- Mar 2023
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www.liberation.fr www.liberation.fr
- Nov 2022
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stats.stackexchange.com stats.stackexchange.com
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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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- Sep 2022
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Khinchin, Aleksandr Yakovlevich. Continued Fractions. 3rd ed. Chicago: University of Chicago Press, 1964.
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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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- Aug 2022
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securingdemocracy.gmfus.org securingdemocracy.gmfus.org
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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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Hunderte Menschen protestierten gegen Corona-Maßnahmen. (2021, November 28). www.t-online.de. https://www.t-online.de/-/91225726
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Pagel, C., & McKee, M. (2021, October 7). Why is England doing worse against Covid than its European neighbours? The Guardian. https://www.theguardian.com/commentisfree/2021/oct/07/england-vaccine-just-plus-europe-covid
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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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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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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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twitter.com twitter.com
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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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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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- Mar 2022
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Brupbacher, M., & Vögeli, P. (2022, January 31). Wegen Corona-Massnahmen – Die Masern sind in der Schweiz verschwunden – bleibt das so? Tages-Anzeiger. https://www.tagesanzeiger.ch/die-masern-sind-in-der-schweiz-verschwunden-bleibt-das-so-164628795215
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- Feb 2022
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www.youtube.com www.youtube.com
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indie_SAGE. (2022, February 11). Indie_SAGE 11.02.2022. https://www.youtube.com/watch?v=0GC9QsKfrNc
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gothamist.com gothamist.com
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How cherry-picking science became the center of the anti-mask movement. (2022, February 14). Gothamist. https://gothamist.com
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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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Andersen, D. B., Petersen, M. B., Midtgaard, S. F., Højlund, A.-S. G., Lippert-Rasmussen, K., & Pedersen, V. M. L. (2021). Collective paternalism and vaccination programmes. PsyArXiv. https://doi.org/10.31234/osf.io/5hvqc
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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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Sridhar, D. (2021, December 9). Plan B shows Britain is still chasing Covid. Here’s how we can get ahead of it. The Guardian. https://www.theguardian.com/commentisfree/2021/dec/09/plan-b-britain-covid-omicron-vaccinating-travel
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Gregory, A., & editor, A. G. H. (2021, November 18). Mask-wearing cuts Covid incidence by 53%, says global study. The Guardian. https://www.theguardian.com/world/2021/nov/17/wearing-masks-single-most-effective-way-to-tackle-covid-study-finds
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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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Writer, L. M. H. S. (2021, March 25). Harvard experts discuss the history of social distancing. Harvard Gazette. https://news.harvard.edu/gazette/story/2021/03/harvard-experts-discuss-the-history-of-social-distancing/
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Manoochehri, M., Šrol, J., Asl, F. A., Mehdinasab, M., & Akhoundi, Z. (2021). Association of Mental Fatigue due to Long-term Restrictive Measures with Reasoning: A COVID-19 Study. PsyArXiv. https://doi.org/10.31234/osf.io/4yme9
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Tennessee Bills Would Shield Docs Who Spread COVID Misinformation. (2021, October 29). https://www.medpagetoday.com/special-reports/exclusives/95355
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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
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Reicher, S. (2021, October 28). Covid measures give us choice. They are not restrictions on British life. The Guardian. https://www.theguardian.com/commentisfree/2021/oct/28/covid-measures-choice-restrictions-science-freedom-infections-safe
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Sonabend, R., Whittles, L. K., Imai, N., Perez-Guzman, P. N., Knock, E. S., Rawson, T., Gaythorpe, K. A. M., Djaafara, B. A., Hinsley, W., FitzJohn, R. G., Lees, J. A., Kanapram, D. T., Volz, E. M., Ghani, A. C., Ferguson, N. M., Baguelin, M., & Cori, A. (2021). Non-pharmaceutical interventions, vaccination, and the SARS-CoV-2 delta variant in England: A mathematical modelling study. The Lancet, 0(0). https://doi.org/10.1016/S0140-6736(21)02276-5
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Meuris, C., Kremer, C., Geerinck, A., Locquet, M., Bruyère, O., Defêche, J., Meex, C., Hayette, M.-P., Duchene, L., Dellot, P., Azarzar, S., Maréchal, N., Sauvage, A.-S., Frippiat, F., Giot, J.-B., Léonard, P., Fombellida, K., Moutschen, M., Durkin, K., … Darcis, G. (2021). Transmission of SARS-CoV-2 After COVID-19 Screening and Mitigation Measures for Primary School Children Attending School in Liège, Belgium. JAMA Network Open, 4(10), e2128757. https://doi.org/10.1001/jamanetworkopen.2021.28757
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Davis, N. (2021, October 22). Act early on rising UK Covid cases or face harsher measures, Sage experts warn. The Guardian. https://www.theguardian.com/politics/2021/oct/22/act-early-on-rising-uk-covid-cases-rise-or-face-harsher-measures-experts-warn
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Peeples, L. (2021). COVID reinfections likely within one or two years, models propose. Nature. https://doi.org/10.1038/d41586-021-02825-8
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Mlcochova, P., Kemp, S. A., Dhar, M. S., Papa, G., Meng, B., Ferreira, I. A. T. M., Datir, R., Collier, D. A., Albecka, A., Singh, S., Pandey, R., Brown, J., Zhou, J., Goonawardane, N., Mishra, S., Whittaker, C., Mellan, T., Marwal, R., Datta, M., … Gupta, R. K. (2021). SARS-CoV-2 B.1.617.2 Delta variant replication and immune evasion. Nature, 1–6. https://doi.org/10.1038/s41586-021-03944-y
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Chu, J. S. G., & Evans, J. A. (2021). Slowed canonical progress in large fields of science. Proceedings of the National Academy of Sciences, 118(41). https://doi.org/10.1073/pnas.2021636118
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www.acpjournals.org www.acpjournals.org
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Moghadas, S. M., Sah, P., Shoukat, A., Meyers, L. A., & Galvani, A. P. (2021). Population Immunity Against COVID-19 in the United States. Annals of Internal Medicine. https://doi.org/10.7326/M21-2721
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- Aug 2021
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psyarxiv.com psyarxiv.com
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Constant, A., Conserve, D. F., Gallopel-Morvan, K., & Raude, J. (2021). Acceptance of COVID-19 preventive measures as a tradeoff between health and social outcomes. PsyArXiv. https://doi.org/10.31234/osf.io/ytz8p
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- Jul 2021
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psyarxiv.com psyarxiv.com
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Warmerdam, R., Wiersma, H. H., Lanting, P., Dijkema, M. X. L., Vonk, J. M., Boezen, M. H. M., Deelen, P., & Franke, L. (2021). Increased genetic contribution to wellbeing during the COVID-19 pandemic. PsyArXiv. https://doi.org/10.31234/osf.io/uksxt
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Is Your Office Safe from COVID? What to Know Now That Your Boss Wants You Back—Scientific American. (n.d.). Retrieved July 15, 2021, from https://www.scientificamerican.com/article/is-your-office-safe-from-covid-what-to-know-now-that-your-boss-wants-you-back1/
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- Jun 2021
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psyarxiv.com psyarxiv.com
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Mills, F., Symons, D. C., & Carter, D. H. (2021). Exploring the Role of Enforcement in Promoting Adherence with Protective Behaviours During COVID-19. PsyArXiv. https://doi.org/10.31234/osf.io/89sj3
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www.medrxiv.org www.medrxiv.org
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V Shah, A. S., Gribben, C., Bishop, J., Hanlon, P., Caldwell, D., Wood, R., Reid, M., McMenamin, J., Goldberg, D., Stockton, D., Hutchinson, S., Robertson, C., McKeigue, P. M., Colhoun, H. M., & McAllister, D. A. (2021). Effect of vaccination on transmission of COVID-19: An observational study in healthcare workers and their households [Preprint]. Public and Global Health. https://doi.org/10.1101/2021.03.11.21253275
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Soo, Z. (2021, May 14). Singapore tightens COVID-19 measures, travel bubble unlikely. Coronavirus. https://www.ctvnews.ca/health/coronavirus/singapore-tightens-covid-19-measures-travel-bubble-unlikely-1.5427967
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twitter.com twitter.com
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Dr. Tom Frieden. (2021, April 30). Globally, the end of the pandemic isn’t near. More than a million lives depend on improving our response quickly. Don’t be blinded by the light at the end of the tunnel. There isn’t enough vaccine and the virus is gathering strength & speed. Global cooperation is crucial. 1/ [Tweet]. @DrTomFrieden. https://twitter.com/DrTomFrieden/status/1388172436999376899
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twitter.com twitter.com
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Kit Yates. (2021, May 1). Just hypothetically, if it were possible, if you could go back to January 2020 would you prefer to do what New Zealand did or what the U.K. did to ‘manage the pandemic’? [Tweet]. @Kit_Yates_Maths. https://twitter.com/Kit_Yates_Maths/status/1388375916900933633
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science.sciencemag.org science.sciencemag.org
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Pavelka, M., Van-Zandvoort, K., Abbott, S., Sherratt, K., Majdan, M., Group†, C. C.-19 working, Analýz†, I. Z., Jarčuška, P., Krajčí, M., Flasche, S., & Funk, S. (2021). The impact of population-wide rapid antigen testing on SARS-CoV-2 prevalence in Slovakia. Science, 372(6542), 635–641. https://doi.org/10.1126/science.abf9648
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Pancevski, B. (2020, December 6). Long a Holdout From Covid-19 Restrictions, Sweden Ends Its Pandemic Experiment. Wall Street Journal. https://www.wsj.com/articles/long-a-holdout-from-covid-19-restrictions-sweden-ends-its-pandemic-experiment-11607261658
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www.theguardian.com www.theguardian.com
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Gayle, C. S. (now); D., Marsh, S., Rawlinson, K., Sullivan, H., Doherty (earlier), B., Boffey, D., & Smith, H. (2020, November 16). Sweden limits public gatherings to eight people – as it happened. The Guardian. https://www.theguardian.com/world/live/2020/nov/16/coronavirus-live-news-us-confirms-1m-cases-in-a-week-boris-johnson-selfisolating?page=with%3Ablock-5fb28c538f082807d3a4fcaa
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www.swissinfo.ch www.swissinfo.ch
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Keystone-SDA/gw. (n.d.). Emergency care workers urge Swiss government to act as Covid cases soar. SWI Swissinfo.Ch. Retrieved 1 March 2021, from https://www.swissinfo.ch/eng/emergency-care-workers-urge-swiss-government-to-act-as-covid-cases-surge/46103186
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www.reuters.com www.reuters.com
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Revill, J. (2020, October 18). Swiss tighten measures to combat COVID-19 second wave. Reuters. https://www.reuters.com/article/uk-health-coronavirus-swiss-idUKKBN2730G4
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twitter.com twitter.com
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Covid One Year Ago on Twitter. (n.d.). Twitter. Retrieved 3 March 2021, from https://twitter.com/YearCovid/status/1367044325054423041
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- Mar 2021
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