Conditional probability ($P(A\mid B)$)

Conditional probability is $P(A\mid B)=P(A\cap B)/P(B)$ when $P(B)>0$.

When all conditioning events have positive probability, repeatedly apply the definition of conditional probability to factor a joint event into successive conditional factors. For a sequence of discrete responses, this yields the product of conditional masses along the actual outcome path. It requires no unconditional independence assumption.

For a probability measure on a standard Borel space and a measurable map $\pi$ to another standard Borel space, there is an almost-everywhere unique measurable probability kernel $y\mapsto {\mu }_y$ supported on ${\pi }^{-1}\{y\}$. It integrates to $\mu$, and $\int fd{\mu }_{\pi (x)}$ realizes conditional expectation given $\pi$. These measures are the conditional measures of a factor.

The fiber-supported measurable probability kernel supplied by the disintegration theorem for a probability measure for a factor map. For integrable $f$, $\mathbb{E}[f\mid {\phi }^{-1}{\mathcal{B}}_Y](x)=\int fd{\mu }_{\phi (x)}$ almost everywhere. The same construction applies to general measurable maps and countably generated sub-sigma-algebras modulo null sets.

A family of probability measures $K_y$ on a measurable space for which $y\mapsto K_y(A)$ is measurable for every measurable set $A$. This combines measurable dependence on a parameter with a probability measure at each parameter. The conditional measures of a factor form such a kernel, with the additional fiber-support property from disintegration.

For nontrivial events, strict attraction under conditional probability is equivalent to a positive covariance of their indicator variables. This condition is symmetric in the two events. Complementing exactly one of the events reverses the strict inequality. This two-event condition should be distinguished from association of an entire family of random variables.

If $d\tilde{\mathbb{P}}=Zd\mathbb{P}$ and $Z_t=\mathbb{E}[Z\mid {\mathcal{F}}_t]>0$, then

For a finite partition $(B_j)$ with positive probabilities,

Let $N$ have the positive-integer geometric distribution with success parameter $p\in (0,1)$, and conditionally toss $N$ independent coins with head probability $\theta \in (0,1)$. Exactly one head has conditional probability $n\theta (1-\theta {)}^{n-1}$. Combining this with the prior by Bayes theorem gives the displayed posterior with $r=(1-p)(1-\theta )$; its normalization uses ${\sum }_{n\ge 1}n{r}^{n-1}=(1-r{)}^{-2}$. For $p=\theta =1/2$, the posterior is $9n/4^{n+1}$. It is also a shifted negative binomial distribution, with $N-1$ equal in law to the number of failures before two successes of probability $1-r$.

If every model's likelihood is multiplied by the same positive factor independent of its parameter or model index, Bayes theorem cancels that factor from the normalized posterior. Fixed-count Bernoulli sampling and sampling stopped at a specified positive head count give such proportional likelihoods for the same total heads and tails. This conclusion uses their actual sampling likelihoods; it is not a claim that every selection or stopping rule can be ignored.

If a hidden coin is selected once with prior probability $\pi$ of coin $B$, repeated tosses are independent conditional on that coin. Bayes theorem weights the prior by each all-heads likelihood. They need not be independent without conditioning on the chosen coin.

A conditional distribution is the probability law of one random quantity given information about another.

Where the marginal density $f_X(x)$ is positive, a joint probability density determines the conditional densityIt integrates to one in $y$, and supplies conditional expectations by integrating against it. Its values on a probability-zero set of conditioning points are not fixed by the joint probability distribution.

A full conditional distribution is the conditional law of one component of a joint model given all the other components and the observed data. A Gibbs sampler updates components by drawing from these laws.