 Nov 2022

stats.stackexchange.com stats.stackexchange.com

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 higherlevel 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 sigmafield 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 sigmaalgebra yields a coherent notion of measureable
A sigmaalgebra \(\cal{F}\) on a set \(\Omega\) is defined somewhat close to the definition of a topology \(\tau\) on some space \(X\). They're both collections of subcollections 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 sigmaalgebra'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 sigmaalgebra 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 BanachTarski 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 sigmafields 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 yaxis/codomain instead of on the xaxis/domain, maybe?)
And with that, I think I've got some intuition about how fundamental sigmaalgebras 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 subcollections, \(\cal{F}\), 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 sigmaalgebra 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 sigmaalgebra). 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 realvalued 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 sigmaalgebra, 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 sigmaalgrebra?
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 sigmaalgebra (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 sigmaalgebras 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.

 Sep 2022

Local file Local file

B. V. Gnf.di.xko
Boris Vladimirovich Gnedenko https://en.wikipedia.org/wiki/Boris_Vladimirovich_Gnedenko
He wrote with/worked with Khinchin and Kolmogorov.

A . Y A . K H I N C H I N
Aleksandr Yakovlevich Khinchin https://en.wikipedia.org/wiki/Aleksandr_Khinchin

 Mar 2021

www.namics.nysaes.cornell.edu www.namics.nysaes.cornell.edu

In each of the games moves are entirely determined by chance; there is no opportunity to make decisions regarding play. (This, of course, is one reason why most adults with any intellectual capacity have little interest in playing the games for extended times, especially since no money or alcohol is involved.)
The entire motivation for this study.



Devriendt, K., MartinGutierrez, S., & Lambiotte, R. (2020). Variance and covariance of distributions on graphs. ArXiv:2008.09155 [Physics, Stat]. http://arxiv.org/abs/2008.09155



<small><cite class='hcite via'>ᔥ <span class='pauthor hcard'>hyperlink.academy</span> in The Future of Textbooks (<time class='dtpublished'>03/18/2021 23:54:19</time>)</cite></small>


arxiv.org arxiv.org

Hota, Ashish R., Tanya Sneh, and Kavish Gupta. ‘Impacts of GameTheoretic Activation on Epidemic Spread over Dynamical Networks’. ArXiv:2011.00445 [Physics], 1 November 2020. http://arxiv.org/abs/2011.00445.


danallosso.substack.com danallosso.substack.com

He introduces the idea of the apophatic: what we can't put into words, but is important and vaguely understood. This term comes from Orthodox theology, where people defined god by saying what it was not.
Too often as humans we're focused on what is immediately in front of us and not what is missing.
This same thing plagues our science in that we're only publishing positive results and not negative results.
From an information theoretic perspective, we're throwing away half (or more?) of the information we're generating. We might be able to go much farther much faster if we were keeping and publishing all of our results in better fashion.
Is there a better word for this negative information? #openquestions


kids.frontiersin.org kids.frontiersin.org
 Dec 2020

cims.nyu.edu cims.nyu.edu

gromov
Mikhael Gromov's docs at NYU
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 Oct 2020

parenting.nytimes.com parenting.nytimes.com

How this phenomenon translates into absolute, rather than relative, risk, however, is a bit thorny. A large study published in 2018, for instance, found that among women who had children between 34 and 47, 2.2 percent developed breast cancer within three to seven years after they gave birth (among women who never had children, the rate was 1.9 percent). Over all, according to the American Cancer Society, women between 40 and 49 have a 1.5 percent chance of developing breast cancer.
The rates here are so low as to be nearly negligible on their face. Why bother reporting it?

 Jan 2020

levels.io levels.io

My friend Marc again to the rescue. He suggested that since there was 10,000+ people RT'ing and following, I could just pick a random follower from my current total follower list (78,000 at this point), then go to their profile to check if they RT'd it and see. If they didn't, get another random follower and repeat, until you find someone. With 78,000 followers this should take about ~8 tries.
Technically he said it would be random among those who retweeted, but he's chose a much smaller subset of people who are BOTH following him and who retweeted it. Oops!
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 Dec 2019

colinwalker.blog colinwalker.blog

Many people luck out like me, accidentally. We recognize what particular path to mastery we’re on, long after we actually get on it.
Far too many people luck out this way and we all perceive them as magically talented when in reality, they're no better than we, they just had better circumstances or were in the right place at the right time.
