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Conditional Probability

Also Conditional PDF · Conditional Density · Joint Probability · Bedingte Wahrscheinlichkeit

The conditional probability P(A|B) is the probability that event A occurs given that event B has occurred, P(A|B) = P(A, B) / P(B); for densities p(x|y) = p(x, y) / p(y).

Know these firstRandom Variable

Properties#

Product rule: $P(A, B) = P(A|B)\,P(B) = P(B|A)\,P(A)$ . Equating the two factorizations gives Bayes’ Theorem.
Independence: $A$ , $B$ are independent iff $P(A|B) = P(A)$ ; for densities iff $p(x|y) = p(x)$ .
Order matters: $P(A|B) \ne P(B|A)$ in general. In the antique store example of Bayes’ Theorem, $P(B|A) = 0.2$ but $P(A|B) = 2/3$ .

Example#

Sources#

MSI script p. 13 (events, §2.1), MSI Slides 02 > 2.1 Random Variables and Probabilities (not published)
MSI script p. 14 (conditional PDF, §2.2), MSI Slides 02 > 2.2 Scalar Random Variables and Probability Density Functions (not published)

Connections

Prerequisites know first1

Random Variable

Mentioned in inline links9

Hidden Markov ModelDBIS
The factorization follows from two independence assumptions (Conditional Probability, Statistical Independence): §
Naive Bayes ClassifierDBIS
Derivation. By Bayes’ Theorem, $\Pr(c \mid d) = \Pr(d \mid c)\,\Pr(c) / \Pr(d)$ . The denominator $\Pr(d)$ is the same for all classes and drops out of the $\arg\max$ . The naive assumption is that the words of $d$ are independent given the… §
Naive Bayes ClassificationDBIS
Prerequisites from the foundations: Bayes’ Theorem, Conditional Probability, Statistical Independence. §
POS-TaggingDBIS
Prerequisites from the foundations: Conditional Probability, Statistical Independence. §
VL02 – Probability and Statistics in a NutshellMSI
The section introduces a Random Variable $X$ as a description of an experiment’s outcomes. $X$ has no value itself and only takes values $x$ . It then gives the rules for probabilities of events: the sum for mutually exclusive events and th… §
Conditional Probability: $P(A|B) = P(A,B)/P(B)$ , also for PDFs §
Bayes' Theorem
$P(A)$ is the a-priori (prior) probability of $A$ , and $P(A|B)$ is the a-posteriori (posterior) conditional probability of $A$ after the evidence $B$ has occurred. §
Probability Density Function
Conditional PDF: $p_{X|Y}(x|y) = \dfrac{p_{X,Y}(x, y)}{p_Y(y)}$ , short $p(x|y) = \dfrac{p(x, y)}{p(y)}$ (Conditional Probability). §
Statistical Independence
Conditioning changes nothing: for independent $X, Y$ , the conditional PDF is $p_{X|Y}(x|y) = \frac{p_X(x)\,p_Y(y)}{p_Y(y)} = p_X(x)$ (Conditional Probability); likewise $P(A|B) = P(A)$ for independent events with $P(B) > 0$ . §
Random Variables and Distributions
Conditional Probability — probabilities and densities given an observed event or value §

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