Open thread, 23-29 June 2014

I would prefer posts like that to stand on their own in discussion and not be posted in an open thread.

Hi, gwern it's awesome you are grappling with these issues. Here are some jambling responses.

You might enjoy Sander Greenland's essay here:

http://bayes.cs.ucla.edu/TRIBUTE/festschrift-complete.pdf

Sander can be pretty bleak!

But does the number of causal relationships go up just as fast? I don't think so (although at the moment I can't prove it).

I am not sure exactly what you mean, but I can think of a formalization where this is not hard to show. We say A "structurally causes" B in a DAG G if and only if there is a directed path from A to B in G. We say A is "structurally dependent" with B in a DAG G if and only if there is a marginal d-connecting path from A to B in G.

A marginal d-connecting path between two nodes is a path with no consecutive edges of the form -> <- * (that is, no colliders on the path). In other words all directed paths are marginal d-connecting but the opposite isn't true.

The justification for this definition is that if A "structurally causes" B in a DAG G, then if we were to intervene on A, we would observe B change (but not vice versa) in "most" distributions that arise from causal structures consistent with G. Similarly, if A and B are "structurally dependent" in a DAG G, then in "most" distributions consistent with G, A and B would be marginally dependent (e.g. what you probably mean when you say 'correlations are there').

I qualify with "most" because we cannot simultaneously represent dependences and independences by a graph, so we have to choose. People have chosen to represent independences. That is, if in a DAG G some arrow is missing, then in any distribution (causal structure) consistent with G, there is some sort of independence (missing effect). But if the arrow is not missing we cannot say anything. Maybe there is dependence, maybe there is independence. An arrow may be present in G, and there may still be independence in a distribution consistent with G. We call such distributions "unfaithful" to G. If we pick distributions consistent with G randomly, we are unlikely to hit on unfaithful ones (subset of all distributions consistent with G that is unfaithful to G has measure zero), but Nature does not pick randomly.. so unfaithful distributions are a worry. They may arise for systematic reasons (maybe equilibrium of a feedback process in bio?)

If you accept above definition, then clearly for a DAG with n vertices, the number of pairwise structural dependence relationships is an upper bound on the number of pairwise structural causal relationships. I am not aware of anyone having worked out the exact combinatorics here, but it's clear there are many many more paths for structural dependence than paths for structural causality.

But what you actually want is not a DAG with n vertices, but another type of graph with n vertices. The "Universe DAG" has a lot of vertices, but what we actually observe is a very small subset of these vertices, and we marginalize over the rest. The trouble is, if you start with a distribution that is consistent with a DAG, and you marginalize over some things, you may end up with a distribution that isn't well represented by a DAG. Or "DAG models aren't closed under marginalization."

That is, if our DAG is A -> B <- H -> C <- D, and we marginalize over H because we do not observe H, what we get is a distribution where no DAG can properly represent all conditional independences. We need another kind of graph.

In fact, people have come up with a mixed graph (containing -> arrows and <-> arrows) to represent margins of DAGs. Here -> means the same as in a causal DAG, but <-> means "there is some sort of common cause/confounder that we don't want to explicitly write down." Note: <-> is not a correlative arrow, it is still encoding something causal (the presence of a hidden common cause or causes). I am being loose here -- in fact it is the absence of arrows that means things, not the presence.

I do a lot of work on these kinds of graphs, because these are graphs are the sensible representation of data we typically get -- drawn from a marginal of a joint distribution consistent with a big unknown DAG.

But the combinatorics work out the same in these graphs -- the number of marginal d-connected paths is much bigger than the number of directed paths. This is probably the source of your intuition. Of course what often happens is you do have a (weak) causal link between A and B, but a much stronger non-causal link between A and B through an unobserved common parent. So the causal link is hard to find without "tricks."

The dependence that arises from a conditioned common effect (simplest case A -> [C] <- B) that people have brought up does arise in practice, usually if your samples aren't independent. Typical case: phone surveys are only administered to people with phones. Or case control studies for rare diseases need to gather one arm from people who are actually already sick (called "outcome dependent sampling.")

Sterner measures might be needed: could we draw causal nets with not just arrows showing influence but also another kind of arrow showing correlations?

Phil Dawid works with DAG models that are partially causal and partially statistical. But I think we should first be very very clear on exactly what a statistical DAG model is, and what a causal DAG model is, and how they are different. Then we could start combining without confusion!

If you have a prior over DAG/mixed graph structures because you are Bayesian, you can obviously have beliefs about a causal relationship between A and B vs a dependent relationship between A and B, and update your beliefs based on evidence, etc.. Bayesian reasoning about causality does involve saying at some point "I have an assumption that is letting me draw causal conclusions from a fact I observed about a joint distribution," which is not a trivial step (this is not unique to B of course -- anyone who wants to do causality from observational data has to deal with this).

what's the psychology of this?

Pearl has this hypothesis that a lot of probabilistic fallacies/paradoxes/biases are due to the fact that causal and not probabilistic relationships are what our brain natively thinks about. So e.g. Simpson's paradox is surprising because we intuitively think of a conditional distribution (where conditioning can change anything!) as a kind of "interventional distribution" (no Simpson's type reversal under interventions: http://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf).

This hypothesis would claim that people who haven't looked into the math just interpret statements about conditional probabilities as about "interventional probabilities" (or whatever their intuitive analogue of a causal thing is).

This post is a good example of why LW is dying. Specifically, that it was posted as a comment to a garbage-collector thread in the second-class area. Something is horribly wrong with the selection mechanism for what gets on the front page.

I think that deserves its own post.

It would be interesting to try to come up with good priors for random causal networks.

A simple informal reason for giving a small probability to "A causes B" could be this:

For any fixed values A and B, there is only one explanation "A causes B", one explanation "B causes A", and many explanations "C causes A and B" (for many different values of C). If we split 100% between all these explanations, the last group gets most of the probability mass.

And as you said, the more complex given field is, the more realistic values of C there are.

You are leaving out one type of causation. It is possible that you are conditioning on some common effect C of A and B.

You may argue that this does not actually give a correlation between A and B, it only give a correlation between A given C and B given C. However, in real life, there will always be things you condition on whenever you collect data, so you cannot completely remove this possibility.

Regarding the psychology of why people overestimate the correlation-causation link, I was just recently reading this, and something vaguely relevant struck my eye:

Later, Johnson-Laird put forward the theory that individuals reason by carrying out three fundamental steps [21]:

1. They imagine a state of affairs in which the premises are true – i.e. they construct a mental model of them.

2. They formulate, if possible, an informative conclusion true in the model.

3. They check for an alternative model of the premises in which the putative conclusion is false.

If there is no such model, then the conclusion is a valid inference from the premises.

Johnson-Laird and Steedman implemented the theory in a computer program that made deductions from singly-quantified assertions, and its predictions about the relative difficulty of such problems were strikingly confirmed: the greater the number of models that have to be constructed in order to draw the correct conclusion, the harder the task [25]. Johnson-Laird concluded [22] that comprehension is a process of constructing a mental model, and set out his theory in an influential book [23]. Since then he has applied the idea to reasoning about Boolean circuitry [3] and to reasoning in modal logic [24].

It's hard to say because how would you measure this other than directly, and to measure this directly you need a clear set of correlations which are proposed to be causal, randomized experiments to establish what the true causal relationship is, and both categories need to be sharply delineated in advance to avoid issues of cherrypicking and retroactively confirming a correlation so you can say something like '11 out of the 100 proposed A->B causal relationships panned out'. This is pretty rare, although the few examples I've found from medicine tend to indicate under 10%. Not great. And we can't explain all of this away as the result of illusory correlations being throw up by the standard statistical problems with findings such as small n/sampling error, selection bias, publication bias, etc.

Say you do this, and you find that about 10% of all correlations in a dataset are shown to have a causal link. Can you then look for a correlation between certain aspects of a correlation (such as coefficient, field of study) and those correlations which are causal?

Building on this, you might establish something like "correlations at .9 are more likely to be causal than correlations at .7" and establish a causal mechanism for this. Alternatively, you might find that "correlations from the field of farkology are more often causal than correlations from spleen medicine", and find a causal explanation for this.

Part or all of this explanation might involve the size of the causal network. It could well be that both correlation coefficients and field of study are just proxy variables to describe the size of a network, and that's the only important factor in the ratio of correlations to causal links, but it might be the case that there is more to it.

This could lead to quite a bit of trouble in academic literature, as measures of what evidence a correlation is for causation will become dependent on a set of variables about the context you're working in, and this could potentially be gamed. In fact, that could be the case even with gwern's original proposition -- claiming you're working with a small causal net could be enough to lend strong evidence to a causal claim based on correlation, and it's only by having someone point out that your causal net is lacking that this evidence can have its weighting adjusted.

All these thoughts are sketchy outlines of an extension of what gwern's brought up. More considered comment is welcome.

If the falling price of gene sequencing lets us determine a lot about how genes influence human behavior social scientists, I predict, will get a lot better at figuring out the causal effects of social programs.

After all, you can more or less divvy up the possibilities as:

A causes B B causes A both A and B are caused by C

There are at least two more possibilities: A and B are unrelated, but happen to be in sync for a while, and the data was collected wrong in some way.

Test reply please ignore

This seems deserving of at the very least a full post, possibly in main.

..oh, that wasn't a delete button. Not actually retracted.

When it comes to the replication of those breakthrough cancer results I think you can't forget publication bias. A lab runs an experiment 6 times. If it produces results in one of those trials they write a paper.

That's an interesting example of the power of the feed-forward branch in the memory-percpetion loop.

I remember having the same negative association with cauliflower, although not as strong as yours: could it be because of the strong sulphurous smell? Or the very bland taste and consistency? Surely the coupling of these two factors doesn't help.

I remember my negative association dissolved one day when I tried deep fried cauliflower, they tasted delicious. Only a lot later I would discover the equally delicious pickled caulis.

Culinary wise, the bland taste is perfect to carry over the salty taste of the batter when deep fried or the acidic taste when pickled.

The first LessWrong arch-villain.

FURIOUSLY DOWNVOTING in UNQUENCHABLE NERD RAGE.

Y'know, there's someone who literally can't think of anything better to do for the cause of existential risk. CLICK! CLICK! AHAHAHA! CLICK!!

Some areas that I have gaps in that I would like to fill with actionable steps:

• Networking

• Finding a reputable lawyer (for just in case)

• Signing up for cryonics

• Index funds is a great one

the proposed solution was someone assembling a 15-minute talk on index funds, along with a step-by-step guide to explain all the actions required to get from not having an index-fund product to having one.

Oh, please make it also a text. With pictures (activating near mode thinking).

Alternatively, a LW wiki page. (Then it's okay without pictures.)

Have you heard the slogan, "The truth is too complicated to fit on a bumper sticker"? I'd wholeheartedly endorse that if it's brevity didn't make me suspicious.

Firstly, it is an outrageous slur to think that the right-wing equivalent of Upworthy is the BNP. The right-wing equivalent is, of course, the Daily Mail, which has so mastered the art of click-baitery as to become the most read newspaper in the world (as of 2013 - may no longer be true).

Secondly, the social media cluster to which Less Wrong belongs is, obviously, Salon, Slate, and works of that ilk. Yes, Caliban, it's true.

And no, you cannot raise the sanity waterline with social media. All you will get is (as the joke goes) people enthusiastically retweeting a study that finally proves what they'd always believed about confirmation bias.

There have been some attempts to make "rationalist memes", but they're bland for people who are already aspiring rationalists and not funny enough for those that aren't.

The best thing of the sort that I've seen has been Pretty Rational and that has stopped updating for quite some time now.

... "actually, I think you'll find it's a bit more complicated than that" ... Is it just too difficult to spin a populist narrative for this sort of material?

Yes. Populist narratives are aimed at people with the attention span of a goldfish who like to get the feeling of them having been right all along within the first few seconds.

"A bit more complicated than that" is a non-starter.

tl;dr: can we raise the sanity waterline with clever use of social media?

Like "perpetual motion machine", the conjunction of "raise sanity waterline" and "use of social media" in the same sentence is immediately activating my red flags for "oxymoron; doomed to failure".

V sbyybjrq n srj bs gur fbheprf naq vg frrzrq gb or n fznyy zvabevgl bs ohgguheg qbpgbef engure guna gur "zrqvpny rfgnoyvfuzrag". Vf gung gur chapuyvar?

Quick little trick I do (this works if you can feel the annoyance manifest in your body, which is what happens for me):

1. I breath in slowly, and imagine the annoyance feeling I have as a red light, being sucked up into my forhead or into my threat
2. I breath out, imaging the red feeling either shooting out of my forhead or being expelled from my throat.
3. If the annoyance starts to come back, I'll repeat this two or three times.

One, I have these things too. I have not yet found a clear trigger for these and no clear category to which these annoyances belong. Two, people on Reddit in askreddit commonly describe thinking back to some event and cringing. So it does seem to be not that uncommon.

Meditation?

Try having a pleasant mental image you can quickly contemplate as soon as you start to think about the bad event. If the bad event involved someone else doing something bad to you, forgive them.

I have this problem and do intermittent fasting. I never before thought there might be a causal connection.

I get these "annoyance flashbacks" now and then and it seems like they are more frequent now that I calorie restrict.

Nominated for the funniest case of 'fake causality' this week.

Edit: MattG makes a fine point. Retracted.

I was getting this about a couple of specific events so I tried journaling about it: Pennebaker method, (http://en.wikipedia.org/wiki/Writing_therapy) - basically, 20 minutes non-stop writing, focusing on both thoughts and feelings about what happened, for three or four consecutive days - I find three is enough. If it's working, the narrative will change. There was a noticeable reduction in the flashbacks and also a lift in mood. Probably doesn't work for everyone, though.

I have the experience. No particular solution.

I used to get these split second emotional flashbacks all the time, primarily when stressed. They were usually about embarrassing or frustrating moments, and would very frequently cause some kind of verbal twitch - usually these compulsive phrases like "I want to go home" or "I hate this place" or "I hate myself". Very embarassing in the rare instances it occured around other people.

Meditation seems to have brought down their frequency a lot. Or at least, of the many things that have changed, the frequency with which I meditate seems to correlate most closely.

Yeah. I push them out of my mind or take a break to do something I enjoy.

This describes my feelings about job-hunting pretty perfectly. The only way I've found to deal with this is to push the toughts out of my mind, which isn't the most healthy solution.

Well, a popular solution in some circles is to declare the reminders "triggers" and insist that people avoid mentioning them in your presence. ;)

I've already looked over Alcor's webpage and the Cryonics Institute

Don't forget Oregon Cryonics.

This is an interesting example because it's not actually as susceptible to what Christian proposes as many other forms of assault, since laser pointers are silent and can have a very long range. and, if you're blinded, you're not exactly in a position to see who did it, the way you might be with kids throwing rocks. Sunglasses will probably keep you safe from most publicly available lasers but it's not exactly convenient if you don't like wearing them already.

Using Dark Arts for a good cause: Let's invent an urband legend about a psychopathic killer who murdered five kids who pointed at him laser pointers. Then spread the legend to make sure most of the kids in your environment (and their parents) know it.

Yeah, technically there always is a risk that this story could inspire a real mentally unstable person, but... torture versus laser pointers... I say let's do it.

It's a job for the police. If someone is intentionally harming others in a way that can cause lasting damage, then they need to be dragged in front of a court.

In the US you have a second amendment but in Germany we wouldn't have any issue with simply outlawing or restricting strong laser pointers in a way that prevents kids from possessing them if there's political will to do so.

What are you trying to optimize for? Are you sure the experience you're having now isn't the whole point of the thing?

Read reviews of the book rather than the book. If you are concerned the reviews of the book will not be as accurate as the book, or in some sense there is something to the book you won't get from the reviews I suggest

1) The reviews will tell you why the book is famous, which is probably why you set about reading the book in the first place. The reviews of the book will tell you more efficiently why the book is famous than you could ever expect to gain from reading the book itself. If you determine the the reasons the book is famous are sufficient for you to want to read the book, it will be easier to tolerate your own anger as you read it.

2) On the off chance that the reviews of the book somehow miss the true literary point of the book, there is a vanishingly small probability that you will repair that deficit in your own reading.

3) On the chance that the book is only thought to be good but is not actually good (whatever that means), you will at least know why it was thought to be good, which is ultimately what brought you to the book in the first place.

Just as with anything else modern, modern literature, at least some of it, is written for a small audience made up of the kind of people who like modern literature and read a lot of it. Its like reading a physics paper not being a physics graduate student, or looking at abstract modern art not being an insider on that particular thing.

If you can find a decent study guide (online, or, if there's a physical edition, then secondhand copies of it will doubtless cheaply available on Amazon from students who are done with them), then reading that along with the book isn't cheating. Reading notes for something which is fiction and therefore ostensibly 'leisure' reading may seem a bit absurd, but I think it can perhaps be justified. Aside from anything else, it can supply useful context not otherwise easily available to those not living in early twentieth century America and/or part of the high modern literati.

Whether or not you want to invest that kind of effort over and above what you're already doing is your call, though.

It is a hard book. I read it when I was a rather high-minded teenager, surely understanding very little of it, but it's actually a little hard for me to conceive of myself reading something so difficult now.

Build up to it by reading other books by the same author or in the same style. As I Lay Dying is supposed to be easier. Woolf is also stream of consciousness. Henry James is modern but less stream of consciousness. Or you could try his brother, William.

Short answer: this popped up on r/programming the other day. Lots of interesting questions there, and they don't come with answers. This will force you to solve them yourself, without spoilers, which is an incredibly valuable exercise which I strongly recommend for any questions you ask.

Long answer: you're going to have to unpack your intentions a little. You only have an hour (or less!), and you want to provide the maximum possible resolving power, so to do the best possible job you must know what your company's decision criteria are for this employee, and also what kinds of evidence your fellow interviewers are going to provide. "Ask technical questions" is too broad a mandate to be really ideal, so here are some questions which might help:

Do you need to do a basic competence check? This is a concrete coding question which a minimally capable developer can solve on a whiteboard, with fully correct syntax, in five minutes or less. Fizzbuzz is the canonical example; I've also seen "write a function to determine whether an input string is a palindrome" used to good effect. The point of this question is to efficiently divide your candidate pool into "people who maybe ask a few clarifying questions and then write out a correct, compilable answer as quickly as they can move the whiteboard marker" and "people who fumble around for half an hour and eventually come to something that's kinda right". Senior candidates will sometimes balk at this, especially if they're older than you are; I've found it works well to present the problem as a warmup, before you get to the more interesting stuff.

Do they have to already know the languages/tools you use? If so, then you have to drill down on that. Write down a bunch of things which you do in your env every day, pick a random subset, and ask your candidate how to do them. Java people need to know the difference between overloading and overriding, Javascript people had better be able to explain callbacks, and so forth.

How important is it that the candidate be able to solve new problems from scratch? To evaluate this, you want a problem which is novel and not easily solved with a standard algorithm. From the link above, "Add two strings. For example ""423" + "99" = "522"" is a good example of this sort of problem, I think (or else I just don't know the standard algorithm). For contrast, question #46 is not good for this: some candidates will have read about reader/writer locks and will immediately start reciting various algorithms and tradeoffs, while others won't have and are stuck trying to solve what was once an open research problem in the final half hour of your interview.

What algorithms/data structures must they already have mastered? Everybody needs arrays and hash tables, most people need sets and graphs. Good is to ask people to explain them, better is to ask a question that's easily solved using them.

Do they need to understand the hardware? Probably not, but maybe you're developing for a console, or doing embedded work, or something.

That got really long, and I don't have time to make it shorter. Sorry. tl;dr be mindful: know what your criteria are before you ask the questions, not afterward when you're trying to judge the answers.

If your applicant can solve googlable questions, s/he is probably way above the average. A version of Fizzbuzz is likely to trip up a lot of weak applicants.

What I used to do when interviewing software engineers is letting them write a simple (but complete) program (not on paper but in actual IDE, compile, run, debug etc.). I used a command line game I invented where the player navigates a simple "labyrinth" of squares, but feel free to invent your own. Typically, the candidates took 1.5-2.5 hours to complete it.

This gives you a way better idea than programming puzzles: you see how they write real live code, how well they understand the requirements, how they test their code, how they use the debugger etc.

I could google for some, but they would be the ones that everyone has seen before

You might be surprised. I'm always astonished at how many people I interview don't have a clue how to solve interview questions I Googled up or remembered from my own interview process, even the older-than-dirt ones like "how do you detect a cycle in a singly linked list?"

Besides, effective use of Google and literature is also a useful engineering skill. So is willingness to front-load work in preparation for an expected challenge.

I'm curious if anyone has made substantial effort to reach a 'flow' state in tasks outside of coding, like reading or doing math etc etc., and what they learned. Are there easy tricks? Is it possible? Is flow just a buzzword that doesn't really mean anything?

I find reading is just about the easiest activity to get into that state with. I routinely get so absorbed in a book that I forget to move. And I think that's the experience of most readers. It's a little harder with programming actually, since there are all these pauses while I wait for things to compile or run, and all these times when I have to switch to a web browser to look something up. With reading, you can just keep turning pages.

Awhile ago on my blog I broke the process down into three steps that seem to work for me:

1. Empty your head -Write down distracting thoughts, make important decisions, etc.

2. Focus your thoughts -Minimize distractions, relaxation techniques, etc.

3. Engage Your Action Mind -Use triggers, exercise, or a shock to your system.

Seems to work well for me but YMMV.

On an unrelated note it se,ems flow is actually the great state for peak performance, but it turns out to be a poor ideal for learning because it's antithetical to interleaved practice.

Everything I've read about it says that flow results from working at a challenge that's not too hard and not too easy, and that you enjoy (not so sure about that last one though). Seems to work for me.

An important part of "flow" is temporarily forgetting about the rest of the world. Not sure if you can reach this state artificially, but you certainly can be artificially removed from it.

For me, being alone and in silence seems to be an important factor.

Can you explain what the diagram means? I haven't been able to come up with a good guess as to what the arrows mean, or how the principles govern what.

I coincidentally read that paper today (confession: I am not a physicist yet, still a student), and I am really suspicious of his use of unitary transformations. A transformation is unitary if and only if it preserves the l^2-norm, which is precisely what the Born rule describes (i.e. that the l^2-norm is the correct norm on wavefunctions). I asked myself which step would break down if rather than the Born rule the actual probability was the amplitude to the power 4, and I haven't found it yet (provided we also redefine 'unitary'). But (hopefully) I'm just misunderstanding the problem..?

I think it is correct, but works only in a limited setup, and the authors claim

it seems reasonable to conclude that the Born Rule is established

is rather optimistic. I'm going to write an explanatory post in before the weekend hits.

Tox may provide the service you're looking for. http://tox.im/

I don't work at MIRI but: in the movie the guy cranks out a novel in like one night. That's years of work compressed into a few hours. He then proceeds to understand enough about markets to become extremely wealthy (thus negating the time travel informed betting angle), itself a many-year task. Most importantly: NZT-48 lets him figure out how to make MORE NZT-48 with fewer side effects, thus ensuring an indefinite supply. The NZT-48 is definitely the correct choice.

Approaching the question from the other direction: How much has MIRI really accomplished in the last 5 years? I think it's safe to say that a large part of what they've achieved is in terms of popularization and awareness raising rather than actual research/information generated. If they sent back a flash-drive to five years ago, that wouldn't jumpstart them anywhere near 5 years in terms of effort invested.

You could get the textbooks you used in school, and read them again. This time, noticing when you are confused.

Or use alternative textbooks, if available. That could reduce the feeling of "I have already seen this".

Or take a Coursera or Udacity lesson on the subject. They are usually full of exercises = quick feedback.

It's also good to have a positive attitude about learning outside the school. I mean, it's sad that the school didn't teach you better, but in any case, the science goes forward, the school time is limited, so everyone who really wants to understand their subject needs to study more after they leave school.

I think it's very important to be conscious about your questions. If you don't know something, don't go for the first answer but write the question down and revisit it regularly.

Have something you want to actually do/accomplish with the knowledge, and start doing it as soon as you can. Worked for me with programming (unlike physics where advanced topics largely went in one ear and out the other).

If all you have is a hammer, then you should be hammering on quite a few things that aren't nails. Heck, I've fixed a propane regulator by whacking it with a hammer. It's just that you should also.. you know.. buy some more tools.

If you're aware of the non-nailness of the thing in question, I don't see any problem with trying out your new tool on it - how else are you to learn its limits!? Using standard tools in a domain where it's not normally applied is often a source of fresh insight.

Life is like an adventure game. If all you have is a hammer, use your hammer on everything. Once you get a screwdriver, use it on everything, since you already tried the hammer.

Factor Analysis is a very flawed technique for many circumstances; in particular, "independent" and "not linearly correlated" are wildly different.

So yes, try to be acutely aware that not everything is a nail. Be aware of your hammer's limitations, and what it's effectiveness is based on.

That said, I've been guilty of this too. And certainly, we rely on people pushing their hammers in strange directions; some esoteric new hammering technique for metaphorical screws can open the floodgates of innovation. That said, don't confuse the fact that some people have managed to hone their skills with one tool with that tool being a panacea.

OCEAN personality factors and g, all of which are the recult of factor analysis

Factor analysis and principal components analysis are related but subtly different techniques.

I think most people realize these problems can have emotional roots. Some are just inclined to try to treat their own emotional state mechanistically (increase exercise, decrease expectations -> lower stress, become happier about doing work).

I have heard (I have no citation and it's probably apocryphal, but I found the anecdote enlightening) that Enrico Fermi's way of reading articles was to read the abstract, put the paper away, do the maths by himself and once he was done, compare his results with the article. That's probably a bit hardcore, but you should be able to start from somewhere in the paper's reasoning and do a few steps forward.

But where are you in your paper reading at the moment ? Is there a particular problem that spurred this question ?

but would like to know if I just really need to bite the bullet and do it this way (Latex or similar).

If you ever want to communicate with others, LaTeX is the lingua franca of mathematics.

LaTex is for typesetting. I know of nobody who "does the math" in LaTex; they do the math on paper and write it up in LaTex for presentation or publication (or if they need to ask a question on MathOverflow, or something like that).

That said, you should learn LaTex if you ever want to do research in anything mathematical.

Last year I made a deliberate choice to produce all my maths assignments in LaTeX, the upshot of which is that I'm now pretty comfortable with LaTeX. I'm pretty damn sure you wouldn't want to use it as substitute to pencil-and-paper working, though.

It's neither of these. The first condition is not necessary for self-similarity, and the second is not sufficient.

Consider an archetypical example of a self-similar structure, the Sierpinski triangle. Looking at that picture, you can see that not every part of the triangle looks like the triangle. There are parts of the triangle that look like two triangles side by side, for instance. So it's not necessary that every part of the whole be identical to the whole.

On the other hand, it is also not sufficient that at least one part of the whole be identical to the whole. First of all (and slightly pedantically), this would make any structure trivially self-similar, since according to the standard axioms of mereology every whole is a part of itself.

More substantively, even if you modified your definition to say "proper part" instead of just "part", it still wouldn't be sufficient. You can see the mathematical definition of self-similarity here. The definition is slightly opaque, but basically what it's saying is that a bounded set is self-similar if it can be built up as a finite union of smaller-scale copies of itself. There are sets where some proper part resembles the whole, but which still cannot be built up in this way, so a proper part resembling the whole is not sufficient for self-similarity.

As an example, consider this picture. If the process it shows goes on to infinity, then there will be a proper part of the picture that will be a smaller scale version of the picture as a whole. However, the picture as a whole is not a finite union of such parts, so it is not self-similar, unlike the Sierpinski triangle.

So here's a colloquial definition of self-similarity that captures the idea: A structure is self-similar if it can be exhaustively divided into a finite number of parts (greater than 1) such that each part exactly resembles the whole (except for scale). The Sierpinski triangle, for instance, can be exhaustively divided into three parts, all of which are smaller-scale versions of the triangle itself.

"Good" and "water-resistant" pretty much don't occur together in practice, sadly.

(So, I want a Natural-shaped keyboard ... with clicky keys ... that's water-resistant.)

I am a fan of Cal Newport, whose advice is especially relevant in college.

Your Brain at Work is far and away the best productivity book I've ever read, at least for me. It's also recommended by CFAR.

What are you struggling with? What would you most like to improve about your productivity?