Conditional probability

Given two events A and B from the sigma-field of a probability space, with the unconditional probability of B being greater than zero (i.e., P(B)>0), the conditional probability of A given B is defined to be the quotient of the probability of the joint of events A and B, and the probability of B:[3][6][7]

${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}},}$

where ${\displaystyle P(A\cap B)}$ is the probability that both events A and B occur. This may be visualized as restricting the sample space to situations in which B occurs. The logic behind this equation is that if the possible outcomes for A and B are restricted to those in which B occurs, this set serves as the new sample space.

Note that the above equation is a definition—not a theoretical result. We just denote the quantity ${\displaystyle {\frac {P(A\cap B)}{P(B)}}}$ as ${\displaystyle P(A\mid B)}$, and call it the conditional probability of A given B.

Some authors, such as de Finetti, prefer to introduce conditional probability as an axiom of probability:

${\displaystyle P(A\cap B)=P(A\mid B)P(B)}$

Although mathematically equivalent, this may be preferred philosophically; under major probability interpretations, such as the subjective theory, conditional probability is considered a primitive entity. Further, this "multiplication axiom" introduces a symmetry with the summation axiom for mutually exclusive events:[8]

${\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}$

Conditional probability can be defined as the probability of a conditional event ${\displaystyle A_{B}}$. The Goodman–Nguyen–Van Fraassen conditional event can be defined as

${\displaystyle A_{B}=\bigcup _{i\geq 1}\left(\bigcap _{j<i}{\overline {B}}_{j},A_{i}B_{i}\right).}$

It can be shown that

${\displaystyle P(A_{B})={\frac {P(A\cap B)}{P(B)}}}$

which meets the Kolmogorov definition of conditional probability.

If P(B)=0, then according to the definition, P(A|B) is undefined.

The case of greatest interest is that of a random variable Y, conditioned on a continuous random variable X resulting in a particular outcome x. The event ${\displaystyle B=\{X=x\}}$ has probability zero and, as such, cannot be conditioned on.

Instead of conditioning on X being exactly x, we could condition on it being closer than distance ${\displaystyle \epsilon }$ away from x. The event ${\displaystyle B=\{x-\epsilon <X<x+\epsilon \}}$ will generally have nonzero probability and hence, can be conditioned on. We can then take the limit

${\displaystyle \lim _{\epsilon \to 0}P(A\mid x-\epsilon <X<x+\epsilon ).}$

For example, if two continuous random variables X and Y have a joint density ${\displaystyle f_{X,Y}(x,y)}$, then by L'Hôpital's rule:

${\displaystyle {\begin{aligned}\lim _{\epsilon \to 0}P(Y\in U\mid x_{0}-\epsilon <X<x_{0}+\epsilon )&=\lim _{\epsilon \to 0}{\frac {\int _{x_{0}-\epsilon }^{x_{0}+\epsilon }\int _{U}f_{X,Y}(x,y)\mathrm {d} y\mathrm {d} x}{\int _{x_{0}-\epsilon }^{x_{0}+\epsilon }\int _{\mathbb {R} }f_{X,Y}(x,y)\mathrm {d} y\mathrm {d} x}}\\&={\frac {\int _{U}f_{X,Y}(x_{0},y)\mathrm {d} y}{\int _{\mathbb {R} }f_{X,Y}(x_{0},y)\mathrm {d} y}}.\end{aligned}}}$

The resulting limit is the conditional probability distribution of Y given X and exists when the denominator, the probability density ${\displaystyle f_{X}(x_{0})}$, is strictly positive.

It is tempting to define the undefined probability ${\displaystyle P(A|X=x)}$ using this limit, but this cannot be done in a consistent manner. In particular, it is possible to find random variables X and W and values x, w such that the events ${\displaystyle \{X=x\}}$ and ${\displaystyle \{W=w\}}$ are identical but the resulting limits are not:[9]

${\displaystyle \lim _{\epsilon \to 0}P(A\mid x-\epsilon \leq X\leq x+\epsilon )\neq \lim _{\epsilon \to 0}P(A\mid w-\epsilon \leq W\leq w+\epsilon ).}$

The Borel–Kolmogorov paradox demonstrates this with a geometrical argument.

Let X be a random variable and its possible outcomes denoted V. For example, if X represents the value of a rolled die then V is the set ${\displaystyle \{1,2,3,4,5,6\}}$. Let us assume for the sake of presentation that X is a discrete random variable, so that each value in V has a nonzero probability.

For a value x in V and an event A, the conditional probability is given by ${\displaystyle P(A\mid X=x)}$. Writing

${\displaystyle c(x,A)=P(A\mid X=x)}$

for short, we see that it is a function of two variables, x and A.

For a fixed A, we can form the random variable ${\displaystyle Y=c(X,A)}$. It represents an outcome of ${\displaystyle P(A\mid X=x)}$ whenever a value x of X is observed.

The conditional probability of A given X can thus be treated as a random variable Y with outcomes in the interval ${\displaystyle [0,1]}$. From the law of total probability, its expected value is equal to the unconditional probability of A.

The partial conditional probability ${\displaystyle P(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})}$ is about the probability of event ${\displaystyle A}$ given that each of the condition events ${\displaystyle B_{i}}$ has occurred to a degree ${\displaystyle b_{i}}$ (degree of belief, degree of experience) that might be different from 100%. Frequentistically, partial conditional probability makes sense, if the conditions are tested in experiment repetitions of appropriate length ${\displaystyle n}$.[10] Such ${\displaystyle n}$-bounded partial conditional probability can be defined as the conditionally expected average occurrence of event ${\displaystyle A}$ in testbeds of length ${\displaystyle n}$ that adhere to all of the probability specifications ${\displaystyle B_{i}\equiv b_{i}}$, i.e.:

${\displaystyle P^{n}(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})=\operatorname {E} ({\overline {A}}^{n}\mid {\overline {B}}_{1}^{n}=b_{1},\ldots ,{\overline {B}}_{m}^{n}=b_{m})}$[10]

Based on that, partial conditional probability can be defined as

${\displaystyle P(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})=\lim _{n\to \infty }P^{n}(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m}),}$

where ${\displaystyle b_{i}n\in \mathbb {N} }$[10]

Jeffrey conditionalization[11][12] is a special case of partial conditional probability, in which the condition events must form a partition:

${\displaystyle P(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})=\sum _{i=1}^{m}b_{i}P(A\mid B_{i})}$

A geometric visualisation 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 P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = P(B|A) P(A)/P(B) . Similar reasoning can be used to show that P(Ā|B) = P(B|Ā) P(Ā)/P(B) etc.

In general, it cannot be assumed that P(A|B) ≈ P(B|A). This can be an insidious error, even for those who are highly conversant with statistics.[13] The relationship between P(A|B) and P(B|A) is given by Bayes' theorem:

${\displaystyle {\begin{aligned}P(B\mid A)&={\frac {P(A\mid B)P(B)}{P(A)}}\\\Leftrightarrow {\frac {P(B\mid A)}{P(A\mid B)}}&={\frac {P(B)}{P(A)}}\end{aligned}}}$

That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B).

In general, it cannot be assumed that P(A) ≈ P(A|B). These probabilities are linked through the law of total probability:

${\displaystyle P(A)=\sum _{n}P(A\cap B_{n})=\sum _{n}P(A\mid B_{n})P(B_{n}).}$

where the events ${\displaystyle (B_{n})}$ form a countable partition of ${\displaystyle \Omega }$.

This fallacy may arise through selection bias.[14] For example, in the context of a medical claim, let S_C be the event that a 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_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_C) is high. The actual probability observed by the doctor is P(S_C|H).

Not taking prior probability into account partially or completely is called base rate neglect. The reverse, insufficient adjustment from the prior probability is conservatism.