degrees-of-freedom   39

China’s Ideological Spectrum
We find that public preferences are weakly constrained, and the configuration of preferences is multidimensional, but the latent traits of these dimensions are highly correlated. Those who prefer authoritarian rule are more likely to support nationalism, state intervention in the economy, and traditional social values; those who prefer democratic institutions and values are more likely to support market reforms but less likely to be nationalistic and less likely to support traditional social values. This latter set of preferences appears more in provinces with higher levels of development and among wealthier and better-educated respondents.
pdf  study  economics  polisci  sociology  politics  ideology  coalitions  china  asia  things  phalanges  dimensionality  degrees-of-freedom  markets  democracy  capitalism  communism  authoritarianism  government  leviathan  tradition  values  correlation  exploratory  nationalism-globalism  heterodox  sinosphere 
march 2018 by nhaliday
Uniformitarianism - Wikipedia
Uniformitarianism, also known as the Doctrine of Uniformity,[1] is the assumption that the same natural laws and processes that operate in the universe now have always operated in the universe in the past and apply everywhere.[2][3] It refers to invariance in the principles underpinning science, such as the constancy of causality, or causation, throughout time,[4] but it has also been used to describe invariance of physical laws through time and space.[5] Though an unprovable postulate that cannot be verified using the scientific method, uniformitarianism has been a key first principle of virtually all fields of science.[6]

In geology, uniformitarianism has included the gradualistic concept that "the present is the key to the past" (that events occur at the same rate now as they have always done); many geologists now, however, no longer hold to a strict theory of gradualism.[7] Coined by William Whewell, the word was proposed in contrast to catastrophism[8] by British naturalists in the late 18th century, starting with the work of the geologist James Hutton. Hutton's work was later refined by scientist John Playfair and popularised by geologist Charles Lyell's Principles of Geology in 1830.[9] Today, Earth's history is considered to have been a slow, gradual process, punctuated by occasional natural catastrophic events.
concept  axioms  jargon  homo-hetero  wiki  reference  science  the-trenches  philosophy  invariance  universalism-particularism  time  spatial  religion  christianity  theos  contradiction  noble-lie  thinking  metabuch  reason  rigidity  flexibility  analytical-holistic  systematic-ad-hoc  degrees-of-freedom  absolute-relative  n-factor  explanans  the-great-west-whale  occident  sinosphere  orient  truth  earth  conceptual-vocab  metameta  history  early-modern  britain  anglo  anglosphere  roots  forms-instances  volo-avolo  deep-materialism  new-religion  logos 
january 2018 by nhaliday
The Grumpy Economist: Bitcoin and Bubbles
Bitcoin is not a very good money. It is a pure fiat money (no backing), whose value comes from limited supply plus these demands. As such it has the huge price fluctuations we see. It's an electronic version of gold, and the price variation should be a warning to economists who long for a return to  gold. My bet is that stable-value cryptocurrencies, offering one dollar per currency unit and low transactions costs, will prosper in the role of money. At least until there is a big inflation or sovereign debt crisis and a stable-value cryptocurrency not linked to government debt emerges.

https://twitter.com/GarettJones/status/939242620869660672
https://archive.is/Rrbg6
The Kareken-Wallace Cryptocurrency Price Indeterminacy theorem will someday receive the attention it deserves

https://www.mercatus.org/system/files/cryptocurrency-article.pdf
Cryptocurrencies also raise in a new way questions of exchange rate indeterminacy. As Kareken and Wallace (1981) observed, fiat currencies are all alike: slips of paper not redeemable for anything. Under a regime of floating exchange rates and no capital controls, and assuming some version of interest rate parity holds, there are an infinity of exchange rates between any two fiat currencies that constitute an equilibrium in their model.

The question of exchange rate indeterminacy is both more and less striking between cryptocurrencies than between fiat currencies. It is less striking because there are considerably more differences between cryptocurrencies than there are between paper money. Paper money is all basically the same. Cryptocurrencies sometimes have different characteristics from each other. For example, the algorithm used as the basis for mining makes a difference – it determines how professionalised the mining pools become. Litecoin uses an algorithm that tends to make mining less concentrated. Another difference is the capability of the cryptocurrency’s language for programming transactions. Ethereum is a new currency that boasts a much more robust language than Bitcoin. Zerocash is another currency that offers much stronger anonymity than Bitcoin. To the extent that cryptocurrencies differ from each other more than fiat currencies do, those differences might be able to pin down exchange rates in a model like Kareken and Wallace’s.

On the other hand, exchange rate indeterminacy could be more severe among cryptocurrencies than between fiat currencies because it is easy to simply create an exact copy of an open source cryptocurrency. There are even websites on which you can create and download the software for your own cryptocurrency with a few clicks of a mouse. These currencies are exactly alike except for their names and other identifying information. Furthermore, unlike fiat currencies, they don’t benefit from government acceptance or optimal currency area considerations that can tie a currency to a given territory.

Even identical currencies, however, can differ in terms of the quality of governance. Bitcoin currently has high quality governance institutions. The core developers are competent and conservative, and the mining and user communities are serious about making the currency work. An exact Bitcoin clone is likely to have a difficult time competing with Bitcoin unless it can promise similarly high-quality governance. When a crisis hits, users of identical currencies are going to want to hold the one that is mostly likely to weather the storm. Consequently, between currencies with identical technical characteristics, we think governance creates something close to a winner-take-all market. Network externalities are very strong in payment systems, and the governance question with respect to cryptocurrencies in particular compounds them.

https://twitter.com/GarettJones/status/939259281039380480
https://archive.is/ldof8
Explaining a price rise via future increases in the asset's value isn't good economics. The invisible hand should be pushing today's price up to the point where it earns normal expected returns. +
I don't doubt the likelihood of a future cryptocurrency being widely used, but that doesn't pin down the price of any one cryptocurrency as the Kareken-Wallace result shows. There may be a big first mover advantage for Bitcoin but ease of replication makes it a fragile dominance.

https://twitter.com/netouyo_/status/939566116229218306
https://archive.is/CtE6Q
I actually can't believe governments are allowing bitcoin to exist (they must be fully on board with going digital at some point)

btc will eventually come in direct competition with national currencies, which will have to raise rates dramatically, or die

http://www.thebigquestions.com/2017/12/08/matters-of-money/
The technology of Bitcoin Cash is very similar to the technology of Bitcoin. It offers the same sorts of anonymity, security, and so forth. There are some reasons to believe that in the future, Bitcoin Cash will be a bit easier to trade than Bitcoin (though that is not true in the present), and there are some other technological differences between them, but I’d be surprised to learn that those differences are accounting for any substantial fraction of the price differential.

The total supplies of Bitcoins and of Bitcoin Cash are currently about equal (because of the way that Bitcoin Cash originated). In each case, the supply will gradually grow to 21 million and then stop.

Question 1: Given the near identical properties of these two currencies, how can one sell for ten times the price of the other? Perhaps the answer involves the word “bubble”, but I’d be more interested in answers that assume (at least for the sake of argument) that the price of Bitcoin fairly reflects its properties as a store of value. Given that assumption, is the price differential entirely driven by the fact that Bitcoin came first? Is there that much of a first-mover advantage in this kind of game?

Question 2: Given the existence of other precious metals (e.g. platinum) what accounts for the dominance of gold as a physical store of value? (I note, for example, that when people buy gold as a store of value, they don’t often hesitate out of fear that gold will be displaced by platinum in the foreseeable future.) Is this entirely driven by the fact that gold happened to come first?

Question 3: Are Questions 1 and 2 the same question? Are the dominance of Bitcoin in the digital store-of-value market and the dominance of gold in the physical store-of-value market two sides of the same coin, so to speak? Or do they require fundamentally different explanations?

https://twitter.com/GarettJones/status/944582032780382208
https://archive.is/kqTXg
Champ/Freeman in 2001 explain why the dollar-bitcoin exchange rate is inherently unstable, and why the price of cryptocurrencies is indeterminate:

https://twitter.com/GarettJones/status/945046058073071617
https://archive.is/Y0OQB
Lay down a marker:
And remember that the modern macro dogma is that monetary systems matter little for prosperity, once bare competence is achieved.
econotariat  randy-ayndy  commentary  current-events  trends  cryptocurrency  bitcoin  money  monetary-fiscal  economics  cycles  multi  twitter  social  garett-jones  pdf  white-paper  article  macro  trade  incentives  equilibrium  backup  degrees-of-freedom  uncertainty  supply-demand  markets  gnon  🐸  government  gedanken  questions  comparison  analogy  explanans  fungibility-liquidity 
december 2017 by nhaliday
Karl Pearson and the Chi-squared Test
Pearson's paper of 1900 introduced what subsequently became known as the chi-squared test of goodness of fit. The terminology and allusions of 80 years ago create a barrier for the modern reader, who finds that the interpretation of Pearson's test procedure and the assessment of what he achieved are less than straightforward, notwithstanding the technical advances made since then. An attempt is made here to surmount these difficulties by exploring Pearson's relevant activities during the first decade of his statistical career, and by describing the work by his contemporaries and predecessors which seem to have influenced his approach to the problem. Not all the questions are answered, and others remain for further study.

original paper: http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf

How did Karl Pearson come up with the chi-squared statistic?: https://stats.stackexchange.com/questions/97604/how-did-karl-pearson-come-up-with-the-chi-squared-statistic
He proceeds by working with the multivariate normal, and the chi-square arises as a sum of squared standardized normal variates.

You can see from the discussion on p160-161 he's clearly discussing applying the test to multinomial distributed data (I don't think he uses that term anywhere). He apparently understands the approximate multivariate normality of the multinomial (certainly he knows the margins are approximately normal - that's a very old result - and knows the means, variances and covariances, since they're stated in the paper); my guess is that most of that stuff is already old hat by 1900. (Note that the chi-squared distribution itself dates back to work by Helmert in the mid-1870s.)

Then by the bottom of p163 he derives a chi-square statistic as "a measure of goodness of fit" (the statistic itself appears in the exponent of the multivariate normal approximation).

He then goes on to discuss how to evaluate the p-value*, and then he correctly gives the upper tail area of a χ212χ122 beyond 43.87 as 0.000016. [You should keep in mind, however, that he didn't correctly understand how to adjust degrees of freedom for parameter estimation at that stage, so some of the examples in his papers use too high a d.f.]
nibble  papers  acm  stats  hypothesis-testing  methodology  history  mostly-modern  pre-ww2  old-anglo  giants  science  the-trenches  stories  multi  q-n-a  overflow  explanation  summary  innovation  discovery  distribution  degrees-of-freedom  limits 
october 2017 by nhaliday
Gimbal lock - Wikipedia
Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space.

The word lock is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes there is no gimbal available to accommodate rotation along one axis.

https://blender.stackexchange.com/questions/469/could-someone-please-explain-gimbal-lock
https://computergraphics.stackexchange.com/questions/4436/how-to-achieve-gimbal-lock-with-euler-angles
Now this is where most people stop thinking about the issue and move on with their life. They just conclude that Euler angles are somehow broken. This is also where a lot of misunderstandings happen so it's worth investigating the matter slightly further than what causes gimbal lock.

It is important to understand that this is only problematic if you interpolate in Euler angles**! In a real physical gimbal this is given - you have no other choice. In computer graphics you have many other choices, from normalized matrix, axis angle or quaternion interpolation. Gimbal lock has a much more dramatic implication to designing control systems than it has to 3d graphics. Which is why a mechanical engineer for example will have a very different take on gimbal locking.

You don't have to give up using Euler angles to get rid of gimbal locking, just stop interpolating values in Euler angles. Of course, this means that you can now no longer drive a rotation by doing direct manipulation of one of the channels. But as long as you key the 3 angles simultaneously you have no problems and you can internally convert your interpolation target to something that has less problems.

Using Euler angles is just simply more intuitive to think in most cases. And indeed Euler never claimed it was good for interpolating but just that it can model all possible space orientations. So Euler angles are just fine for setting orientations like they were meant to do. Also incidentally Euler angles have the benefit of being able to model multi turn rotations which will not happen sanely for the other representations.
nibble  dirty-hands  physics  mechanics  robotics  degrees-of-freedom  measurement  gotchas  volo-avolo  duplication  wiki  reference  multi  q-n-a  stackex  graphics  spatial  direction  dimensionality  sky 
september 2017 by nhaliday
How & Why Solar Eclipses Happen | Solar Eclipse Across America - August 21, 2017
Cosmic Coincidence
The Sun’s diameter is about 400 times that of the Moon. The Sun is also (on average) about 400 times farther away. As a result, the two bodies appear almost exactly the same angular size in the sky — about ½°, roughly half the width of your pinky finger seen at arm's length. This truly remarkable coincidence is what gives us total solar eclipses. If the Moon were slightly smaller or orbited a little farther away from Earth, it would never completely cover the solar disk. If the Moon were a little larger or orbited a bit closer to Earth, it would block much of the solar corona during totality, and eclipses wouldn’t be nearly as spectacular.

https://blogs.scientificamerican.com/life-unbounded/the-solar-eclipse-coincidence/
nibble  org:junk  org:edu  space  physics  mechanics  spatial  visuo  data  scale  measure  volo-avolo  earth  multi  news  org:mag  org:sci  popsci  sky  cycles  pro-rata  navigation  degrees-of-freedom 
august 2017 by nhaliday
America's Ur-Choropleths
Gabriel Rossman remarked to me a while ago that most choropleth maps of the U.S. for whatever variable in effect show population density more than anything else. (There’s an xkcd strip about this, too.) The other big variable, in the U.S. case, is Percent Black. Between the two of them, population density and percent black will do a lot to obliterate many a suggestively-patterned map of the United States. Those two variables aren’t explanations of anything in isolation, but if it turns out it’s more useful to know one or both of them instead of the thing you’re plotting, you probably want to reconsider your theory.

https://www.nytimes.com/interactive/2016/12/26/upshot/duck-dynasty-vs-modern-family-television-maps.html
https://www.nytimes.com/interactive/2017/08/07/upshot/music-fandom-maps.html
scitariat  sociology  data  maps  usa  visualization  dataviz  within-group  degrees-of-freedom  roots  regularizer  population  density  race  demographics  things  phalanges  society  multi  news  org:rec  culture  tv  media  politics  american-nations  org:data  urban  polarization  class-warfare  music  urban-rural 
august 2017 by nhaliday
Overcoming Bias : High Dimensional Societes?
I’ve seen many “spatial” models in social science. Such as models where voters and politicians sit at points in a space of policies. Or where customers and firms sit at points in a space of products. But I’ve never seen a discussion of how one should expect such models to change in high dimensions, such as when there are more dimensions than points.

In small dimensional spaces, the distances between points vary greatly; neighboring points are much closer to each other than are distant points. However, in high dimensional spaces, distances between points vary much less; all points are about the same distance from all other points. When points are distributed randomly, however, these distances do vary somewhat, allowing us to define the few points closest to each point as that point’s “neighbors”. “Hubs” are closest neighbors to many more points than average, while “anti-hubs” are closest neighbors to many fewer points than average. It turns out that in higher dimensions a larger fraction of points are hubs and anti-hubs (Zimek et al. 2012).

If we think of people or organizations as such points, is being a hub or anti-hub associated with any distinct social behavior?  Does it contribute substantially to being popular or unpopular? Or does the fact that real people and organizations are in fact distributed in real space overwhelm such things, which only only happen in a truly high dimensional social world?
ratty  hanson  speculation  ideas  thinking  spatial  dimensionality  high-dimension  homo-hetero  analogy  models  network-structure  degrees-of-freedom 
july 2017 by nhaliday
Is Pharma Research Worse Than Chance? | Slate Star Codex
Here’s one hypothesis: at the highest level, the brain doesn’t have that many variables to affect, or all the variables are connected. If you smack the brain really really hard in some direction or other, you will probably treat some psychiatric disease. Drugs of abuse are ones that smack the brain really hard in some direction or other. They do something. So find the psychiatric illness that’s treated by smacking the brain in that direction, and you’re good.

Actual carefully-researched psychiatric drugs are exquisitely selected for having few side effects. The goal is something like an SSRI – mild stomach discomfort, some problems having sex, but overall you can be on them forever and barely notice their existence. In the grand scheme of things their side effects are tiny – in most placebo-controlled studies, people have a really hard time telling whether they’re in the experimental or the placebo group.

...

But given that we’re all very excited to learn about ketamine and MDMA, and given that if their original promise survives further testing we will consider them great discoveries, it suggests we chose the wrong part of the tradeoff curve. Or at least it suggests a different way of framing that tradeoff curve. A drug that makes you feel extreme side effects for a few hours – but also has very strong and lasting treatment effects – is better than a drug with few side effects and weaker treatment effects. That suggests a new direction pharmaceutical companies might take: look for the chemicals that have the strongest and wackiest effects on the human mind. Then see if any of them also treat some disease.

I think this is impossible with current incentives. There’s too little risk-tolerance at every stage in the system. But if everyone rallied around the idea, it might be that trying the top hundred craziest things Alexander Shulgin dreamed up on whatever your rat model is would be orders of magnitude more productive than whatever people are doing now.
ratty  yvain  ssc  reflection  psychiatry  medicine  pharma  drugs  error  efficiency  random  meta:medicine  flexibility  outcome-risk  incentives  stagnation  innovation  low-hanging  tradeoffs  realness  perturbation  degrees-of-freedom  volo-avolo  null-result 
june 2017 by nhaliday
The Distance Between Mars and Venus: Measuring Global Sex Differences in Personality
something other than Big Five

http://www.bbc.com/future/story/20161011-do-men-and-women-really-have-different-personalities
In an email, Del Giudice explained his approach to me with an analogy. “Gender differences in personality are very much like gender differences in facial appearance,” he said. “Each individual trait (nose length, eye size, etc) shows small differences between men and women, but once you put them all together... differences become clear and you can distinguish between male and female faces with more than 95% accuracy.”

Gender Differences in Personality across the Ten Aspects of the Big Five: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3149680/
Replicating previous findings, women reported higher Big Five Extraversion, Agreeableness, and Neuroticism scores than men. However, more extensive gender differences were found at the level of the aspects, with significant gender differences appearing in both aspects of every Big Five trait. For Extraversion, Openness, and Conscientiousness, the gender differences were found to diverge at the aspect level, rendering them either small or undetectable at the Big Five level.

some moderation by ethnicity and aging
study  psychology  cog-psych  personality  data  gender  gender-diff  psych-architecture  multi  news  org:rec  summary  evopsych  org:anglo  similarity  comparison  dimensionality  effect-size  degrees-of-freedom  race  aging  canada  anglo  self-report  discipline  extra-introversion  pop-diff  chart 
april 2017 by nhaliday
interpretation - How to understand degrees of freedom? - Cross Validated
From Wikipedia, there are three interpretations of the degrees of freedom of a statistic:

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.

Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df). In general, the degrees of freedom of an estimate of a parameter is equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself (which, in sample variance, is one, since the sample mean is the only intermediate step).

Mathematically, degrees of freedom is the dimension of the domain of a random vector, or essentially the number of 'free' components: how many components need to be known before the vector is fully determined.

...

This is a subtle question. It takes a thoughtful person not to understand those quotations! Although they are suggestive, it turns out that none of them is exactly or generally correct. I haven't the time (and there isn't the space here) to give a full exposition, but I would like to share one approach and an insight that it suggests.

Where does the concept of degrees of freedom (DF) arise? The contexts in which it's found in elementary treatments are:

- The Student t-test and its variants such as the Welch or Satterthwaite solutions to the Behrens-Fisher problem (where two populations have different variances).
- The Chi-squared distribution (defined as a sum of squares of independent standard Normals), which is implicated in the sampling distribution of the variance.
- The F-test (of ratios of estimated variances).
- The Chi-squared test, comprising its uses in (a) testing for independence in contingency tables and (b) testing for goodness of fit of distributional estimates.

In spirit, these tests run a gamut from being exact (the Student t-test and F-test for Normal variates) to being good approximations (the Student t-test and the Welch/Satterthwaite tests for not-too-badly-skewed data) to being based on asymptotic approximations (the Chi-squared test). An interesting aspect of some of these is the appearance of non-integral "degrees of freedom" (the Welch/Satterthwaite tests and, as we will see, the Chi-squared test). This is of especial interest because it is the first hint that DF is not any of the things claimed of it.

...

Having been alerted by these potential ambiguities, let's hold up the Chi-squared goodness of fit test for examination, because (a) it's simple, (b) it's one of the common situations where people really do need to know about DF to get the p-value right and (c) it's often used incorrectly. Here's a brief synopsis of the least controversial application of this test:

...

This, many authorities tell us, should have (to a very close approximation) a Chi-squared distribution. But there's a whole family of such distributions. They are differentiated by a parameter νν often referred to as the "degrees of freedom." The standard reasoning about how to determine νν goes like this

I have kk counts. That's kk pieces of data. But there are (functional) relationships among them. To start with, I know in advance that the sum of the counts must equal nn. That's one relationship. I estimated two (or pp, generally) parameters from the data. That's two (or pp) additional relationships, giving p+1p+1 total relationships. Presuming they (the parameters) are all (functionally) independent, that leaves only k−p−1k−p−1 (functionally) independent "degrees of freedom": that's the value to use for νν.

The problem with this reasoning (which is the sort of calculation the quotations in the question are hinting at) is that it's wrong except when some special additional conditions hold. Moreover, those conditions have nothing to do with independence (functional or statistical), with numbers of "components" of the data, with the numbers of parameters, nor with anything else referred to in the original question.

...

Things went wrong because I violated two requirements of the Chi-squared test:

1. You must use the Maximum Likelihood estimate of the parameters. (This requirement can, in practice, be slightly violated.)
2. You must base that estimate on the counts, not on the actual data! (This is crucial.)

...

The point of this comparison--which I hope you have seen coming--is that the correct DF to use for computing the p-values depends on many things other than dimensions of manifolds, counts of functional relationships, or the geometry of Normal variates. There is a subtle, delicate interaction between certain functional dependencies, as found in mathematical relationships among quantities, and distributions of the data, their statistics, and the estimators formed from them. Accordingly, it cannot be the case that DF is adequately explainable in terms of the geometry of multivariate normal distributions, or in terms of functional independence, or as counts of parameters, or anything else of this nature.

We are led to see, then, that "degrees of freedom" is merely a heuristic that suggests what the sampling distribution of a (t, Chi-squared, or F) statistic ought to be, but it is not dispositive. Belief that it is dispositive leads to egregious errors. (For instance, the top hit on Google when searching "chi squared goodness of fit" is a Web page from an Ivy League university that gets most of this completely wrong! In particular, a simulation based on its instructions shows that the chi-squared value it recommends as having 7 DF actually has 9 DF.)
q-n-a  overflow  stats  data-science  concept  jargon  explanation  methodology  things  nibble  degrees-of-freedom  clarity  curiosity  manifolds  dimensionality  ground-up  intricacy  hypothesis-testing  examples  list  ML-MAP-E  gotchas 
january 2017 by nhaliday
teaching - Intuitive explanation for dividing by $n-1$ when calculating standard deviation? - Cross Validated
The standard deviation calculated with a divisor of n-1 is a standard deviation calculated from the sample as an estimate of the standard deviation of the population from which the sample was drawn. Because the observed values fall, on average, closer to the sample mean than to the population mean, the standard deviation which is calculated using deviations from the sample mean underestimates the desired standard deviation of the population. Using n-1 instead of n as the divisor corrects for that by making the result a little bit bigger.

Note that the correction has a larger proportional effect when n is small than when it is large, which is what we want because when n is larger the sample mean is likely to be a good estimator of the population mean.

...

A common one is that the definition of variance (of a distribution) is the second moment recentered around a known, definite mean, whereas the estimator uses an estimated mean. This loss of a degree of freedom (given the mean, you can reconstitute the dataset with knowledge of just n−1 of the data values) requires the use of n−1 rather than nn to "adjust" the result.
q-n-a  overflow  stats  acm  intuition  explanation  bias-variance  methodology  moments  nibble  degrees-of-freedom  sampling-bias  generalization  dimensionality  ground-up  intricacy 
january 2017 by nhaliday

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