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1. Last 7 days
1. if we lose the Green  and Ice Sheet, or the AMOC, it would be a complete disaster. So, you cannot measure  it economically, it's an infinite parameter. So then, if the probability, even if the  probability is low, if you multiply a low probability with an infinite impact,  then risks are also infinitely high.

for - planetary emergency - risk analysis

planetary emergency - risk analysis - risk = probability x impact - If impact is high, then even low probability x high impact means high risk - If AMOC or Greenland icesheet melts, the impact is so high that it is not even economically measurable

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3. Mar 2023
4. www.liberation.fr www.liberation.fr

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5. Nov 2022
6. 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.

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

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9. Local file Local file
1. The mostcommon relative poverty measure is one that counts individuals as poor if theyfall below one-half of a country’s median income.4

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10. Aug 2022
11. securingdemocracy.gmfus.org securingdemocracy.gmfus.org
1. Schafer, B. (2021, October 5). RT Deutsch Finds a Home with Anti-Vaccination Skeptics in Germany. Alliance For Securing Democracy. https://securingdemocracy.gmfus.org/rt-deutsch-youtube-antivaccination-germany/

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12. www.t-online.de www.t-online.de
1. Hunderte Menschen protestierten gegen Corona-Maßnahmen. (2021, November 28). www.t-online.de. https://www.t-online.de/-/91225726

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13. www.theguardian.com www.theguardian.com
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14. www.spectator.co.uk www.spectator.co.uk
1. Kulldorff, M. (2021, October 12). Covid, lockdown and the retreat of scientific debate | The Spectator. https://www.spectator.co.uk/article/covid-lockdown-and-the-retreat-of-scientific-debate

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15. 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

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1. ReconfigBehSci. (2021, December 16). RT @MadsAlbertsen85: #Omicron update from Denmark. Omicron cases on the 12th of December adjusted up to 20.5% (+2%). Omicron is now having… [Tweet]. @SciBeh. https://twitter.com/SciBeh/status/1471516083819462670

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1. ReconfigBehSci. (2021, December 22). RT @chrishendel: We need increased targeted measures now to slow the spread of #Omicron by @DrSarahHal via @bmj_latest https://t.co/3Tx… [Tweet]. @SciBeh. https://twitter.com/SciBeh/status/1473987200509521925

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18. www.bbc.co.uk www.bbc.co.uk
1. Covid: Dutch go into Christmas lockdown over Omicron wave. (2021, December 18). BBC News. https://www.bbc.com/news/world-europe-59713503

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19. Mar 2022
20. www.tagesanzeiger.ch www.tagesanzeiger.ch
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21. Feb 2022
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23. gothamist.com gothamist.com
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1. Peter R. Hansen. (2022, February 3). Weighting, is the answer. The only study to find lockdowns ⬆️mortality is given weight 91.8% = 7390/8030, and then you get -0.2% to be the estimate. To summarize: -0.2% META-STUDY ESTIMATE is based on 91.8% ONE STUDY and 8.2% ALL OTHER STUDIES. https://t.co/j6e7ziPNAI [Tweet]. @ProfPHansen. https://twitter.com/ProfPHansen/status/1489366528956919808

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25. psyarxiv.com psyarxiv.com
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26. Jan 2022
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1. Barry McAree 💙. (2022, January 6). Teachers on these islands will get FFP2(rightly so).Healthcare workers on other parts of these islands..nah!..Surgical masks/spit guards/not PPE,for working with COVID-positive patients risking other patient’s, our own & our family’s health.”Protect the NHS”🤔⁦@CMO_England⁩ https://t.co/OngrD5BBPU [Tweet]. @BarryMcAree. https://twitter.com/BarryMcAree/status/1478883258305814536

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29. Dec 2021
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32. Nov 2021
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1. WHO/Europe. (2021, October 19). Are you a Mask Master? 😷 Take this quick quiz to find out 👉 https://bit.ly/3jbn5iS Wearing a well-fitted mask, along with practicing other prevention measures, is an important part of slowing the spread of #COVID19 High quiz scores = Mask Master badge 🎖 https://t.co/0PzSsgsBfD [Tweet]. @WHO_Europe. https://twitter.com/WHO_Europe/status/1450451677098790916

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36. news.harvard.edu news.harvard.edu
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37. psyarxiv.com psyarxiv.com
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1. ReconfigBehSci. (2021, October 30). Does there maybe need to be more distinction between points raised for discussion and any actual decision? Without knowing about votes etc., it’s maybe a bit strong to say ‘JCVI wanted x...’? I’ve sat on many bodies with minutes documenting positions I disagreed with [Tweet]. @SciBeh. https://twitter.com/SciBeh/status/1454488759785897987

40. Oct 2021

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