Conditional Probability

Conditional probability is the probability of an event, calculated on the assumption that another event has already happened or is known to be true. It answers questions of the form "how likely is A, given B?" and is written $P(A \mid B)$. New information narrows the set of possible outcomes, and the conditional probability is the chance of A within that narrower set.

The concept is central to probability theory and to any reasoning from evidence. In finance, most useful statements are conditional: the chance of a loss given that volatility is high, the chance of default given a credit downgrade, the chance that a price rises given a trading signal. A trading signal has value only if the conditional probability it produces differs from the unconditional one.

The definition is:

• $P(A \mid B)$ is the probability of A given B.
• $P(A \cap B)$ is the probability that both A and B occur.
• $P(B)$ is the probability of B, which must be greater than zero.

Consider 1,000 trading days. The market rose on 540 of them, so the unconditional probability of a rise is 0.54. A certain indicator gave a buy signal on 200 days, and the market rose on 130 of those. The probability of a rise and a signal together is 130 ÷ 1,000 = 0.13, and the probability of a signal is 0.20. The conditional probability of a rise given a signal is 0.13 ÷ 0.20 = 0.65. On the 800 days without a signal, the market rose on 410, a conditional probability of about 0.51. In this sample the signal carries information.

Two events are independent when knowing one does not change the probability of the other, that is, when $P(A \mid B) = P(A)$. Had the market risen on 108 of the 200 signal days, the conditional probability would have been 0.54, the same as the base rate, and the signal would be worthless.

The probability of A given B is generally not equal to the probability of B given A. Bayes' theorem links the two:

Suppose a warning indicator has flashed before 90 percent of past market crashes. Crashes occur in 2 percent of months, and the indicator also flashes in 10 percent of months with no crash. The probability of a flash in any month is (0.90 × 0.02) + (0.10 × 0.98) = 0.018 + 0.098 = 0.116. The probability of a crash given a flash is 0.018 ÷ 0.116 = 0.155, or about 16 percent. An indicator that catches nine crashes in ten still gives a false alarm more than five times in six, because crashes are rare.

• Risk measurement. Value at risk gives a loss level that is exceeded with a small probability. Expected shortfall, also called conditional value at risk, is the average loss given that this level has been exceeded.
• Volatility models. A GARCH model estimates the variance of tomorrow's return conditional on recent returns. The word conditional in its name refers to this.
• Credit risk. Lenders estimate the probability of default conditional on a borrower's characteristics and on the state of the economy.
• Signals and classifiers. A classifier used for trading or fraud detection outputs the probability of an outcome conditional on the input features.
• Statistical tests. A p-value is a conditional probability: the chance of data at least as extreme as that observed, given that the null hypothesis is true.
• Options. The probability of profit of an option position is updated as the price moves and time passes, each update being conditional on the new price.

The most frequent mistake is reversing the condition. The crash example shows the gap: 90 percent of crashes were preceded by a flash, but only 16 percent of flashes were followed by a crash. Reading a p-value as the probability that the null hypothesis is true is the same error in another form.

A second mistake is ignoring the base rate. When an event is rare, even an accurate test produces mostly false positives. A third is conditioning on information that was not available at the time. A backtest that selects stocks using data published later suffers from look-ahead bias, and a study limited to companies that still exist today conditions on survival, which is survivorship bias. Both inflate apparent success rates.

Conditional probabilities in markets are estimated from limited history, and narrowing the condition shrinks the sample. A rule tested on 2,000 days may fire on only 40, and a success rate measured on 40 cases has a wide margin of error. Adding more conditions makes each estimate rest on fewer observations and raises the risk of finding patterns that arose by chance. Estimates also drift: a relationship that held in one period can weaken once it becomes widely known and traded upon. For these reasons analysts check conditional estimates on separate data and attach a confidence interval to them.