for example, they produce sequences where the proportion of heads in any short segment stays far closer to 0.5 than the laws of chance would predict (Tune, 1964) // Thus, each segment of the response sequence is highly representative of the "fairness" of the coin

• Could this be a nice idea to have student predict 10 consecutive coin toss outcomes and compare to a simulation?
• Explain concepts of probability, intuition of "representative" sampling

Subjects act as if every segment of the random sequence must reflect the true proportion: if the sequence has strayed from the population proportion, a corrective bias in the other direction is expected. This has beenc alled the gambler's fallacy.

How to win in Lotto every time:

This is a good example of the fuzzy sorts of boundaries created by adding probabilities to individuals and putting them into (equivalence) classes. They can provide distributions of likelihoods.

This expands on: https://hypothes.is/a/3FVi6JtXEe2Xwp_BIaCv5g

Reframing signal relationships into probability spaces may mean that signal relationships are symmetric.

How far can this be pressed? They'll also likely be reflexive and transitive (though the probability may be smaller here) and thus make an equivalence relation.

How far can we press this idea of equivalence relations here with respect to our work? Presumably it would work to the level of providing at least good general distribution?

Amos Tversky's famous "The Hot Hand in Basketball: On the Misperception of Random Sequences".

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.

Boris Vladimirovich Gnedenko https://en.wikipedia.org/wiki/Boris_Vladimirovich_Gnedenko

He wrote with/worked with Khinchin and Kolmogorov.

Aleksandr Yakovlevich Khinchin https://en.wikipedia.org/wiki/Aleksandr_Khinchin

EVM execution is deterministic. How to account for randomness?

Pesudo random generator, probability distribution.

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.

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

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.

Lai, J., German, J., Hong, F., Tai, S.-H. S., McPhaul, K. M., Milton, D. K., & Group, for the U. of M. S. R. (2021). Comparison of Saliva and Mid-Turbinate Swabs for Detection of COVID-19 (p. 2021.12.01.21267147). https://doi.org/10.1101/2021.12.01.21267147

Washington Post. ‘Opinion | Remaining Unvaccinated in Public Should Be Considered as Bad as Drunken Driving’, 15 September 2021. https://www.washingtonpost.com/opinions/2021/09/15/remaining-unvaccinated-public-should-be-considered-bad-drunken-driving/.

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

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

Sun, Q., Lu, J., Zhang, H., & Liu, Y. (2021). Social Distance Reduces the Biases of Overweighting Small Probabilities and Underweighting Large Probabilities. Personality and Social Psychology Bulletin, 47(8), 1309–1324. https://doi.org/10.1177/0146167220969051

Moore, D. A., Backus, M., & Little, A. T. (2021). Constraints on Thinking Cause Overprecision [Preprint]. PsyArXiv. https://doi.org/10.31234/osf.io/evcx2

Cheng, Y., Ma, N., Witt, C., Rapp, S., Wild, P. S., Andreae, M. O., Pöschl, U., & Su, H. (2021). Face masks effectively limit the probability of SARS-CoV-2 transmission. Science. https://doi.org/10.1126/science.abg6296

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.

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

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: $$P(A∩B)=0$$

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:

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

The entire motivation for this study.

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

Traczyk, J., Fulawka, K., Lenda, D., & Zaleskiewicz, T. (n.d.). Consistency in probability processing as a function of affective context and numeracy. Journal of Behavioral Decision Making, n/a(n/a). https://doi.org/10.1002/bdm.2206

<small><cite class='h-cite via'>ᔥ <span class='p-author h-card'>hyperlink.academy</span> in The Future of Textbooks (<time class='dt-published'>03/18/2021 23:54:19</time>)</cite></small>

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

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

Unrealistic optimism about future life events: A cautionary note. (n.d.). Retrieved March 4, 2021, from https://psycnet.apa.org/fulltext/2010-22979-001.pdf?auth_token=a25fd4b7f008a50b15fd7b0f1fdb222fc38373f4

Westreich, D., & Iliinsky, N. (2014). Epidemiology Visualized: The Prosecutor’s Fallacy. American Journal of Epidemiology, 179(9), 1125–1127. https://doi.org/10.1093/aje/kwu025

Hanea, A., Wilkinson, D. P., McBride, M., Lyon, A., Ravenzwaaij, D. van, Thorn, F. S., Gray, C. T., Mandel, D. R., Willcox, A., Gould, E., Smith, E., Mody, F., Bush, M., Fidler, F., Fraser, H., & Wintle, B. (2021). Mathematically aggregating experts’ predictions of possible futures. MetaArXiv. https://doi.org/10.31222/osf.io/rxmh7

Mikhael Gromov's docs at NYU

Nodes

Kunin, D. (n.d.). Seeing Theory. Retrieved October 27, 2020, from http://seeingtheory.io

David Spiegelhalter and False Positives. (2020, October 14). https://www.youtube.com/watch?v=XmiEzi54lBI&feature=youtu.be

read "statistical mechanics"

The rates here are so low as to be nearly negligible on their face. Why bother reporting it?

Nick Brown on Twitter. (n.d.). Twitter. Retrieved October 1, 2020, from https://twitter.com/sTeamTraen/status/1311282470084644865

Holman, E. A., Thompson, R. R., Garfin, D. R., & Silver, R. C. (2020). The unfolding COVID-19 pandemic: A probability-based, nationally representative study of mental health in the U.S. Science Advances, eabd5390. https://doi.org/10.1126/sciadv.abd5390

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

Explaining Governors’ Response to the COVID-19 Pandemic in the United States. COVID-19 and the Labor Market. (n.d.). IZA – Institute of Labor Economics. Retrieved August 8, 2020, from https://covid-19.iza.org/publications/dp13137/

Lo, A. W., Siah, K. W., & Wong, C. H. (2020). Estimating Probabilities of Success of Vaccine and Other Anti-Infective Therapeutic Development Programs (Working Paper No. 27176; Working Paper Series). National Bureau of Economic Research. https://doi.org/10.3386/w27176

BBC Radio 4—The Political School, Episode 1. (n.d.). BBC. Retrieved August 2, 2020, from https://www.bbc.co.uk/programmes/m000kv6v

Andia, T., Mantilla, C., Rodriguez-Lesmes, P., Criado, L., Gomez, J. S., Ortiz, S., Quintero, A., Rincón, H., & Romero, S. (2020). Mentioning anosmia improves how community pharmacies handle phone call requests during the COVID-19 pandemic: An audit study in Colombia [Preprint]. SocArXiv. https://doi.org/10.31235/osf.io/s2z47

Block, P., Hoffman, M., Raabe, I. J., Dowd, J. B., Rahal, C., Kashyap, R., & Mills, M. C. (2020). Social network-based distancing strategies to flatten the COVID-19 curve in a post-lockdown world. Nature Human Behaviour, 4(6), 588–596. https://doi.org/10.1038/s41562-020-0898-6

Argument Quality in Real World Argumentation. (2020). Trends in Cognitive Sciences, 24(5), 363–374. https://doi.org/10.1016/j.tics.2020.01.004

ReconfigBehSci on Twitter: “Fellow behavioural scientists, I had a lightbulb moment yesterday. I suspect I might have been unusually slow here, and you all clocked this months ago, but thought I’d share nevertheless: I’ve lost count of how often I’ve been told ‘You only have a 1/100 chance of dying’” / Twitter. (n.d.). Twitter. Retrieved June 27, 2020, from https://twitter.com/scibeh/status/1276434777856446464

Veltri, G. A., Prof, Lupiáñez-Villanueva, F., Folkvord, F., Theben, A., & Gaskell, G. (2020, April 29). The impact of online platform transparency of information on consumer’s choices. https://doi.org/10.31234/osf.io/htja5

Attali, Y., Budescu, D., & Arieli-Attali, M. (2020). An item response approach to calibration of confidence judgments. Decision, 7(1), 1–19. https://doi.org/10.1037/dec0000111

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

Erev, I., Wallsten, T. S., & Budescu, D. V. (1994). Simultaneous over- and underconfidence: The role of error in judgment processes. Psychological Review, 101(3), 519–527. https://doi.org/10.1037/0033-295X.101.3.519

[Jdm-society] Decision on re-opening economies. (n.d.). Retrieved April 21, 2020, from http://www.sjdm.org/mail-archive/jdm-society/2020-April/008496.html

Dhami, M. K., & Mandel, D. R. (2020). UK and US policies for communicating probability in intelligence analysis: A review [Preprint]. PsyArXiv. https://doi.org/10.31234/osf.io/kuyhb

Moore, D. A., & Wallsten, T. S. (2020). Rolling Forecasts. https://doi.org/10.31234/osf.io/ryvg3

Dr Muge Cevik on Twitter

Thread by @taaltree: Antibody tests are coming online. Never before have humans needed to understand Bayes rule more. Let’s talk about why it’s critical NOT to a…. (n.d.). Retrieved April 21, 2020, from https://threadreaderapp.com/thread/1248467731545911296.html

Mandel, D. R., Wallsten, T. S., & Budescu, D. (2020, April 20). Numerically-Bounded Language Schemes Are Unlikely to Communicate Uncertainty Effectively. https://doi.org/10.31234/osf.io/9f6ev

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:

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

Fischhoff, B., de Bruin, W. B., Güvenç, Ü., Caruso, D., & Brilliant, L. (2006). Analyzing disaster risks and plans: An avian flu example. Journal of Risk and Uncertainty, 33(1–2), 131–149. https://doi.org/10.1007/s11166-006-0175-8

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!

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.

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.

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.

Low probability and high impact but not a black swan

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

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.

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

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.

hyperuniform distribution - Appears random at smaller scales, but more predictable at larger scales.

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.

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.

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

From chapter 2.