Conditional Probability, Bayes & Expectation

Everything you believe about a market is conditional on something. When you say an asset has a 60% chance of finishing the week higher, that number is never floating free — it sits on a pile of facts you're holding fixed: today's price, this week's volatility, that earnings already came out. Change any one and the 60% changes with it. This is the most important idea in the probability of returns, and it has a precise name: conditional probability — the probability of one thing given that you already know another. This chapter builds it into the full predictive engine — conditional expectation — that 102's regression and forecasting all rest on.

Conditioning is throwing away worlds

Build the intuition before the symbols. Imagine all the ways next week could unfold as a giant bag of possible worlds. Unconditionally, you reach into the whole bag. The moment you learn something true — say, the price is already up 2% by Tuesday — you throw away every world inconsistent with that fact and reach only into the survivors. The probability of a Friday close above some level, given the Tuesday move, is the fraction of those surviving worlds in which it happens. Conditioning is throwing away worlds. That is all it is. The formula says exactly that:

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

You take the slice where both are true and re-weight by how big the $B$-slice was, because $B$ is your new universe — dividing by $P(B)$ makes the survivors add back to one.

Joint, marginal, conditional

To handle two uncertain quantities at once — say a Tuesday move $X$ and a Friday return $Y$ — you need their joint distribution $f(x,y)$, the probability of each pair. From it, two things drop out. The marginal is what you get by ignoring one variable (sum or integrate it away): $f(x) = \int f(x,y)\,dy$ — the distribution of $X$ alone. The conditional is what you get by fixing one variable: $f(y \mid x) = f(x,y)/f(x)$ — the distribution of $Y$ once $X = x$ is known. Marginal is forgetting; conditional is learning. The whole chapter is the difference between those two operations.

Bayes' rule

From the one definition comes the most useful updating tool, Bayes' rule. It answers the question markets actually pose, which runs backwards from how data arrives: you observe an effect — a sharp move, a volume spike — and want the probability of a cause (the trend has genuinely turned). What you can estimate is the other direction: if the trend had turned, how likely was this move? Bayes flips one into the other —

$\underbrace{P(\text{cause} \mid \text{evidence})}_{\text{posterior}} \;=\; \frac{\overbrace{P(\text{evidence} \mid \text{cause})}^{\text{likelihood}}\;\overbrace{P(\text{cause})}^{\text{prior}}}{P(\text{evidence})} \;\propto\; \text{likelihood} \times \text{prior}.$

The piece beginners forget is the prior, the base rate. A signal that's right 80% of the time when a real breakout is underway sounds impressive — until you remember breakouts are rare, maybe one day in twenty. Run it through Bayes and the chance this firing is real can still be modest, because the tiny prior drags it down (you proved this numerically in Tier 100 ch4). Before acting on a number, always ask what it was conditional on, and what the base rate was underneath.

Conditional expectation: the best forecast

Alongside conditional probability sits the idea this course is really after: conditional expectation. Where conditional probability gives the chance of an event among the surviving worlds, conditional expectation gives the average value of some quantity across them:

$\mathbb{E}[X \mid Y = y] = \int x\, f(x \mid y)\,dx.$

Fix $Y$, restrict to the worlds where it holds, average $X$ over what remains. This is the single best forecast of $X$ once $Y$ is known — "best" in the precise sense that it minimizes average squared error. No other function of $Y$ predicts $X$ with smaller expected squared miss. That is a strong claim, and it is why conditional expectation is the theoretical target of all of forecasting.

Now the subtle point that elevates this above ch1's expectation. $\mathbb{E}[X \mid Y = y]$ is a number once you plug in a specific $y$ — but before you observe $Y$, it is a function of $Y$, and therefore itself a random variable, written $\mathbb{E}[X \mid Y]$. Its value is whatever the conditional mean happens to be once $Y$ lands. Read as a function of the conditioning variable, $\mathbb{E}[Y \mid X]$ is exactly the regression function — the curve that gives the best $Y$-prediction at each $X$ — which is why estimating it from data is the entire subject of Tier 200's regression.

The tower rule: iterated expectations

Conditional and unconditional expectations are tied by one elegant identity, the law of iterated expectations (the "tower rule"):

$\mathbb{E}\big[\,\mathbb{E}[Y \mid X]\,\big] = \mathbb{E}[Y].$

In words: the average of the conditional averages is the grand average. Average the Friday return within each possible Tuesday move, then average those across all Tuesday moves weighted by how often each occurs, and you recover the plain expected Friday return. It is both a sanity check (conditioning can't manufacture or destroy the overall mean) and a practical computing trick — hard expectations often crack open when you compute them in two stages, inner then outer.

How much did conditioning actually help? Total variance

Iterated expectations says conditioning leaves the mean alone. The deeper question is what conditioning does to uncertainty — and the answer is the law of total variance, the most important equation in this chapter:

$\mathrm{Var}(Y) = \underbrace{\mathbb{E}\big[\mathrm{Var}(Y \mid X)\big]}_{\text{unexplained}} + \underbrace{\mathrm{Var}\big(\mathbb{E}[Y \mid X]\big)}_{\text{explained}}.$

Total uncertainty in $Y$ splits cleanly into two pieces. The unexplained part is the average leftover scatter within each $X$-slice — the noise conditioning can't remove. The explained part is how much the conditional mean itself moves as $X$ varies — the uncertainty conditioning resolves. A predictor that genuinely knows something makes $\mathbb{E}[Y\mid X]$ swing widely (large explained) and leaves little residual scatter (small unexplained). The fraction $\mathrm{Var}(\mathbb{E}[Y\mid X])/\mathrm{Var}(Y)$ is exactly the R² you'll meet in regression — the share of variance a signal accounts for.

The limiting case names itself: if $X$ and $Y$ are independent, then $\mathbb{E}[Y\mid X] = \mathbb{E}[Y]$ — a flat line, no rise — so the explained variance is zero. Conditioning on an independent variable tells you nothing, and the cone doesn't budge. Independence is the formal statement of "this signal is useless."

The cone is conditional belief, drawn

This is the exact machinery under one of SVQ's most-used outputs.

And here is the deeper lever, straight from the total-variance law. The standard cone conditions only on price and volatility. Feed it a genuinely informative signal — condition on something that moves $\mathbb{E}[Y\mid X]$ — and the band should tighten, because you've converted unexplained variance into explained.