Probability and Expectation

A probability model assigns chances to outcomes under stated assumptions. Its usefulness depends on whether those assumptions fit the problem.

Expected value

For discrete outcomes $x_i$ with probabilities $p_i$ ,

$\mathbb{E}[X] = \sum_i p_i x_i.$

The expectation is a weighted average. It is not the most likely outcome and need not be an outcome that can occur in one trial.

Account for the size of losses as well as the chance of winning. A high win rate alone says little about expected profit. Fees and other costs belong in the outcome being averaged.

Variance and dependence

Variance measures squared distance from the expectation. Standard deviation has the same units as the original quantity. Neither describes every feature of a distribution, especially its tails.

Expectations add even when variables are dependent. Variances generally need covariance terms. Treating correlated positions as independent can understate how much they move together.

Conditional probability describes an event after restricting attention to outcomes consistent with given information. Bayes' rule relates the likelihood of that information to the probability of competing explanations.

Check the base rate before interpreting a signal. An observation can be common under one explanation without making that explanation likely overall.

A simple signal model

Define a hidden state as up, flat or down. Give each state an initial probability and define how often a signal is emitted in each state. On receiving a signal, multiply each prior by its signal likelihood and normalise the three weights.

Use the updated probabilities to calculate an expected move. This defines an original model with inspectable assumptions. It does not establish that real prices follow those states or signal frequencies.

Calibration

When a model repeatedly assigns the same probability to an event, compare that probability with the observed frequency across similar predictions. A confidence score and a probability are not automatically the same thing.

Keep evaluation data separate from data used to fit the model. A model can appear well calibrated on the observations that shaped it and fail on later ones.

See Backtesting for evaluation over time and Market Making for acting on an estimated value.