Independence and Conditioning

Motivation: why this matters in quant finance

Independence and conditioning are the two basic ways information changes probability. Independence says new information does not matter. Conditioning says it does, and tells you how to update the model once that information is known.

Brownian motion is useful in finance because its future increments are independent of the past. Bayesian credit models, filtering, scenario analysis, and conditional distributions of future prices are useful because many events are not independent. A pricing model needs both: independence for tractable dynamics, conditioning for time- $t$ valuation.

Bertsekas teaches conditioning first as a new probability law on a restricted universe, then uses it to build multiplication rules, total probability, Bayes' rule, conditional PMFs, and conditional expectations. That route is important here. Conditioning is not a slogan about "updating beliefs"; it is a precise way to replace one probability law by another after information arrives.

The informal idea

If event $B$ has occurred, outcomes outside $B$ are no longer relevant. Conditional probability renormalises the original probability law inside $B$ :

\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}.

Independence is the case where this renormalisation changes nothing:

\mathbb{P}(A\mid B)=\mathbb{P}(A).

For random variables, the same ideas apply to numerical observations. Conditioning on $Y=y$ slices the joint distribution along the value $y$ and renormalises the slice. Independence says every slice looks like the original marginal distribution.

Formal definitions

Conditional probability

For events $A,B\in\mathcal{F}$ with $\mathbb{P}(B)>0$ ,

\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}.

For fixed $B$ , the map $A\mapsto \mathbb{P}(A\mid B)$ is itself a probability law.

Independence of events

Events

A

and

B

are independent when

\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B).

For several events, mutual independence requires the same product rule for every subcollection, not just every pair.

Independence of random variables

Random variables $X$ and $Y$ are independent if for all Borel sets $A,B$ ,

\mathbb{P}(X\in A, Y\in B)=\mathbb{P}(X\in A)\mathbb{P}(Y\in B).

In density language, this becomes $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ whenever the densities exist.

Conditional PMF and density

For discrete $X,Y$ ,

p_{X\mid Y}(x\mid y)=\frac{p_{X,Y}(x,y)}{p_Y(y)}.

For continuous variables with joint density,

f_{X\mid Y}(x\mid y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}.

Key properties

Multiplication rule

Conditional probability can be rearranged into

\mathbb{P}(A\cap B)=\mathbb{P}(B)\mathbb{P}(A\mid B).

For sequential models, probabilities along a path multiply. This is the arithmetic behind tree models, default cascades, and Bayesian filtering recursions.

Total probability

If $B_1,\ldots,B_n$ form a partition with positive probabilities, then

\mathbb{P}(A)=\sum_i \mathbb{P}(A\mid B_i)\mathbb{P}(B_i).

The unconditional probability is an average of conditional probabilities over scenarios.

Bayes' rule

\mathbb{P}(B_i\mid A)=\frac{\mathbb{P}(A\mid B_i)\mathbb{P}(B_i)}{\sum_j \mathbb{P}(A\mid B_j)\mathbb{P}(B_j)}.

This converts prior probabilities and likelihoods into posterior probabilities.

Factorisation under independence

If $X$ and $Y$ are independent and integrable where needed, then

\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y], \qquad \text{Cov}(X,Y)=0.

The converse is false in general.

Conditional averages produce unconditional averages

Bertsekas' total expectation theorem says that unconditional averages can be computed by averaging conditional averages:

\mathbb{E}[X]=\sum_i \mathbb{P}(B_i)\mathbb{E}[X\mid B_i]

for a finite partition. The sigma-algebra version becomes the tower property in conditional expectation.

Worked examples

Example 1: posterior default probability

A borrower has prior default probability $\mathbb{P}(D)=2\%$ . A severe equity drop occurs with probability $60\%$ given default and $5\%$ given no default. Bayes' rule gives

\mathbb{P}(D\mid \text{drop})= \frac{0.60\cdot0.02}{0.60\cdot0.02+0.05\cdot0.98}=19.7\%.

The event is strong evidence, but the posterior is not $60\%$ because the prior default rate is small.

Example 2: conditional law of a terminal price

Under geometric Brownian motion, conditional on $S_t=s$ ,

\log S_T \mid S_t=s \sim \mathcal{N}\left(\log s + \left(\mu-\frac12\sigma^2\right)(T-t),\;\sigma^2(T-t)\right).

The future distribution depends on the current level $s$ but not on the path before $t$ . That statement uses both conditioning and independent increments.

Example 3: diversification and failed independence

If $R_1,\ldots,R_n$ are i.i.d. with variance $\sigma^2$ , an equally weighted portfolio has variance $\sigma^2/n$ . If every pair has common correlation $\rho$ , the variance tends to $\rho\sigma^2$ as $n$ grows. The independent case diversifies away; the common-factor component does not.

Example 4: total probability in stochastic volatility

Suppose an option price conditional on average variance $V$ is $C(V)$ . The unconditional price is

\mathbb{E}[C(V)]=\int C(v)f_V(v)\,dv.

This is total probability / total expectation in pricing form: condition on a scenario, price inside the scenario, then average over scenarios.

Common confusions and pitfalls

"Pairwise independence is enough." It is not. Mutual independence requires product rules for every subcollection. Pairwise checks can miss higher-order dependence.

" $\mathbb{P}(A\mid B)=\mathbb{P}(B\mid A)$ ." These are different questions. Confusing likelihood with posterior probability is the base-rate error behind many bad risk inferences.

"Uncorrelated means independent." Zero covariance removes only linear co-movement. Independence is a statement about all events generated by the variables.

"Conditioning is always on a positive-probability event." Elementary conditioning uses

\mathbb{P}(B)>0

. Conditioning on

Y=y

for a continuous random variable needs densities or the more general conditional-expectation framework.

"Independent increments means no uncertainty." Future increments are still random. Independence says the past gives no extra information about their distribution.

Where this goes next

Conditional Expectation: Generalises conditioning from events and random-variable values to sigma-algebras.
Filtrations and Information: Formalises "past information" so independence of future increments can be stated correctly.
Brownian Motion: Uses independent increments as a defining axiom.
Central Limit Theorem: Shows why sums of independent terms become normal after scaling.
Moment Generating Functions: Independent sums turn into products of MGFs.

References

Bertsekas, D. P., & Tsitsiklis, J. N. (2008). Introduction to Probability (2nd ed.). Athena Scientific. Ch. 1 §1.3 (Conditional Probability), §1.4 (Total Probability Theorem and Bayes' Rule), §1.5 (Independence), Ch. 2 §2.6 (Conditioning), §2.7 (Independence).