I too spent a few years with a similar desire to understand probability and statistics at a deeper level, but we might have been stuck on different things. Here's an explanation:

Suppose you have 37 numbers. Purchase a massless ruler and 37 identical weights. For each of your numbers, find the number on the ruler and glue a weight there. You now have a massless ruler with 37 weights glued onto it.

Now try to balance the ruler sideways on a spike sticking out of the ground. The mean of your numbers will be the point on the ruler where it balances.

Now spin the ruler on the spike. It's easy to speed up or slow down the spinning ruler if the weights are close together, but more force is required if the weights are far apart. The variance of your numbers is proportional to the amount the ruler resists changes to its angular velocity -- how hard you have to twist the ruler to make it spin, or to make it stop spinning.

"I'd like to understand this more deeply" is a thought that occurs to people at many levels of study, so this explanation could be too high or low. Where did my comment hit?

I don't have a good resource for you - I've had too much math education to pin down exactly where I picked up this kind of logic. I'd recommend set theory in general for getting an understanding of how math works and how to talk and read precisely in mathematics.

For your specific question about the mean, it's the only number such that the sum of all (samples - mean) equals zero. Go ahead and play with the algebra to show it to yourself. What it means is that if you go off of the mean, you're going to be more positive of the numbers in the sample than you are negative, or more negative than positive.

Not really - but I do agree that it's absolutely vital to understand the basic concepts or terms. I think that's a major reason why people fail to learn - they just don't really grasp the most vital concepts. That's especially true of fields with lots of technical terms. If you don't understand the terms you'll struggle to follow even basic lines of reasoning.

For this reason I sometimes provide students with a list of central terms, together with comprehensive explanations of what they mean, when I teach.

When you have thousands of different pieces of data, to grasp it mentally, you need to replace them with some simplification. For example, instead of a thousand different weights you could imagine a thousand identical weights, such that the new set is somehow the same as the original set; and then you would focus on the individual weight from the new set.

What precisely does "somehow the same as the original set" mean? Well, it depends on what did the numbers from the original set do; how exactly they join together.

For example, if we speak about weights, the natural way of "joining together" is to add their weight. Thus the new set of the identical weights is equivalent to the original set if the sum of the new set is the same as sum of the old set. The sum of the new set = number of pieces × weight of one piece. Therefore the weight of the piece in the new set is the sum of the pieces in the original set divided by their number; the "sum/n".

Specifically, if addition is the natural thing to do, the set 3, 4, 8 is equivalent to 5, 5, 5, because 3 + 4 + 8 = 5 + 5 + 5. Saying that "5 is the mean of the original set" means "the original set behaves (with regards to the natural thing to do, i.e. addition) as if it was composed of the 5's".

There are situations where some other operation is the natural thing to do. Sometimes it is multiplication. For example, if you multiply some original value with 2, and they you multiply it by 8, the result of these two operations is the same as if you would multiply it twice by 4. In this case it's called geometric mean, and it's a root of product.

It can be even more complicated, so it doesn't necessarily have a name, but the idea is always replacing the original set with a set of identical values such that in the original context they would behave the same way. For example, the example above could be described as a 100% growth (multiplication by 2) and 700% growth (multiplication by 8), and you need to get a result 300% (multiplication by 4); in which case it would be "root of (product of (Xi + 100%)) - 100%".

If there is no meaningful operation in the original set, if the set can be ordered, we can pick the median. If the set can't even be ordered, if there are discrete values, we can pick the most frequent value as the best approximation of the original set.

I want a deeper understanding of the basic concepts. Like, mean is an indicator of the central tendency of a sample. Intuitively, it makes sense. But why this particular formula of sum/n? You can apply all kinds of mathematical stuff to the sample.

• The mean of the sum of two random variables is the sum of the means (ditto with the variances); there's no similarly simple formula for the median. (See ChristianKl's comment for why you'd care about the sum.)

• The mean if the value of x that minimizes SUM_i (x - x_i)^2; if you have to approximate all elements in your sample with the same value and the cost of an imperfect approximation is the square distance from the exact value (and any smooth function looks like the square when you're sufficiently close to the minimum), then you should use the mean.

• The mean and variance are jointly sufficient statistics for the normal distribution

• Possibly something else which doesn't come to my mind at the moment.

Here, have a book!

http://www-biba.inrialpes.fr/Jaynes/prob.html

Attending a CFAR workshop and session on Bayes (the 'advanced' session) helped me understand a lot of things in an intuitive way. Reading some online stuff to get intuitions about how Bayes' theorem and probability mass work was helpful too. I took an advanced stats course right after doing these things, and ended up learning all the math correctly, and it solidified my intuitions in a really nice way. (Other students didn't seem to have as good a time without those intuitions.) So that might be a good order to do things in.

Some multidimensional calc might be helpful, but other than that, I think you don't need too much other math to support learning more probability and stats.

I asked a similar question a while back, and I was directed to this book, which I found to be incredibly useful. It is written at an elementary level, has minimal little maths, yet is still technical, and brings across so many central ideas in very clear, Bayesian, terms. It is also on Lukeprog's CSA book recommendations for 'Become Smart Quickly'.

Note: this is the only probability textbook I have read. I've glanced through the openings of others, and they've tended to be above my level. I am sixteen.

This isn't at introductory level, but try exploring the ideas around Fisher information -- it basically ties together information theory and some important statistical concepts.

Read Edwin Jaynes.

The problem with most Probability and Statistics courses is the axiomatic approach. Purely formalism. Here are the rules - you can play by them if you want to.

Jaynes was such a revelation for me, because he starts with something you want, not arbitrary rules and conventions. He builds probability theory on basic desiredata of reason that you that make sense. He had reasons for my "whys?".

Also, standard statistics classes always seemed a bit perverse to me - logically backward. They always just felt wrong. Jaynes approach replaced that tortured backward thinking with clear, straight lines going forward. You're always asking the same basic question "What is the probability of A given that I know B?"

And he also had the best notation. Even if I'm not going to do any math, I'll often formulate a problem using his notation to clarify my thinking.

As a first step, I suggest Dennis Lindley's Understanding Uncertainty. It's written for the layperson, so there's not much in the way of mathematical detail, but it is very good for clarifying the basic concepts, and covers some surprisingly sophisticated topics.

ETA: Ah, I didn't notice that Benito had already recommended this book. Well, consider this a second opinion then.

I don't think that's really what means are. That intuition might fit the median better. One reason means are nice is that they have really nice properties, e.g. they're linear under addition of random variables. That makes them particularly easy to compute with and/or prove theorems about. Another reason means are nice is related to betting and the interpretation of a mean as an expected value; the theorem justifying this interpretation is the law of large numbers.

Nevertheless in many situations the mean of a random variable is a very bad description of it (e.g. mean income is a terrible description of the income distribution and median would be much more appropriate).

Edit: On the other hand, here's one very undesirable property of means: they're not "covariant under increasing changes of coordinates," which on the other hand is true of medians. What I mean is the following: suppose you decide to compute the mean population of all cities in the US, but later decide this is a bad idea because there are some really big cities. If you suspect that city populations grow multiplicatively rather than additively (e.g. the presence of good thing X causes a city to be 1.2x bigger than it otherwise would, as opposed to 200 people bigger), you might decide that instead of looking at population you should look at log population. But the mean of log population is not the log of mean population!

On the other hand, because log is an increasing function, the median of log population is still the log of median population. So taking medians is in some sense insensitive to these sorts of decisions, which is nice.

Corrected url: Foundation Training

I tried the video at the url, and it seemed a lot more like straining (little pun about the mistaken url), but that might not be a fair test.

The basic idea of getting hip mobility seems sound, but I recommend Scott Sonnon's Ageless Mobility and IntuFlow, and the The Five Tibetan Rites -- sorry for the cheesy name on the latter, but they're a cross between yoga and calisthenics with a lot of emphasis on getting backwards/forwards pelvis mobility.

Just to be clear: you're worried that you aren't sloppy enough?

If so, for us non-artists, can you explain how 'sloppiness' can be a good thing?

Seems to me that that's likely a self-fulfilling prophecy, which I subjectively estimate is at least as likely to prevent you from doing better as an actual biological problem. Maybe try to think of more ways to get better at it - perhaps some different kind of exercises - and do your best at those, before drawing any conclusions about your fundamental limits... because those conclusions themselves will limit you even more.

I would guess that you try to exert too much control. The kind of "sloppiness" that's useful for creativity is about letting things go.

Meditation might help.

As you are female, dancing a partner dance where you have to follow and can't control everything might be useful. Letting go of trying to control is lesson 101 for a lot of woman who pick up Salsa dancing.

Some guesses on my part-

• Maybe your tendency towards precision is at the wrong times? If practicing, for example, it might be counterproductive since you probably want quantity instead of quality, or maybe you're trying to get everything down precisely too early on and it's making your work stiff.

• Manfred's point is good- "metaphor that captures the scene without the need for detail."... If you render background details overmuch, they can distract the viewer from the focal point of the work. Maybe put some effort into looking at how the "metaphors" of different things work? For example, how more skilled artists draw/paint grass in the distance, or whatnot.

• I think it's a common thing to sort of notice something wrong in an area, and to spend a lot of time on that area in hopes of fixing it, which would make it less sloppy... Maybe sketch that thing a lot for practice.

• If you're drawing from life, it's possible that lack of sloppiness comes from not making sense of the gestalt, so to speak. I'd think that understanding the form of the subject and how the lighting on it works means you can simplify things away. I don't do much (read: any) figure drawings from life, but I'd imagine that understanding the figure and what's important and what isn't would be helpful. Maybe doing some master copies of skilled, more abstract drawings of the figure would help. Maybe look up a comic artist or cartoonist you like and look at what they do.

ETA:

To address your actual question, I'd say I don't know any particular evidence for why that should be so.

Rationality-technique-wise: It's good that you asked people, since that would bring you evidence of the idea being true or false. In the future it might be even more useful to suppress hypothesizing until some more investigating has gone on- "biological limit" is the sort of thing that feels true if you don't understand how to do something or how to understand how to do something. I think there's a post about this, or something; let me see if I can find it... ETA2: The exact anecdote I was thinking of doesn't apply as much as I thought it did, but maybe the post "Fake Explanations" or something applies?

I have never biked twenty miles in one go.
It could be that this reflects some inherent limit.
Or it could be that I just haven't tried yet.

If I believe that it is an inherent limit, how might I test my belief?
Only by trying anyway.
If I try and succeed, then I will update.

If I believe that it is not an inherent limit, how might I test my belief?
Only by trying anyway.
If I try and fail, then I will update.

In either case, the test of my ability
Is not in contemplating what mechanisms of self might limit me,
But in trying anyway, when I have the opportunity to do so,
And seeing what happens.

If other people working the same craft have managed to achieve precision, it's very unlikely to be a biological limit, right? The resolution of human fine motor skills is really high.

You didn't mention what the craft was or the nature of the sloppiness, but have you considered using simple tools to augment technical skills? Perhaps a magnifying glass, rulers. pieces of string/clay or other suitably shaped objects to guide the hand, etc?

You could try doing something that gives immediate feedback for sloppiness, like simple math problems for example. You might gain some generalizable insight like that speed affects sloppiness. Since you already practice meditation, it should be easier to become aware of the specific failure modes that contribute to sloppiness, which doesn't seem to be a well defined thing in itself.

I don't think you understand signaling well.

Eliezer managed signaling well enough to get a billionaire to fund him on his project. A billionaire who fund people who drop out of college systematically in projects like his 20 Under 20 program.

Trying to go the traditional route wouldn't fit into the highly effective image that he already signals.

Has anyone toyed around with the idea of sending him off to get a math degree somewhere?

I think the bigger issue w/ people not taking EY seriously is he does not communicate (e.g. publish peer reviewed papers). Facebook stream of consciousness does not count. Conditional on great papers, credentials don't mean that much (otherwise people would never move up the academic status chain).

Yes it is too bad that writing things down clearly takes a long time.

Here's your tuition and an extra sum of money to cover the opportunity cost of your time

If you buy into the “crunch time” narrative, that's a lot of opportunity cost.

4 years (or even 1 year if you are super hard-core) of time is a pretty non-trivial investment. I was 2 classes away from a second degree and declined to take them, because the ~100 hours of work it would have taken wasn't worth the additional letters after my name. I also just really don't know anyone relevant who thinks that a college degree or lack thereof particularly matters (although the knowledge and skills acquired in the course of pursuing said degree may matter a lot). Good people will judge you by what you've done to demonstrate skill, not based on a college diploma.

I think IlyaShpitser's comment pretty much nails it.

This might have been a good call 10 years ago but nowadays Eliezer is participating in regular face to face meetings with skilled mathematicians and scientists in the context of constructing and analyzing theorems and decision strategies. This means that for a large amount of the people who are most important to convince, he gets to screen out all the "evidence" of not having a degree. And to a large extent, someone having the respect of a bunch of math phds is more important a qualifier of talent than having that phd themselves.

There's theoretically still the problem of selling Eliezer to the muggles but I don't think that's anywhere near as important as getting serious thinkers on board.

Would getting more citations partly nullify the lack of formal education?

Different target groups may use different signals.

For example, for a scientist the citations may be more important than formal education. For an ordinary person with a university diploma who never published anything anywhere, formal education will probably remain the most important signal, because that's what they use. A smart sponsor may instead consider the ability of getting things done. And the New Age fans will debate about how much Eliezer fits the definitions of an "indigo child".

If the goal is to impress people for whom having an university diploma is the most important signal (they are a majority of the population), the best way would be to find an university which gives the diploma for minimum time and energy spent. Perhaps one where you just pay some money (hopefully not too much), take a few easy exams, and that's it; you don't have to spend time at the lessons. After this, no one can technically say that Eliezer has "no formal education". (And if they start discussing the quality of the university, then Eliezer can point to his citations.) The idea is to do this as easily as possible... assuming it's even worth doing.

There are also other things to consider, such as the fact that other people working with Eliezer do have formal education... so why exactly is it a problem if Eliezer doesn't? Does MIRI seem from outside like one man show? Maybe that should be fixed.

I think the non-intuitive nature of the A choice is because we naturally think of utilons as "things". For any valuable thing (money, moments of pleasure, whatever) anybody who is minimally risk adverse would choose B. But utllons are not things, they are abstractions defined by one's preferences. So that A is the rational choice is a tautology, in the standard versions of utility theory.

It may help to think it the other way around, starting from the actual preference. You would choose a 99.9% chance of losing ten cents and 0.1% chance of winning 10000 dollars over winning one cent with certainty, right? So then perhaps, as long as we don't think of other bets and outcomes, we can map winning 1 cent to +1 utilon, losing 10 cents to -100 utilons and winning 10000 dollars to +10000 utilons. Then we can refine and extend the "outcomes <=> utilons" map by considering your actual preferences under more and more bets. As long as your preferences are self-consistent in the sense of the VNM axioms, then there will a mapping that can be constructed.

ETA: of course, it is possible that your preferences are not self-consistent. The Allais paradox is an example where many people's intuitive preferences are not self-consistent in the VNM sense. But constructing such a case is more complicated that just considering risk-aversion on a single bet.

I'd flip that around. Whatever action you end up choosing reveals what you think has highest utility, according to the information and utility function you have at the time. It's almost a definition of what utility is - if you consistently make choices that rank lower according to what you think your utility function is, then your model of your utility function is wrong.

If the utility function you think you have prefers B over A, and you prefer A over B, then there's some fact that's missing from the utility function you think you have (probably related to risk).

I've recently come to terms with how much fear/anxiety/risk avoidance is in my revealed preferences. I'm working on working with that to do effective long-term planning -- the best trick I have so far is weighing "unacceptable status quo continues" as a risk. That, and making explicit comparisons between anticipated and experienced outcomes of actions (consistently over-estimating risks doesn't help any, and I've been doing that).

A, which grants -100 utilons with 99.9% chance and +10000 utilons with 0.1%

A has an expected utility of +900.1 utilons

Um, A actually has a utility of -89.9.

That explains why it seems less appealing!

Is it always correct to choose that action with the highest expected utility?

People who play with money don't like high variance, and sometimes trade off some of the mean to reduce variance.

Depending on your preferred framework, this is in some sense backwards: utility is, by definition, that thing which it is always correct to choose the action with the highest expected value of (say, in the framework of the von Neumann-Morgenstern theorem).

I don't know what you mean precisely by confusion, but I personally can't always control what my immediate primal level response is to certain situations. If I try to strictly avoid certain feelings, I usually end up convincing myself that I'm not feeling that way when actually I am. I'd rather notice what I'm feeling and then move on from there, it's probably easier to control your thinking that way. Just because you're angry doesn't mean you have to act like angry.

A quick look at the first page of your recent comments shows most of your recent activity to have been in the recent "Is Less Wrong too scary to marginalized groups?" firestorm.

One of the most recent users to complain about mass downvoting also cited participation in flame-bait topics (specifically gender).

Gah... This is becoming way too common, and it seems like there's pretty good evidence (further supported in this instance) regarding the responsible party. I wish someone with the power to do so would do something about it.

I have blindly upvoted your 10 most recent comments. This is meant as consolation but likely a one-time action .

a comment where I said "thanks" when somebody pointed out a formatting error in my comments is now at -1.

That sounds like a pretty low value comment. Is it beneficial to third parties to be able to read it? If not, just make the correction and PM your thanks. Otherwise you're unnecessarily wasting everyone's time in the guise of politeness.

For context, link to past discussion of mass-downvoting.

I got a seemingly one-time hit of this about a week ago. For what it's worth I had just been posting comments on the subject of rape, but a whole bunch of my unrelated comments got it too.

(Since then it's been having an obnoxious deterrent effect on my commenting, because I feel so precariously close to just accumulating negative karma every time I post, leaving readers with the impression that my ideas have all been identified as worthless by someone probably cleverer than themselves. I'm now consciously trying to avoid thinking like this)

On September 26, 1983, Soviet officer Stanislav Petrov saved the world. (Nov 2011)

Eh, not really.

The Wiki link in the linked LW post seems to be closer to "Stanislav Petrov saved the world" than "not really":

Petrov judged the report to be a false alarm, and his decision is credited with having prevented an erroneous retaliatory nuclear attack

...

His colleagues were all professional soldiers with purely military training and, following instructions, would have reported a missile strike if they had been on his shift.

...

Petrov, as an individual, was not in a position where he could single-handedly have launched any of the Soviet missile arsenal. ... But Petrov's role was crucial in providing information to make that decision. According to Bruce Blair, a Cold War nuclear strategies expert and nuclear disarmament advocate, formerly with the Center for Defense Information, "The top leadership, given only a couple of minutes to decide, told that an attack had been launched, would make a decision to retaliate."

A closely related article says:

Petrov's responsibilities included observing the satellite early warning network and notifying his superiors of any impending nuclear missile attack against the Soviet Union. If notification was received from the early warning systems that inbound missiles had been detected, the Soviet Union's strategy was an immediate nuclear counter-attack against the United States (launch on warning), specified in the doctrine of mutual assured destruction.

That he didn't literally have his finger on the "Smite!" button, or that the SU might still not have retaliated if he'd raised the alarm, is not the point.

previous eras' low life expectancy was mostly due to high child mortality.

I have long thought that the very idea of "life expectancy at birth" is a harmful one, because it encourages exactly that sort of confusion. It lumps together two things (child mortality and life expectancy once out of infancy) with sufficiently different causes and sufficiently different effects that they really ought to be kept separate.

On the AIXI and such... you see, its just hard to appreciate just how much training it takes to properly understand something like that. Very intelligent people, with very high mental endurance, train for decades, to be able to mentally manipulate the relevant concepts at their base level. Now, let's say someone only spent a small fraction of the time - either because they've pursued a wrong topic through the most critical years, or because they have low mental endurance. Unless they're impossibly intelligent, they have no chance of forming even a merely good understanding.

Smart move not only to review but post the results. Shows humbleness and at the same time prevents being called on it later.

This is an approach I'd like to see more often. Maybe you should add it to the http://lesswrong.com/lw/h7d/grad_student_advice_repository/ or some such.

Huh. I followed the link to the correction of the Petrov story, and found I'd already upvoted it.

But if you'd asked me yesterday for examples of heroes yesterday, I'd have cited Petrov immediately. S

hows how hard it is to unlearn false information once you've learned it.

Heretical? Well, considering that 'heretic' means 'someone who thinks on their own', I'm not sure how we're supposed to interpret that negatively.

I assume however that you meant 'disagreeing with common positions displayed on LW' - which of those common positions did you differ on, and why, and just how homogeneous do you think LW is on those?

Yes, although it would help if you could be a bit more specific, the term is somewhat overloaded.

As for the strategy, depends. Find a better community (than the one you feel alienated from) in the sense of better matching values? We both seem to feel quite at home in this one (for me, if not for the suffocating supremacy of EA).

Does anyone else experience the feeling of alienation?

But of course.

And does anyone have a good strategy for dealing with it?

Accept that you're not average and not even typical.

Feeling usually become a problem when you resist them.

My general approach with feelings:

1. Find someone towards which you can express the content behind the feeling. This works best in person. Online communication isn't good for resolving feelings. Speak openly about whatever comes to mind.

2. Track the feeling down in your body. Be aware where it happens to be. Then release it.

I think that feeling is more common than you might think. Especially if you deviate enough from the societal norm (which Less Wrong generally does).

My general strategy for dealing with is social interaction with people who'll probably understand. Just talk it over with them. It's best if you do this with people you care about. It doesn't have to be in person, if you've got someone relevant on Skype, that works as well.

I think this feeling arises from social norms feeling unnatural to you. This feeling should be expected if your interests are relevant to this site, since people are not trying to be rational by default.

The difference between a pathetic misfit and and an admirable eccentric is their level of awesomeness. If you become good enough at anything relevant to other people, you don't have to live through their social expectations. Conform to the norms or rise above them.

Note that I think most social norms are nice to have, but this doesn't mean there aren't enough of the kind that make me feel alienated. It could be that the feeling of alienation is a necessary side effect of some beneficial cognitive change, in which case I'd try to cherish the feeling. I've found that rising to a leadership position diminishes the feeling significantly, however.

In my experience memorization often comes for free when you strive for fluency through repetition. You end up remembering the quadratic formula after solving a few hundred quadratic equations. Same with the trig identities. I probably still remember all the most common identities years out of school, owing to the thousands (no exaggeration) of trig problems I had to solve in high school and uni. And can derive the rest in under a minute.

Memorization through solving problems gives you much more than anki decks, however: you end up remembering the roads, not just the signposts, so to speak, which is important for solving test problems quickly.

You are right that "the reduction in mental effort required on basic operations will rapidly compound to allow for much greater fluency with harder problems", I am not sure that anki is the best way to achieve this reduction, though it is certainly worth a try.

Nice, and good luck! I'm glad to see that my post resonated with someone. For rhetorical purposes, I didn't temper my recommendations as much as I could have -- I still think building mental models through deliberate practice in solving difficult problems is at the core of physics education.

I treat even "signpost" flashcards as opportunities to rehearse a web of connections rather than as the quiz "what's on the other side of this card?" If an angle-addition formula came up, I'd want to recall the easy derivation in terms of complex exponentials and visualize some specific cases on the unit circle, at least at first. I also use cards like that in addition to cards which are themselves mini-problems.

In general there the core principle of spaced repetition that you don't put something into the system that you don't already understand.

When trying to memorize mathematical results make sure that you only add cards when you really have a mental understanding. Using Anki to avoid forgetting basic operations is great. If you however add a bunch of information that's complex, you will forget it and waste a lot of time.

"Sexy" isn't signaling -- it's a characteristic that people (usually) try to signal, more or less successfully. "I'm sexy" basically means "You want me" : note the difference in subjects :-)

Being sexy signals health, youth, and fertility. This is quite well supported by evidence and discussed in many books and articles.

I would agree with what Lumifer says below, but I think sexy can be signalling when many people are involved: look at the sexy people I hang out with. Being with sexy people brings high status because it's high status.

Sexy is a quite broad word that probably used by different people in different ways. I think for most people it about what they feel when looking at the person. Those feeling where set up by evolution over large time frames.

Evolution doesn't really care about whether you get a fun intercourse partner.

But it's not only evolution. It also has a lot to do with culture. Culture also doesn't care about whether you get a fun intercourse partner. People who watch a lot of TV get taught that certain characteristics are sexy.

For myself I would guess that most of my cultural imprint regarding what I find sexy comes from dancing interactions. If a woman moves in a way that suggests that she doesn't dance well, that will reduce her sex appeal to me more than it probably does with the average male.

Not much. The kind of probability relevant to MIRI's interests is not the kind of probability you need calculus to understand (the random variables are usually discrete, etc.). The closest thing to needing a calculus background is maybe numerical analysis (I suspect it would be helpful to at least have the intuition that derivatives measure the sensitivity of a function to changes in its input), but even then I think that's more algorithms. Not an expert on numerical analysis by any means, though.

If you have a general interest in mathematics, I still recommend that you learn some calculus because it's an important foundation for other parts of mathematics and because people, when explaining things to you, will often assume that you know calculus after a certain point and use that as a jumping-off point.

I don't, but in Tim Ferris' book Four-Hour Work Week, I think I recall him recommending them. I think this was the one he recommended: https://www.yourmaninindia.com/.

Let me know if you come across some good findings on this. If effective, virtual assistants could be very useful, and thus they're something I'm interested in. On that note, it'd probably be worth writing a post about them.

Is there a good way of finding what kind of job might fit a person?

That's a strange question.

Either you want to know how to pick up the skill of being a career adviser. Alternatively you want to find a job for yourself. You might also be a parent who tries to find a job that fits his child instead of letting the child decide for themselves.

I think the answers to those three possibilities are very different.

I think for most people who ask this question, the range of fitting jobs is much wider than they think. You learn to like what you become good at.

If I were to pick a career right now, I'd just take a long list of reasonably complex jobs and remove any that contain an obvious obstacle like a skill requirement I'm unlikely to improve at. Then from what is left, I'd narrow the choice by some other criteria than perceived fit, income and future employment prospects for example and then pick one of them either by some additional criteria or randomly. I'm confident I'd learn to like almost any job chosen this way.

If you make money you can do whatever you like in the future even if you chose your job poorly in the first place. So please don't choose to become an English major.

There's a proliferation of terminology in this area; I think a lot of these are in some sense equivalent and/or special cases of each other. I guess "Bayesian network" is more consistent with the other Bayes-based vocabulary around here.

Hidden Markov models are a reasoning model to solve a specific problem. If you don't face that specific problem they are no use.

Most of the problems we discuss aren't modeled well with HMMs.