81 Matching Annotations
  1. Nov 2022
    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.

  2. Sep 2022
    1. Running this simulation over many time steps, Lilian Weng of OSoMe found that as agents' attention became increasingly limited, the propagation of memes came to reflect the power-law distribution of actual social media: the probability that a meme would be shared a given number of times was roughly an inverse power of that number. For example, the likelihood of a meme being shared three times was approximately nine times less than that of its being shared once.
  3. Aug 2022
    1. Generating randomness.

      EVM execution is deterministic. How to account for randomness?

      Pesudo random generator, probability distribution.

  4. May 2022
    1. In the case ofLévi-Strauss, meanwhile, the card index continued to serve inimportant ways as a ‘memory crutch’, albeit with a key differencefrom previous uses of the index as an aide-memoire. In Lévi-Strauss’case, what the fallibility of memory takes away, the card index givesback via the workings of chance. As he explains in an interview withDidier Erebon:I get by when I work by accumulating notes – a bitabout everything, ideas captured on the fly,summaries of what I have read, references,quotations... And when I want to start a project, Ipull a packet of notes out of their pigeonhole anddeal them out like a deck of cards. This kind ofoperation, where chance plays a role, helps merevive my failing memory. (Cited in Krapp, 2006:361)For Krapp, the crucial point here is that, through his use of indexcards, Lévi-Strauss ‘seems to allow that the notes may either restorememory – or else restore the possibilities of contingency which givesthinking a chance under the conditions of modernity’ (2006: 361).

      Claude Lévi-Strauss had a note taking practice in which he accumulated notes of ideas on the fly, summaries of what he read, references, and quotations. He kept them on cards which he would keep in a pigeonhole. When planning a project, he would pull them out and use them to "revive [his] failing memory."


      Questions: - Did his system have any internal linkages? - How big was his system? (Manageable, unmanageable?) - Was it only used for memory, or was it also used for creativity? - Did the combinatorial reshufflings of his cards provide inspiration a la the Llullan arts?


      Link this to the ideas of Raymond Llull's combinatorial arts.

  5. Apr 2022
    1. ReconfigBehSci. (2021, February 1). @islaut1 @richarddmorey I think of strength of inference resting on P(not E|not H) (for coronavirus case). Search determines the conditional probability (and by total probability of course prob of evidence) but it isn’t itself the evidence. So, was siding with R. against what I thought you meant ;-) [Tweet]. @SciBeh. https://twitter.com/SciBeh/status/1356216290847944706

  6. Jan 2022
    1. If the round pea parent is heterozygous, there is a one-eighth probability that a random sample of three progeny peas will all be round.

      Please clarify how you calculated 1:8 ratio. One plant has rr and the other plant has Rr. You create a table and get two Rr and two rr which makes the probability of getting three Rr or RR zero.

  7. Dec 2021
  8. Oct 2021
  9. Sep 2021
    1. Sam Wang on Twitter: “These are risk levels that you pose to other people. They’re compared with you as—A nonsmoker—A sober driver—A vaccinated person. Unvaccinated? 5x as likely to get sick, for 3x as long. Total risk to others? 15x a vaccinated person Details:https://t.co/ckTWaivK8n https://t.co/PhpLvX2dsm” / Twitter. (n.d.). Retrieved September 19, 2021, from https://twitter.com/SamWangPhD/status/1438361144759132167

  10. Aug 2021
    1. Tim Plante, MD MHS on Twitter: “Just reported: About half of recent ICU patients with #Covid19 in #Vermont are vaccinated. Sounds like the vaccines aren’t working, right? WRONG. Vaccines are working and here’s why. But first, let’s talk a bit about unprotected sex. A thread. (Refs at the end.) 1/n https://t.co/iyQcfCDAfh” / Twitter. (n.d.). Retrieved August 27, 2021, from https://twitter.com/tbplante/status/1430222978961317896

  11. Jul 2021
  12. Jun 2021
  13. May 2021
    1. think of your career as a series of experiments designed to help you learn about yourself and test out potentially great longer-term paths

      I wonder if there's a connection here to Duke, A. (2019). Thinking in Bets: Making Smarter Decisions When You Don’t Have All the Facts. Portfolio.

      I haven't read the book but it's on my list.

    1. Dr Ellie Murray. (2021, May 7). I’m seeing a lot of “these people are over-estimating risk” chatter that doesn’t acknowledge that the probability you die if you get covid is always less than the probability anyone dies if you get covid. It’s not “over-estimation” to consider community impacts. [Tweet]. @EpiEllie. https://twitter.com/EpiEllie/status/1390792624777334797

  14. Apr 2021
    1. Events AAA and BBB are mutually exclusive (cannot both occur at once) if they have no elements in common.

      Events \(A\) and \(B\) are mutually exclusive (cannot both occur at once) if they have no elements in common.


      Events \(A\) and \(B\) are mutually exclusive if: $$P(A∩B)=0$$

    2. The complement of an event AAA in a sample space SSS, denoted AcAcA^c, is the collection of all outcomes in SSS that are not elements of the set AAA. It corresponds to negating any description in words of the event AAA.

      The complement of an event \(A\) in a sample space \(S\), denoted \(A^c\), is the collection of all outcomes in \(S\) that are not elements of the set \(A\). It corresponds to negating any description in words of the event \(A\).


      The complement of an event \(A\) consists of all outcomes of the experiment that do not result in event \(A\).

      Complement formula:

      $$P(A^c)=1-P(A)$$

  15. Mar 2021
    1. 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.

    1. 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

  16. Feb 2021
  17. Dec 2020
  18. Oct 2020
    1. 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?

    1. This result of Erd ̋os [E] is famous not because it has large numbers of applications,nor because it is difficult, nor because it solved a long-standing open problem. Its famerests on the fact that it opened the floodgates to probabilistic arguments in combinatorics.If you understand Erd ̋os’s simple argument (or one of many other similar arguments) then,lodged in your mind will be a general principle along the following lines:if one is trying to maximize the size of some structure under certain constraints, andif the constraints seem to force the extremal examples to be spread about in a uniformsort of way, then choosing an example randomly is likely to give a good answer.Once you become aware of this principle, your mathematical power immediately increases.
  19. Sep 2020
  20. Aug 2020
    1. Ray, E. L., Wattanachit, N., Niemi, J., Kanji, A. H., House, K., Cramer, E. Y., Bracher, J., Zheng, A., Yamana, T. K., Xiong, X., Woody, S., Wang, Y., Wang, L., Walraven, R. L., Tomar, V., Sherratt, K., Sheldon, D., Reiner, R. C., Prakash, B. A., … Consortium, C.-19 F. H. (2020). Ensemble Forecasts of Coronavirus Disease 2019 (COVID-19) in the U.S. MedRxiv, 2020.08.19.20177493. https://doi.org/10.1101/2020.08.19.20177493

  21. Jul 2020
  22. Jun 2020
    1. Winman, A., Hansson, P., & Juslin, P. (2004). Subjective Probability Intervals: How to Reduce Overconfidence by Interval Evaluation. Journal of Experimental Psychology: Learning, Memory, and Cognition, 30(6), 1167–1175. https://doi.org/10.1037/0278-7393.30.6.1167

  23. May 2020
  24. Apr 2020
    1. Therefore, En=2n+1−2=2(2n−1)

      Simplified formula for the expected number of tosses (e) to get n consecutive heads (n≥1):

      $$e_n=2(2^n-1)$$

      For example, to get 5 consecutive heads, we've to toss the coin 62 times:

      $$e_n=2(2^5-1)=62$$


      We can also start with the longer analysis of the 5 scenarios:

      1. If we get a tail immediately (probability 1/2) then the expected number is e+1.
      2. If we get a head then a tail (probability 1/4), then the expected number is e+2.
      3. If we get two head then a tail (probability 1/8), then the expected number is e+2.
      4. If we get three head then a tail (probability 1/16), then the expected number is e+4.
      5. If we get four heads then a tail (probability 1/32), then the expected number is e+5.
      6. Finally, if our first 5 tosses are heads, then the expected number is 5.

      Thus:

      $$e=\frac{1}{2}(e+1)+\frac{1}{4}(e+2)+\frac{1}{8}(e+3)+\frac{1}{16}\\(e+4)+\frac{1}{32}(e+5)+\frac{1}{32}(5)=62$$

      We can also generalise the formula to:

      $$e_n=\frac{1}{2}(e_n+1)+\frac{1}{4}(e_n+2)+\frac{1}{8}(e_n+3)+\frac{1}{16}\\(e_n+4)+\cdots +\frac{1}{2^n}(e_n+n)+\frac{1}{2^n}(n) $$

  25. Jan 2020
    1. 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!

  26. Dec 2019
    1. the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.
    1. 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.

  27. Mar 2019
    1. Special Complexity Zoo Exhibit: Classes of Quantum States and Probability Distributions 24 classes and counting! A whole new phylum of the Complexity kingdom has recently been identified. This phylum consists of classes, not of problems or languages, but of quantum states and probability distributions. Well, actually, infinite families of states and distributions, one for each number of bits n. Admittedly, computer scientists have been talking about the complexity of sampling from probability distributions for years, but they haven't tended to organize those distributions into classes designated by inscrutable sequences of capital letters. This needs to change.
  28. Feb 2019
    1. n they will share similar genes, but it 18is the phenotype –upon which selection acts –which is crucia

      There two important things to note.

      1. If the same genetic programme leads to two phenotypes because of the environment, this falls in the category of epigenetics. Epigenetic processes are usually not tree-like, hence, poorly modelled by inferring a tree.

      2. You implicitly assume (via your R-script) that homoiologies (in a strict sense, i.e. parallelism) are rare and not beneficial (neutral). But if the homoiology is beneficial (i.e. positively selected for), it will be much more common in a clade of close relatives than the primitive phenotype (the symplesiomorphy). We can further assume that beneficial homoiologies will accumulate in the most-derived, advanced, specialised taxa, in the worst case (from the mainstream cladistic viewpoint) mimicking or even outcompeting synapomorphies. A simply thought example: let's say we have a monophylum (fide Hennig) with two sublineages, each sublineage defined by a single synapormorphy. Both sublineages radiate and invade in parallel a new niche (geographically separated from each other) and fix (evolve) a set of homoiologies in adaptation to that new niche. The members of both sublineages with the homoiologies will be resolved as one clade, a pseudo-monophylum, supported by the homoiologies as pseudo-synapomorphies. And the actual synapomorphies will be resolved as plesiomorphies or autapomorphies.

      Without molecular (and sometime even with, many molecular trees are based on plastid in plants and mitochondria in animals, and both are maternally inherited, hence, geographically controlled) or ontological-physiological control it will be impossible to make a call what is derived (hence a potential homoiology) and what ancestral in a group of organisms sharing a relative recent common origin and a still similiar genetic programme.

  29. Oct 2018
    1. In contrast to his concept of a simple circular orbit with a fixed radius, orbitals are mathematically derived regions of space with different probabilities of having an electron.

      In this case, the QM model allows for probabilistic radii, not fixed radii, and the quantization is the energy level. An electron with principal quantum number n = 2 will always have quantized energy corresponding to \( E = R(1/n^2) \), but the exact minimal and maximal radial distance from the nucleus is not specified as in the Bohr model of the atom. Similar to the Bohr model though, the most probable radial distance is quantifiable, and that is the radius the electron is most likely to inhabit, however it will be found elsewhere at other times.

  30. Sep 2017
    1. Terrorist use of an actual nuclear bomb is a low-probability event

      Low probability and high impact but not a black swan

  31. Feb 2017
    1. These two qualities, therefore, PROBABILITY and PLAUSIBILITY

      This is an important set of terms to think through in terms of come to think about and with rhetoric.

    2. CHAPTER VI

      Chapter VII: General Audience Awareness

      But, really, Mere Rhetoric has a nice (I'm assuming she's mostly on point here) summary of some of the concepts to follow.

  32. Jan 2017
    1. Hume considers the possibility that there is, indeed, complete relativism in this matter. But his purpose is to find ways to reduce or eliminate disagreement, to set a standard

      A rhetorical concern dating back to at least Aristotle: how to decide upon things in the realm of the probable rather than the absolute.

  33. Nov 2016
    1. Finally, by assuming the non-detection of a species to indicate absence from a given grid cell, we introduced an extra level of error into our models. This error depends on the probability of false absence given imperfect detection (i.e., the probability that a species was present but remained undetected in a given grid cell [73]): the higher this probability, the higher the risk of incorrectly quantifying species-climate relationships [73].

      This will be an ongoing challenge for species distribution modeling, because most of the data appropriate for these purposes is not collected in such a way as to allow the straightforward application of standard detection probability/occupancy models. This could potentially be addressed by developing models for detection probability based on species and habitat type. These models could be built on smaller/different datasets that include the required data for estimating detectability.

  34. Jul 2016
  35. Feb 2016
    1. Great explanation of 15 common probability distributions: Bernouli, Uniform, Binomial, Geometric, Negative Binomial, Exponential, Weibull, Hypergeometric, Poisson, Normal, Log Normal, Student's t, Chi-Squared, Gamma, Beta.

  36. Jan 2016
    1. P(B|E) = P(B) X P(E|B) / P(E), with P standing for probability, B for belief and E for evidence. P(B) is the probability that B is true, and P(E) is the probability that E is true. P(B|E) means the probability of B if E is true, and P(E|B) is the probability of E if B is true.
    1. paradox of unanimity - Unanimous or nearly unanimous agreement doesn't always indicate the correct answer. If agreement is unlikely, it indicates a problem with the system.

      Witnesses who only saw a suspect for a moment are not likely to be able to pick them out of a lineup accurately. If several witnesses all pick the same suspect, you should be suspicious that bias is at work. Perhaps these witnesses were cherry-picked, or they were somehow encouraged to choose a particular suspect.

  37. Oct 2015
    1. Nearly all ap­pli­ca­tions of prob­a­bil­ity to cryp­tog­ra­phy de­pend on the fac­tor prin­ci­ple (or Bayes’ The­o­rem).

      This is easily the most interesting sentence in the paper: Turing used Bayesian analysis for code-breaking during WWII.

  38. Oct 2013