdot P(A)</math><br /><math>= P(A|\neg B)\cdot P(\neg B)</math>
|style="text-align:center;"| {{tmath|P(A)}}
|-
! {{tmath|\neg A}}<br/>(not {{mvar|A}})
| nowrap|<math>P(B|\neg A)\cdot P(\neg A)</math><br /><math>= P(\neg A|B)\cdot P(B)</math> || nowrap|<math>P(\neg B|\neg A)\cdot P(\neg A)</math><br /><math>= P(\neg A|\neg B)\cdot P(\neg B)</math> || nowrap|<math>P(\neg A)</math>=<br /><math>1-P(A)</math>
|-
| colspan="5" style="padding:0;"|
|-
! Total
| style="text-align:center;" | {{tmath|P(B)}}
| style="text-align:center;" | <math>P(\neg B) = 1-P(B)</math>
| style="text-align:center;" | 1
|}
Another form of Bayes' theorem for two competing statements or hypotheses is:

:<math>P(A| B) = \frac{P(B| A) P(A)}{ P(B| A) P(A) + P(B| \neg A) P(\neg A)}.</math>

For an epistemological interpretation:

For proposition ''A'' and evidence or background ''B'',<ref>{{cite web|title=Bayes' Theorem: Introduction|url=http://www.trinity.edu/cbrown/bayesweb/|website=Trinity University|url-status=dead|archive-url=https://web.archive.org/web/20040821012342/http://www.trinity.edu/cbrown/bayesweb/|archive-date=21 August 2004|access-date=5 August 2014}}</ref>

* <math>P(A)</math> is the [[prior probability]], the initial degree of belief in ''A''.
* <math>P(\neg A)</math> is the corresponding initial degree of belief in ''not-A'', that ''A'' is false, where <math> P(\neg A) =1-P(A) </math>
* <math>P(B| A)</math> is the [[conditional probability]] or likelihood, the degree of belief in ''B'' given that ''A'' is true.
* <math>P(B|\neg A)</math> is the [[conditional probability]] or likelihood, the degree of belief in ''B'' given that ''A'' is false.
* <math>P(A| B)</math> is the [[posterior probability]], the probability of ''A'' after taking into account ''B''.