5 Matching Annotations
1. Nov 2022
2. stats.stackexchange.com stats.stackexchange.com
1. The random process has outcomes

## Notation of a random process that has outcomes

The "universal set" aka "sample space" of all possible outcomes is sometimes denoted by $$U$$, $$S$$, or $$\Omega$$: https://en.wikipedia.org/wiki/Sample_space

## Probability theory & measure theory

From what I recall, the notation, $$\Omega$$, was mainly used in higher-level grad courses on probability theory. ie, when trying to frame things in probability theory as a special case of measure theory things/ideas/processes. eg, a probability space, $$(\cal{F}, \Omega, P)$$ where $$\cal{F}$$ is a $$\sigma\text{-field}$$ aka $$\sigma\text{-algebra}$$ and $$P$$ is a probability density function on any element of $$\cal{F}$$ and $$P(\Omega)=1.$$

Somehow, the definition of a sigma-field captures the notion of what we want out of something that's measurable, but it's unclear to me why so let's see where writing through this takes me.

## Working through why a sigma-algebra yields a coherent notion of measureable

A sigma-algebra $$\cal{F}$$ on a set $$\Omega$$ is defined somewhat close to the definition of a topology $$\tau$$ on some space $$X$$. They're both collections of sub-collections of the set/space of reference (ie, $$\tau \sub 2^X$$ and $$\cal{F} \sub 2^\Omega$$). Also, they're both defined to contain their underlying set/space (ie, $$X \in \tau$$ and $$\Omega \in \cal{F}$$).

Additionally, they both contain the empty set but for (maybe) different reasons, definitionally. For a topology, it's simply defined to contain both the whole space and the empty set (ie, $$X \in \tau$$ and $$\empty \in \tau$$). In a sigma-algebra's case, it's defined to be closed under complements, so since $$\Omega \in \cal{F}$$ the complement must also be in $$\cal{F}$$... but the complement of the universal set $$\Omega$$ is the empty set, so $$\empty \in \cal{F}$$.

I think this might be where the similarity ends, since a topology need not be closed under complements (but probably has a special property when it is, although I'm not sure what; oh wait, the complement of open is closed in topology, so it'd be clopen! Not sure what this would really entail though 🤷‍♀️). Moreover, a topology is closed under arbitrary unions (which includes uncountable), but a sigma-algebra is closed under countable unions. Hmm... Maybe this restriction to countable unions is what gives a coherent notion of being measurable? I suspect it also has to do with Banach-Tarski paradox. ie, cutting a sphere into 5 pieces and rearranging in a clever way so that you get 2 sphere's that each have the volume of the original sphere; I mean, WTF, if 1 sphere's volume equals the volume of 2 sphere's, then we're definitely not able to measure stuff any more.

And now I'm starting to vaguely recall that this what sigma-fields essentially outlaw/ban from being possible. It's also related to something important in measure theory called a Lebeque measure, although I'm not really sure what that is (something about doing a Riemann integral but picking the partition on the y-axis/codomain instead of on the x-axis/domain, maybe?)

And with that, I think I've got some intuition about how fundamental sigma-algebras are to letting us handle probability and uncertainty.

## Back to probability theory

So then events like $$E_1$$ and $$E_2$$ that are elements of the set of sub-collections, $$\cal{F}$$, of the possibility space $$\Omega$$. Like, maybe $$\Omega$$ is the set of all possible outcomes of rolling 2 dice, but $$E_1$$ could be a simple event (ie, just one outcome like rolling a 2) while $$E_2$$ could be a compound(?) event (ie, more than one, like rolling an even number). Notably, $$E_1$$ & $$E_2$$ are NOT elements of the sample space $$\Omega$$; they're elements of the powerset of our possibility space (ie, the set of all possible subsets of $$\Omega$$ denoted by $$2^\Omega$$). So maybe this explains why the "closed under complements" is needed; if you roll a 2, you should also be able to NOT roll a 2. And the property that a sigma-algebra must "contain the whole space" might be what's needed to give rise to a notion of a complete measure (conjecture about complete measures: everything in the measurable space can be assigned a value where that part of the measurable space does, in fact, represent some constitutive part of the whole).

## But what about these "random events"?

Ah, so that's where random variables come into play (and probably why in probability theory they prefer to use $$\Omega$$ for the sample space instead of $$X$$ like a base space in topology). There's a function, that is, a mapping from outcomes of this "random event" (eg, a role of 2 dice) to a space in which we can associate (ie, assign) a sense of distance (ie, our sigma-algebra). What confuses me is that we see things like "$$P(X=x)$$" which we interpret as "probability that our random variable, $$X$$, ends up being some particular outcome $$x$$." But it's also said that $$X$$ is a real-valued function, ie, takes some arbitrary elements (eg, events like rolling an even number) and assigns them a real number (ie, some $$x \in \mathbb{R}$$).

Aha! I think I recall the missing link: the notation "$$X=x$$" is really a shorthand for "$$X(\omega)=x$$" where $$\omega \in \cal{F}$$. But something that still feels unreconciled is that our probability metric, $$P$$, is just taking some real value to another real value... So which one is our sigma-algebra, the inputs of $$P$$ or the inputs of $$X$$? 🤔 Hmm... Well, I guess it has the be the set of elements that $$X$$ is mapping into $$\mathbb{R}$$ since $$X\text{'s}$$ input is a small omega $$\omega$$ (which is probably an element of big omega $$\Omega$$ based on the conventions of small notation being elements of big notation), so $$X\text{'s}$$ domain much be the sigma-algrebra?

Let's try to generate a plausible example of this in action... Maybe something with an inequality like "$$X\ge 1$$". Okay, yeah, how about $$X$$ is a random variable for the random process of how long it takes a customer to get through a grocery line. So $$X$$ is mapping the elements of our sigma-algebra (ie, what customers actually end up experiencing in the real world) into a subset of the reals, namely $$[0,\infty)$$ because their time in line could be 0 minutes or infinite minutes (geesh, 😬 what a life that would be, huh?). Okay, so then I can ask a question like "What's the probability that $$X$$ takes on a value greater than or equal to 1 minute?" which I think translates to "$$P\left(X(\omega)\ge 1\right)$$" which is really attempting to model this whole "random event" of "What's gonna happen to a particular person on average?"

So this makes me wonder... Is this fact that $$X$$ can model this "random event" (at all) what people mean when they say something is a stochastic model? That there's a probability distribution it generates which affords us some way of dealing with navigating the uncertainty of the "random event"? If so, then sigma-algebras seem to serve as a kind of gateway and/or foundation into specific cognitive practices (ie, learning to think & reason probabilistically) that affords us a way out of being overwhelmed by our anxiety or fear and can help us reclaim some agency and autonomy in situations with uncertainty.

#### URL

3. Sep 2022
4. Local file Local file
1. Khinchin, Aleksandr Yakovlevich. Continued Fractions. 3rd ed. Chicago: University of Chicago Press, 1964.

#### Annotators

5. Aug 2022
6. psyarxiv.com psyarxiv.com
1. Gelfand, M., Li, R., Stamkou, E., Pieper, D., Denison, E., Fernandez, J., Choi, V. K., Chatman, J., Jackson, J. C., & Dimant, E. (2021). Persuading Republicans and Democrats to Comply with Mask Wearing: An Intervention Tournament. PsyArXiv. https://doi.org/10.31234/osf.io/6gjh8

#### URL

7. Sep 2021
8. www.frontiersin.org www.frontiersin.org
1. Abadi, D., Arnaldo, I., & Fischer, A. (2021). Anxious and Angry: Emotional Responses to the COVID-19 Threat. Frontiers in Psychology, 12, 676116. https://doi.org/10.3389/fpsyg.2021.676116

#### URL

9. Jul 2020
10. psyarxiv.com psyarxiv.com
1. Pavela Banai, I., Banai, B., & Mikloušić, I. (2020, July 14). Beliefs in COVID-19 conspiracy theories predict lower level of compliance with the preventive measures both directly and indirectly by lowering trust in government medical officials. Retrieved from psyarxiv.com/yevq7