Common sense quantum mechanics

11 dvasya 15 May 2014 08:10PM

Related to: Quantum physics sequence.

TLDR: Quantum mechanics can be derived from the rules of probabilistic reasoning. The wavefunction is a mathematical vehicle to transform a nonlinear problem into a linear one. The Born rule that is so puzzling for MWI results from the particular mathematical form of this functional substitution.

This is a brief overview a recent paper in Annals of Physics (recently mentioned in Discussion):

Quantum theory as the most robust description of reproducible experiments (arXiv)

by Hans De RaedtMikhail I. Katsnelson, and Kristel Michielsen. Abstract:

It is shown that the basic equations of quantum theory can be obtained from a straightforward application of logical inference to experiments for which there is uncertainty about individual events and for which the frequencies of the observed events are robust with respect to small changes in the conditions under which the experiments are carried out.

In a nutshell, the authors use the "plausible reasoning" rules (as in, e.g., Jaynes' Probability Theory) to recover the quantum-physical results for the EPR and SternGerlach experiments by adding a notion of experimental reproducibility in a mathematically well-formulated way and without any "quantum" assumptions. Then they show how the Schrodinger equation (SE) can be obtained from the nonlinear variational problem on the probability P for the particle-in-a-potential problem when the classical Hamilton-Jacobi equation holds "on average". The SE allows to transform the nonlinear variational problem into a linear one, and in the course of said transformation, the (real-valued) probability P and the action S are combined in a single complex-valued function ~P1/2exp(iS) which becomes the argument of SE (the wavefunction).

This casts the "serious mystery" of Born probabilities in a new light. Instead of the observed frequency being the square(d amplitude) of the "physically fundamental" wavefunction, the wavefunction is seen as a mathematical vehicle to convert a difficult nonlinear variational problem for inferential probability into a manageable linear PDE, where it so happens that the probability enters the wavefunction under a square root.

Below I will excerpt some math from the paper, mainly to show that the approach actually works, but outlining just the key steps. This will be followed by some general discussion and reflection.

1. Plausible reasoning and reproducibility

The authors start from the usual desiderata that are well laid out in Jaynes' Probability Theory and elsewhere, and add to them another condition:

There may be uncertainty about each event. The conditions under which the experiment is carried out may be uncertain. The frequencies with which events are observed are reproducible and robust against small changes in the conditions.

Mathematically, this is a requirement that the probability P(x|θ,Z) of observation x given an uncertain experimental parameter θ and the rest of out knowledge Z, is maximally robust to small changes in θ and independent of θ. Using log-probabilities, this amounts to minimizing the "evidence"

for any small ε so that |Ev| is not a function of θ (but the probability is).

2. The EinsteinPodolskyRosenBohm experiment

There is a source S that, when activated, sends a pair of signals to two routers R1,2. Each router then sends the signal to one of its two detectors Di+,– (i=1,2). Each router can be rotated and we denote as θ the angle between them. The experiment is repeated N times yielding the data set {x1,y1}, {x2,y2}, ... {xN,yN} where x and y are the outcomes from the two detectors (+1 or –1). We want to find the probability P(x,y|θ,Z).

After some calculations it is found that the single-trial probability can be expressed as P(x,y|θ,Z) = (1 + xyE12(θ) ) / 4, where E12(θ) = Σx,y=+–1 xyP(x,y|θ,Z) is a periodic function.

From the properties of Bernoulli trials it follows that, for a data set of N trials with nxy total outcomes of each type {x,y},

and expanding this in a Taylor series it is found that

The expression in the sum is the Fisher information IF for P. The maximum robustness requirement means it must be minimized. Writing it down as IF = 1/(1 – E12(θ)2) (dE12(θ)/dθ)2 one finds that E12(θ) = cos(θIF1/2 + φ), and since E12 must be periodic in angle, IF1/2 is a natural number, so the smallest possible value is IF = 1. Choosing φ π it is found that E12(θ) = –cos(θ), and we obtain the result that

which is the well-known correlation of two spin-1/2 particles in the singlet state.

Needless to say, our derivation did not use any concepts of quantum theory. Only plain, rational reasoning strictly complying with the rules of logical inference and some elementary facts about the experiment were used

3. The SternGerlach experiment

This case is analogous and simpler than the previous one. The setup contains a source emitting a particle with magnetic moment S, a magnet with field in the direction a, and two detectors D+ and D.

Similarly to the previous section, P(x|θ,Z) = (1 + xE(θ) ) / 2, where E(θ) = P(+|θ,Z) – P(–|θ,Z) is an unknown periodic function. By complete analogy we seek the minimum of IF and find that E(θ) = +–cos(θ), so that

In quantum theory, [this] equation is in essence just the postulate (Born’s rule) that the probability to observe the particle with spin up is given by the square of the absolute value of the amplitude of the wavefunction projected onto the spin-up state. Obviously, the variability of the conditions under which an experiment is carried out is not included in the quantum theoretical description. In contrast, in the logical inference approach, [equation] is not postulated but follows from the assumption that the (thought) experiment that is being performed yields the most reproducible results, revealing the conditions for an experiment to produce data which is described by quantum theory.

To repeat: there are no wavefunctions in the present approach. The only assumption is that a dependence of outcome on particle/magnet orientation is observed with robustness/reproducibility.

4. Schrodinger equation

A particle is located in unknown position θ on a line segment [–L, L]. Another line segment [–L, L] is uniformly covered with detectors. A source emits a signal and the particle's response is detected by one of the detectors.

After going to the continuum limit of infinitely many infinitely small detectors and accounting for translational invariance it is possible to show that the position of the particle θ and of the detector x can be interchanged so that dP(x|θ,Z)/dθ = –dP(x|θ,Z)/dx.

In exactly the same way as before we need to minimize Ev by minimizing the Fisher information, which is now

However, simply solving this minimization problem will not give us anything new because nothing so far accounted for the fact that the particle moves in a potential. This needs to be built into the problem. This can be done by requiring that the classical Hamilton-Jacobi equation holds on average. Using the Lagrange multiplier method, we now need to minimize the functional

Here S(x) is the action (Hamilton's principal function). This minimization yields solutions for the two functions P(x|θ,Z) and S(x). It is a difficult nonlinear minimization problem, but it is possible to find a matching solution in a tractable way using a mathematical "trick". It is known that standard variational minimization of the functional

yields the Schrodinger equation for its extrema. On the other hand, if one makes the substitution combining two real-valued functions P and S into a single complex-valued ψ,

Q is immediately transformed into F, concluding the derivation of the Schrodinger equation. Incidentally, ψ is constructed so that P(x|θ,Z) = |ψ(x|θ,Z)|2, which is the Born rule.

Summing up the meaning of Schrodinger equation in the present context:

Of course, a priori there is no good reason to assume that on average there is agreement with Newtonian mechanics ... In other words, the time-independent Schrodinger equation describes the collective of repeated experiments ... subject to the condition that the averaged observations comply with Newtonian mechanics.

The authors then proceed to derive the time-dependent SE (independently from the stationary SE) in a largely similar fashion.

5. What it all means

Classical mechanics assumes that everything about the system's state and dynamics can be known (at least in principle). It starts from axioms and proceeds to derive its conclusions deductively (as opposed to inductive reasoning). In this respect quantum mechanics is to classical mechanics what probabilistic logic is to classical logic.

Quantum theory is viewed here not as a description of what really goes on at the microscopic level, but as an instance of logical inference:

in the logical inference approach, we take the point of view that a description of our knowledge of the phenomena at a certain level is independent of the description at a more detailed level.

and

quantum theory does not provide any insight into the motion of a particle but instead describes all what can be inferred (within the framework of logical inference) from or, using Bohr’s words, said about the observed data

Such a treatment of QM is similar in spirit to Jaynes' Information Theory and Statistical Mechanics papers (I, II). Traditionally statistical mechanics/thermodynamics is derived bottom-up from the microscopic mechanics and a series of postulates (such as ergodicity) that allow us to progressively ignore microscopic details under strictly defined conditions. In contrast, Jaynes starts with minimum possible assumptions:

"The quantity x is capable of assuming the discrete values xi ... all we know is the expectation value of the function f(x) ... On the basis of this information, what is the expectation value of the function g(x)?"

and proceeds to derive the foundations of statistical physics from the maximum entropy principle. Of course, these papers deserve a separate post.

This community should be particularly interested in how this all aligns with the many-worlds interpretation. Obviously, any conclusions drawn from this work can only apply to the "quantum multiverse" level and cannot rule out or support any other many-worlds proposals.

In quantum physics, MWI does quite naturally resolve some difficult issues in the "wavefunction-centristic" view. However, we see that the concept wavefunction is not really central for quantum mechanics. This removes the whole problem of wavefunction collapse that MWI seeks to resolve.

The Born rule is arguably a big issue for MWI. But here it essentially boils down to "x is quadratic in t where t = sqrt(x)". Without the wavefunction (only probabilities) the problem simply does not appear.

Here is another interesting conclusion:

if it is difficult to engineer nanoscale devices which operate in a regime where the data is reproducible, it is also difficult to perform these experiments such that the data complies with quantum theory.

In particular, this relates to the decoherence of a system via random interactions with the environment. Thus decoherence becomes not as a physical intrinsically-quantum phenomenon of "worlds drifting apart", but a property of experiments that are not well-isolated from the influence of environment and therefore not reproducible. Well-isolated experiments are robust (and described by "quantum inference") and poorly-isolated experiments are not (hence quantum inference does not apply).

In sum, it appears that quantum physics when viewed as inference does not require many-worlds any more than probability theory does.

[LINK] Utilitarian self-driving cars?

7 V_V 14 May 2014 01:00PM

When a collision is unavoidable, should a self-driving car try to maximize the survival chances of its occupants, or of all people involved?

http://www.wired.com/2014/05/the-robot-car-of-tomorrow-might-just-be-programmed-to-hit-you/

Open Thread, May 5 - 11, 2014

2 Tenoke 05 May 2014 10:35AM

Previous Open Thread

You know the drill - If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

 

Notes for future OT posters:

1. Please add the 'open_thread' tag.

2. Check if there is an active Open Thread before posting a new one.

3. Open Threads should start on Monday, and end on Sunday.

4. Open Threads should be posted in Discussion, and not Main.

Link: Study finds that using a foreign language changes moral decisions

8 Vladimir_Golovin 30 April 2014 05:26AM

In the new study, two experiments using the well-known "trolley dilemma" tested the hypothesis that when faced with moral choices in a foreign language, people are more likely to respond with a utilitarian approach that is less emotional.

The researchers collected data from people in the U.S., Spain, Korea, France and Israel. Across all populations, more participants selected the utilitarian choice -- to save five by killing one -- when the dilemmas were presented in the foreign language than when they did the problem in their native tongue.

The article:
http://www.sciencedaily.com/releases/2014/04/140428120659.htm

The publication:
http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0094842

The effect of effectiveness information on charitable giving

15 Unnamed 15 April 2014 04:43PM

A new working paper by economists Dean Karlan and Daniel Wood, The Effect of Effectiveness: Donor Response to Aid Effectiveness in a Direct Mail Fundraising Experiment.

The Abstract:

We test how donors respond to new information about a charity’s effectiveness. Freedom from Hunger implemented a test of its direct marketing solicitations, varying letters by whether they include a discussion of their program’s impact as measured by scientific research. The base script, used for both treatment and control, included a standard qualitative story about an individual beneficiary. Adding scientific impact information has no effect on whether someone donates, or how much, in the full sample. However, we find that amongst recent prior donors (those we posit more likely to open the mail and thus notice the treatment), large prior donors increase the likelihood of giving in response to information on aid effectiveness, whereas small prior donors decrease their giving. We motivate the analysis and experiment with a theoretical model that highlights two predictions. First, larger gift amounts, holding education and income constant, is a proxy for altruism giving (as it is associated with giving more to fewer charities) versus warm glow giving (giving less to more charities). Second, those motivated by altruism will respond positively to appeals based on evidence, whereas those motivated by warm glow may respond negatively to appeals based on evidence as it turns off the emotional trigger for giving, or highlights uncertainty in aid effectiveness.

In the experimental condition (for one of the two waves of mailings), the donors received a mailing with this information about the charity's effectiveness:

In order to know that our programs work for people like Rita, we look for more than anecdotal evidence. That is why we have coordinated with independent researchers [at Yale University] to conduct scientifically rigorous impact studies of our programs. In Peru they found that women who were offered our Credit with Education program had 16% higher profits in their businesses than those who were not, and they increased profits in bad months by 27%! This is particularly important because it means our program helped women generate more stable incomes throughout the year.

These independent researchers used a randomized evaluation, the methodology routinely used in medicine, to measure the impact of our programs on things like business growth, children's health, investment in education, and women's empowerment.

In the control condition, the mailing instead included this paragraph:

Many people would have met Rita and decided she was too poor to repay a loan. Five hungry children and a small plot of mango trees don’t count as collateral. But Freedom from Hunger knows that women like Rita are ready to end hunger in their own families and in their communities.

The Error of Crowds

13 Eliezer_Yudkowsky 01 April 2007 09:50PM

I've always been annoyed at the notion that the bias-variance decomposition tells us something about modesty or Philosophical Majoritarianism.  For example, Scott Page rearranges the equation to get what he calls the Diversity Prediction Theorem:

Collective Error = Average Individual Error - Prediction Diversity

I think I've finally come up with a nice, mathematical way to drive a stake through the heart of that concept and bury it beneath a crossroads at midnight, though I fully expect that it shall someday rise again and shamble forth to eat the brains of the living.

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Be comfortable with hypocrisy

32 The_Duck 08 April 2014 10:03AM

Neal Stephenson's The Diamond Age takes place several decades in the future and this conversation is looking back on the present day:

"You know, when I was a young man, hypocrisy was deemed the worst of vices,” Finkle-McGraw said. “It was all because of moral relativism. You see, in that sort of a climate, you are not allowed to criticise others-after all, if there is no absolute right and wrong, then what grounds is there for criticism?" [...]

"Now, this led to a good deal of general frustration, for people are naturally censorious and love nothing better than to criticise others’ shortcomings. And so it was that they seized on hypocrisy and elevated it from a ubiquitous peccadillo into the monarch of all vices. For, you see, even if there is no right and wrong, you can find grounds to criticise another person by contrasting what he has espoused with what he has actually done. In this case, you are not making any judgment whatsoever as to the correctness of his views or the morality of his behaviour-you are merely pointing out that he has said one thing and done another. Virtually all political discourse in the days of my youth was devoted to the ferreting out of hypocrisy." [...]

"We take a somewhat different view of hypocrisy," Finkle-McGraw continued. "In the late-twentieth-century Weltanschauung, a hypocrite was someone who espoused high moral views as part of a planned campaign of deception-he never held these beliefs sincerely and routinely violated them in privacy. Of course, most hypocrites are not like that. Most of the time it's a spirit-is-willing, flesh-is-weak sort of thing."

"That we occasionally violate our own stated moral code," Major Napier said, working it through, "does not imply that we are insincere in espousing that code."

I'm not sure if I agree with this characterization of the current political climate; in any case, that's not the point I'm interested in. I'm also not interested in moral relativism.

But the passage does point out a flaw which I recognize in myself: a preference for consistency over actually doing the right thing. I place a lot of stock--as I think many here do--on self-consistency. After all, clearly any moral code which is inconsistent is wrong. But dismissing a moral code for inconsistency or a person for hypocrisy is lazy. Morality is hard. It's easy to get a warm glow from the nice self-consistency of your own principles and mistake this for actually being right.

Placing too much emphasis on consistency led me to at least one embarrassing failure. I decided that no one who ate meat could be taken seriously when discussing animal rights: killing animals because they taste good seems completely inconsistent with placing any value on their lives. Furthermore, I myself ignored the whole concept of animal rights because I eat meat, so that it would be inconsistent for me to assign animals any rights. Consistency between my moral principles and my actions--not being a hypocrite--was more important to me than actually figuring out what the correct moral principles were. 

To generalize: holding high moral ideals is going to produce cognitive dissonance when you are not able to live up to those ideals. It is always tempting--for me at least--to resolve this dissonance by backing down from those high ideals. An alternative we might try is to be more comfortable with hypocrisy. 

 

Related: Self-deception: Hypocrisy or Akrasia?

The Problem of "Win-More"

26 katydee 26 March 2014 06:32PM

In Magic: the Gathering and other popular card games, advanced players have developed the notion of a "win-more" card. A "win-more" card is one that works very well, but only if you're already winning. In other words, it never helps turn a loss into a win, but it is very good at turning a win into a blowout. This type of card seems strong at first, but since these games usually do not use margin of victory scoring in tournaments, they end up being a trap-- instead of using cards that convert wins into blowouts, you want to use cards that convert losses into wins.

This concept is useful and important and you should never tell a new player about it, because it tends to make them worse at the game. Without a more experienced player's understanding of core concepts, it's easy to make mistakes and label cards that are actually good as being win-more.

This is an especially dangerous mistake to make because it's relatively uncommon for an outright bad card to seem like a win-more card; win-more cards are almost always cards that look really good at first. That means that if you end up being too wary of win-more cards, you're going to end up misclassifying good cards as bad, and that's an extremely dangerous mistake to make. Misclassifying bad cards as good is relatively easy to deal with, because you'll use them and see that they aren't good; misclassifying good cards as bad is much more dangerous, because you won't play them and therefore won't get the evidence you need to update your position.

I call this the "win-more problem." Concepts that suffer from the win-more problem are those that-- while certainly useful to an advanced user-- are misleading or net harmful to a less skillful person. Further, they are wrong or harmful in ways that are difficult to detect, because they screen off feedback loops that would otherwise allow someone to realize the mistake.

On not diversifying charity

1 DanielLC 14 March 2014 05:14AM

A common belief within the Effective Altruism movement that you should not diversify charity donations when your donation is small compared to the size of the charity. This is counter-intuitive, and most people disagree with this. A Mathematical Explanation of Why Charity Donations Shouldn't Be Diversified has already been written, but it uses a simplistic model. Perhaps you're uncertain about which charity is best, charities are not continuous, let alone differentiable, and any donation is worthless unless it gives the charity enough money to finally afford another project, your utility function is nonlinear, and to top it all off, rather than accepting the standard idea of expected utility, you are risk-averse.

Standard Explanation:

If you are too lazy to follow the link, or you just want to see me rehash the same argument, here's a summary.

The utility of a donation is differentiable. That is to say, if donating one dollar gives you one utilon, donating another dollar will give you close to one utilon. Not exactly the same, but close. This means that, for small donations, it can be approximated as a linear function. In this case, the best way to donate is to find the charity that has the highest slope, and donate everything you can to it. Since the amount you donate is small compared to the size of the charity, a first-order approximation will be fairly accurate. The amount of good you do with that strategy is close to what you predicted it would do, which is more than you'd predict of any other strategy, which is close to what you'd predict for them, so even if this strategy is sub-optimal, it's at least very close.

Corrections to Account for Reality:

Uncertainty:

Uncertainty is simple enough. Just replace utility with expected utility. Everything will still be continuous, and the reasoning works pretty much the same.

Nonlinear Utility Function:

If your utility function is nonlinear, this is fine as long as it's differentiable. Perhaps saving a million lives isn't a million times better than saving one, but saving the millionth life is about as good as the one after that, right? Maybe each additional person counts for a little less, but it's not like the first million all matter the same, but you don't care about additional people after that.

In this case, the effect of the charity is differentiable with respect to the donation, and the utility is differentiable with respect to the effect of the charity, so the utility is differentiable with respect to the donation.

Risk-Aversion:

If you're risk-averse, it gets a little more complicated.

In this case, you don't use expected utility. You use something else, which I will call meta-utility. Perhaps it's expected utility minus the standard deviation of utility. Perhaps it's expected utility, but largely ignoring extreme tails. What it is is a function from a random variable representing all the possibilities of what could happen to the reals. Strictly speaking, you only need an ordering, but that's not good enough here, since it needs to be differentiable.

Differentiable is more confusing in this case. It depends on the metric you're using. The way we'll be using it here is that having a sufficiently small probability of a given change, or a given probability of a sufficiently small change, counts as a small change. For example, if you only care about the median utility, this isn't differentiable. If I flip a coin, and you win a million dollars if it lands on heads, then you will count that as worth a million dollars if the coin is slightly weighted towards heads, and nothing if it's slightly weighted towards tails, no matter how close it is to being fair. But that's not realistic. You can't track probabilities that precisely. You might care less about the tails, so that only things in the 40% - 60% range matter much, but you're going to pick something continuous. In fact, I think we can safely say that you're going to pick something differentiable. If I add a 0.1% chance of saving a life given some condition, it will make about the same difference as adding another 0.1% chance given the same condition. If you're risk-averse, you'd care more about a 0.1% chance of saving a life it's takes effect during the worst-case scenario than the best-case, but you'd still care about the same for a 0.1% chance of saving a life during the worst case as for upgrading it to saving two lives in that case.

Once you accept that it's continuous, the same reasoning follows as with expected utility. A continuous function of a continuous function is continuous, so the meta-utility of a donation with respect to the amount donated is continuous.

To make the reasoning more clear, here's an example:

Charity A saves one life per grand. Charity B saves 0.9 lives per grand. Charity A has ten million dollars, and Charity B has five million. One or more of these charities may be fraudulent, and not actually doing any good. You have $100, and you can decide where to donate it.

The naive view is to split the $100, since you don't want to risk spending it on something fraudulent. That makes sense if you care about how many lives you save, but not if you care about how many people die. They sound like they're the same thing, but they're not.

If you donate everything to Charity A, it has $10,000,100 and Charity B has $5,000,000. If you donate half and half, Charity A has $10,000,050 and Charity B has $5,000,050. It's a little more diversified. Not much more, but you're only donating $100. Maybe the diversification outweighs the good, maybe not. But if you decide that it is diversifying enough to matter more, why not donate everything to Charity B? That way, Charity A has $10,000,000, and Charity B has $5,000,100. If you were controlling all the money, you'd probably move a million or so from Charity A to Charity B, until it's well and truly diversified. Or maybe it's already pretty close to the ideal and you'd just move a few grand. You'd definitely move more than $100. There's no way it's that close to the optimum. But you only control the $100, so you just do as much as you can with that to make it more diversified, and send it all to Charity B. Maybe it turns out that Charity B is a fraud, but all is not lost, because other people donated ten million dollars to Charity A, and lots of lives were saved, just not by you.

Discontinuity:

The final problem to look at is that the effects of donations aren't continuous. The time I've seen this come up the most is when discussing vegetarianism. If you don't it meat, it's not going to make enough difference to keep the stores from ordering another crate of meat, which means exactly the same number of animals are slaughtered.

Unless, of course, you were the straw that broke the camel's back, and you did keep a store from ordering a crate of meat, and you made a huge difference.

There are times where you might be able to figure that out before-hand. If you're deciding whether or not to vote, and you're not in a battleground state, you know you're not going to cast the deciding vote, because you have a fair idea of who will win and by how much. But you have no idea at what point a store will order another crate of meat, or when a charity will be able send another crate of mosquito nets to Africa, or something like that. If you make a graph of the number of crates a charity sends by percentile, you'll get a step function, where there's a certain chance of sending 500 crates, a certain chance of sending 501, etc. You're just shifting the whole thing to the left by epsilon, so it's a little more likely each shipment will be made. What actually happens isn't continuous with respect to your donation, but you're uncertain, and taking what happens as a random variable, it is continuous.

A few other notes:

Small Charities:

In the case of a sufficiently small charity or large donation, the argument is invalid. It's not that it takes more finesse like those other things I listed. The conclusion is false. If you're paying a good portion of the budget, and the marginal effects change significantly due to your donations, you should probably donate to more than one charity even if you're not risk-averse and your utility function is linear.

I would expect that the next best charity you manage to find would be worse by more than a few percent, so I really doubt it would be worth diversifying unless you personally are responsible for more than a third of the donations.

An example of this is keeping money for yourself. The hundredth dollar you spend on yourself has about a tenth of the effect the thousandth does, and the entire budget is donated by you. The only time you shouldn't diversify is if the marginal benefit of the last dollar is still higher than what you could get donating to charity.

Another example is avoiding animal products. Avoiding steak is much more cost-effective than avoiding milk, but once you've stopped eating meat, you're stuck with things like avoiding milk.

Timeless Decision Theory:

If other people are going to make similar decisions to you, your effective donation is larger, so the caveats about small charities applies. That being said, I don't think this is really much of an issue.

If everyone is choosing independently, even if most of them correlate, the end result will be that the charities get just enough funding that some people donate to some and others donate to others. If this happens, chances are that it would be worth while for a few people to actually split their investments, but it won't make a big difference. They might as well just donate it all to one.

I think this will only become a problem if you're just donating to the top charity on GiveWell, regardless of how closely they rated second place, or you're just donating based purely on theory, and you have no idea if that charity is capable of using more money.

Channel factors

17 benkuhn 12 March 2014 04:52AM

Or, “how not to make a fundamental attribution error on yourself;” or, “how to do that thing that you keep being frustrated at yourself for not doing;” or, “finding and solving trivial but leveraged inconveniences.”

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