2 ms·
In probability theory there's the concept of a filtration. Rigorously speaking, probabilities are not attributed to singular events, but to sets (possibly, in
by thanatropism 8y ago
In probability theory there's the concept of a filtration.
Rigorously speaking, probabilities are not attributed to singular events, but to sets (possibly, in the discrete case, single-element sets such as {heads}, {tails}).
To be even more precise: a sigma-algebra is a concept of set-of-all-sets that avoids continuous-case paradoxes through technical stuff you can ignore.
A filtration is a decreasing sequence of sigma algebras: for example, if your event space is the six numbers a die might show, you could have
F(0) = { {}, {die number is odd}, {die number is even}}
and
F(1) = { {}, {die number is 1},...{die number is 6}, {die number is 1 or 2},...} (a sigma algebra is closed under intersections)
What this means is that earlier probabilities are defined on coarser spaces of sets of events as compared to later probabilities. This is how you get dynamic stochastic processes that don't look into the future.