Martingale: E[X_{n+1} | X_1, X_2, ..., X_n] = X_n
Image: Internet Archive Book Images, No restrictions, via Wikimedia Commons
the optional stopping theorem says about martingales and stopping times
Martingale: E[X_{n+1} | X_1, X_2, ..., X_n] = X_n
A martingale is a stochastic process where the expected value of the next observation, given all prior observations, equals the most recent value. This property makes martingales ideal for modeling fair games, as future expected winnings are equal to the current amount, regardless of past outcomes.
Example
Consider a fair coin-tossing game where X_n represents the player's winnings after n tosses. If X_n = 0 for all n, then X_n is a martingale because E[X_{n+1} | X_1, X_2, ..., X_n] = X_n = 0.
Remember this
Understanding martingales is crucial for modeling fair games and predicting outcomes in stochastic processes.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
Related concepts
Local martingale
E[X_{n+1}|X_1,...,X_n] = X_n
Filtration (mathematics)
How can we predict future events with uncertainty?
Law of large numbers
Law of large numbers: X̄_ n → μ as n → ∞ with probability 1
log-probabilities are used instead of probabilities: avoids numerical underflow
Why can't we just add up tiny chances over time?
the minimax theorem says: in zero-sum games, there's a saddle point strategy
Minimax theorem guarantees a saddle point strategy in zero-sum games
Kolmogorov complexity
Kolmogorov complexity is uncomputable
Swipe through 100 ML concepts daily
Open Pocket Polymath