Conditional probability: Difference between revisions

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Conditional probability can be defined as the probability of a conditional event <math>A_B</math>. The [[Goodman–Nguyen–Van Fraassen algebra|Goodman–Nguyen–Van Fraassen]] conditional event can be defined as:

:<math>A_B = \bigcup_{i \ge 1} \left( \bigcap_{j<i} \overline{B}_j, A_i B_i \right), </math> where <math>A_i </math> and <math>B_i </math> represent states or elements of ''A'' or ''B.'' <ref>{{Cite journal|last1=Flaminio|first1=Tommaso|last2=Godo|first2=Lluis|last3=Hosni|first3=Hykel|date=2020-09-01|title=Boolean algebras of conditionals, probability and logic|url=https://www.sciencedirect.com/science/article/pii/S000437022030103X|journal=Artificial Intelligence|language=en|volume=286|pagesarticle-number=103347|doi=10.1016/j.artint.2020.103347|arxiv=2006.04673|s2cid=214584872 |issn=0004-3702}}</ref>

It can be shown that

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where <math> b_i n \in \mathbb{N}</math><ref name=Draheim2017b />

[[Radical probabilism|Jeffrey conditionalization]]<ref>{{citation|first=Richard C.|last=Jeffrey|title=The Logic of Decision, |edition=2nd edition|publisher=University of Chicago Press|year=1983 |isbn=9780226395821|url=https://books.google.com/books?id=geJ-SwTcmyEC&q=%22conditional+probability%22}}</ref><ref>{{cite web|title=Bayesian Epistemology| url=https://plato.stanford.edu/entries/epistemology-bayesian/|publisher=Stanford Encyclopedia of Philosophy|access-date=December 29, 2017|year=2017 }}</ref>

is a special case of partial conditional probability, in which the condition events must form a [[Partition of a set|partition]]:

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:<math>P(B\mid A) = P(B)</math>

is also equivalent. Although the derived forms may seem more intuitive, they are not the preferred definition as the conditional probabilities may be undefined, and the preferred definition is symmetrical in ''A'' and ''B''. Independence does not refer to a disjoint event.<ref>{{Cite book|last=Tijms|first=Henk|url=https://www.cambridge.org/core/books/understanding-probability/B82E701FAAD2C0C2CF36E05CFC0FF3F2|title=Understanding Probability|date=2012|publisher=Cambridge University Press|isbn=978-1-107-65856-1|edition=33rd|___location=Cambridge|doi=10.1017/cbo9781139206990}}</ref>

It should also be noted that given the independent event pair [''A'',''B''] and an event ''C'', the pair is defined to be [[Conditional independence|conditionally independent]] if<ref>{{Cite book|last=Pfeiffer|first=Paul E.|url=https://www.worldcat.org/oclc/858880328|title=Conditional Independence in Applied Probability|date=1978|publisher=Birkhäuser Boston|isbn=978-1-4612-6335-7|___location=Boston, MA|oclc=858880328}}</ref>

: <math>P(AB \mid C) = P(A \mid C)P(B \mid C).</math>

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=== Assuming conditional probability is of similar size to its inverse ===

{{Main|Confusion of the inverse}}

[[File:Bayes theorem visualisation.svg|thumb|450x450px|A geometric visualization of Bayes' theorem. In the table, the values 2, 3, 6 and 9 give the relative weights of each corresponding condition and case. The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. This shows that <math>P(A|\mid B) P(B) = P(B\mid A) P(A)</math> i.e. <math>P(A\mid B) = \frac{P(B\mid A)} {P(A)\capcdot P(B)}</math> . Similar reasoning can be used to show that <math>P(\bar A\mid B) = \frac{P(B\mid\bar A) P(\bar A)}{P(B)}</math> etc.]]

In general, it cannot be assumed that ''P''(''A''|''B'')&nbsp;≈&nbsp;''P''(''B''|''A''). This can be an insidious error, even for those who are highly conversant with statistics.<ref>{{cite book |last=Paulos, |first=J. A. (|year=1988) ''|title=Innumeracy: Mathematical Illiteracy and its Consequences'', |publisher=Hill and Wang. {{ISBN|isbn=0-8090-7447-8}} (|at=p. 63 ''et seq.'') }}</ref> The relationship between ''P''(''A''|''B'') and ''P''(''B''|''A'') is given by [[Bayes' theorem]]:

:<math>\begin{align}

P(B\mid A) &= \frac{P(A\mid B) P(B)}{P(A)}\\

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where the events <math>(B_n)</math> form a countable [[Partition of a set|partition]] of <math>\Omega</math>.

This fallacy may arise through [[selection bias]].<ref>[[{{cite journal |first=F. Thomas |last=Bruss |authorlink=F. Thomas Bruss]] |title=Der Wyatt-Earp-Effekt oder die betörende Macht kleiner Wahrscheinlichkeiten (in German),|language=de |journal=[[Spektrum der Wissenschaft]] (German Edition of Scientific American), Vol |volume=2, |pages=110–113, (|year=2007). }}</ref> For example, in the context of a medical claim, let ''S''{{sub|''C''}} be the event that a [[sequelae|sequela]] (chronic disease) ''S'' occurs as a consequence of circumstance (acute condition) ''C''. Let ''H'' be the event that an individual seeks medical help. Suppose that in most cases, ''C'' does not cause ''S'' (so that ''P''(''S''{{sub|''C''}}) is low). Suppose also that medical attention is only sought if ''S'' has occurred due to ''C''. From experience of patients, a doctor may therefore erroneously conclude that ''P''(''S''{{sub|''C''}}) is high. The actual probability observed by the doctor is ''P''(''S''{{sub|''C''}}|''H'').

=== Over- or under-weighting priors ===

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Formally, ''P''(''A''&nbsp;|&nbsp;''B'') is defined as the probability of ''A'' according to a new probability function on the sample space, such that outcomes not in ''B'' have probability 0 and that it is consistent with all original [[probability measure]]s.<ref>George Casella and Roger L. Berger (1990), ''Statistical Inference'', Duxbury Press, {{ISBN|0-534-11958-1}} (p. 18 ''et seq.'')</ref><ref name="grinstead">[http://math.dartmouth.edu/~prob/prob/prob.pdf Grinstead and Snell's Introduction to Probability], p. 134</ref>

Let Ω be a discrete [[sample space]] with [[elementary event]]s {''ω''}, and let ''P'' be the probability measure with respect to the [[σ-algebra]] of Ω. Suppose we are told that the event ''B''&nbsp;⊆&nbsp;Ω has occurred. A new [[probability distribution]] (denoted by the conditional notation) is to be assigned on {''ω''} to reflect this. All events that are not in ''B'' will have null probability in the new distribution. For events in ''B'', two conditions must be met: the probability of ''B'' is one and the relative magnitudes of the probabilities must be preserved. The former is required by the [[Probability axioms|axioms of probability]], and the latter stems from the fact that the new probability measure has to be the analog of ''P'' in which the probability of ''B'' is one - andone—and every event that is not in ''B'', therefore, has a null probability. Hence, for some scale factor ''α'', the new distribution must satisfy:

#<math>\omega \in B : P(\omega\mid B) = \alpha P(\omega)</math>