How can we predict future events with uncertainty?
Image: Unknown authorUnknown author, CC BY 4.0, via Wikimedia Commons
How can we predict future events with uncertainty?
Imagine you're planning a picnic and want to know if it will rain tomorrow. You check the weather forecast, but it's not certain.
Think of the weather forecast as a series of clues that get more detailed over time. A stopping time is when you finally get a clear answer about tomorrow's weather.
Example
You check the forecast at 9 AM (F1), 12 PM (F2), and 3 PM (F3). You decide to stop checking when you get a definite answer at 12 PM (F2).
Remember this
A stopping time is when you stop looking for more clues because you've got a definitive answer.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
the optional stopping theorem says about martingales and stopping times
Martingale: E[X_{n+1} | X_1, X_2, ..., X_n] = X_n
log-probabilities are used instead of probabilities: avoids numerical underflow
Why can't we just add up tiny chances over time?
Cross-entropy H(p,q) = -Σ p(x) log q(x) measures how well q approximates p
Ever wondered how well we can guess the outcome of a random event?
Entropy H = -Σ p(x) log₂ p(x) measures average surprise in bits
How do we measure uncertainty in everyday decisions?
Diffusion model
q(x_t|x_{t-1}) adds Gaussian noise at each step
the Dirichlet distribution does: distribution over probability simplices
How do we predict the likelihood of various outcomes in uncertain situations?
Swipe through 100 ML concepts daily
Open Pocket Polymath