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It looks like a good article for those who are already studying theoretical probability. For laypeople, not so much. I haven't yet figured out what the author

It looks like a good article for those who are already studying theoretical probability. For laypeople, not so much. I haven't yet figured out what the author is doing with the sigma algebra machinery. I had thought that the event set of a probability space was intuitively the power set of the sample set, and that the machinery of sigma algebras was only added to avoid pathologies like non-measurable sets (think Banach-Tarski paradox). Maybe there is more to it? I would like to understand this better and I do plan to read the article, but I had found the Wikipedia article on martingales to be reasonably understandable a while back. The good thing about this article is it shows applications.

ajkjk 4y ago
I'm not a layperson (not really, at least) and I strongly feel like the sigma algebra framework is basically a waste of time. Mathematicians love abstracting and formalizing things but it's not useful, paradoxically, if you're trying to actually just do math.
- jhanschoo 4y ago
  Sigma algebras are primarily useful for modeling less well-behaved probability spaces when you aren't dealing with (a product space of) probability spaces of that can be described by a pdf or pmf
slopbop 4y ago
Sigma algebras are useful even in a finite setting where we don't have to worry about pathologies. Think of them as modelling the lack of complete information in a probabilistic setting. If I know exactly which sample point represents my state, then I know the exact value of any random variable. If instead I only know that my state belongs to a given member of my sigma algebra, then I have some information, but not enough to necessarily pinpoint the value of a random variable. In fact, the familiar tools of measure theory can take this intuition further. If a random variable is measurable with respect to a sigma algebra, then knowing which element of that sigma algebra my state is in actually is sufficient to pinpoint the value of a random variable. Maybe to make this more concrete: Let's say I'm going to do two coinflips. My probability space is {HH, HT, TH, TT}. You can check for yourself that the sigma algebra generated by {{HH, HT}}, {TT, TH}} is not the trivial one- this is the sigma algebra that represents "Knowing the value of the first flip, but not the second". If we let X_first and X_second be 1 or 0 if the first or second flip is H or T respectively, then X_first is measurable with respect to this sigma algebra, but X_second is not. With Martingales and other stochastic processes, we don't generally have just one sigma algebra, but a sequence of sigma algebras called a "filtration", where each sigma algebra is finer than the last (ie, contains more sets, therefore gives you more measurable random variables). This filtration sort of defines the stochastic process- it's encoding the slow drip of extra information as the stochastic process evolves over time.

ajkjk 4y ago

I'm not a layperson (not really, at least) and I strongly feel like the sigma algebra framework is basically a waste of time. Mathematicians love abstracting and formalizing things but it's not useful, paradoxically, if you're trying to actually just do math.

jhanschoo 4y ago
Sigma algebras are primarily useful for modeling less well-behaved probability spaces when you aren't dealing with (a product space of) probability spaces of that can be described by a pdf or pmf

slopbop 4y ago

Sigma algebras are useful even in a finite setting where we don't have to worry about pathologies. Think of them as modelling the lack of complete information in a probabilistic setting. If I know exactly which sample point represents my state, then I know the exact value of any random variable. If instead I only know that my state belongs to a given member of my sigma algebra, then I have some information, but not enough to necessarily pinpoint the value of a random variable. In fact, the familiar tools of measure theory can take this intuition further. If a random variable is measurable with respect to a sigma algebra, then knowing which element of that sigma algebra my state is in actually is sufficient to pinpoint the value of a random variable. Maybe to make this more concrete: Let's say I'm going to do two coinflips. My probability space is {HH, HT, TH, TT}. You can check for yourself that the sigma algebra generated by {{HH, HT}}, {TT, TH}} is not the trivial one- this is the sigma algebra that represents "Knowing the value of the first flip, but not the second". If we let X_first and X_second be 1 or 0 if the first or second flip is H or T respectively, then X_first is measurable with respect to this sigma algebra, but X_second is not. With Martingales and other stochastic processes, we don't generally have just one sigma algebra, but a sequence of sigma algebras called a "filtration", where each sigma algebra is finer than the last (ie, contains more sets, therefore gives you more measurable random variables). This filtration sort of defines the stochastic process- it's encoding the slow drip of extra information as the stochastic process evolves over time.