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Assessing oneself
polymer20

Perhaps I should've said, hard in the wrong ways. The long term goal for a good professional programmer seems to be understanding what the client wants. Some math is needed to understand the tools, so you can give some context for options. But I spend most of my creative energy making sure my programs do what I want them to do, and that is really hard when each language has it's own prejudice motivating its design.

I seriously considered looking into real time high risk software applications. But I just decided that instead of learning new languages until I ran out of youth, it'd be more fun learning general relativity, or even measure theory. The ideas in those subjects will probably hold out a lot longer then python.

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Assessing oneself
polymer20

So, my point regarding the speed.

In the middle of working out a problem, I had to find the limit of

S = 1/e + 2/e^2 + ... + n/e^n + ...

I had never seen this sum before, so now cleverness is required. If I assumed guess C was true, that would imply

e/(e - 1) = (e - 1)S

This claim is much easier to check,

(e - 1)S = 1 + 1/e + 1/e^2 + ... = 1/(1 - 1/e) = e/(e - 1)

We know what S is, and the solution to the problem follows. In retrospect, I understand one method for how I could find the answer. But during the test, I can't see through the... (read more)

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Assessing oneself
polymer20

I'm not sure. I'm trying to work towards a career path which uses as much of my ability as I can. The most important job for a professional programmer, was understanding what your client wanted. This is a fine job, but being good at algorithms isn't necessarily a requirement.

When talking to an engineer at Google, I asked what he thought a good career choice was for working on hard problems. His immediate first thought was graduate school, then he sort of mentioned robotics.

My ideal dream isn't being a professor, it's working on something that needs inferen... (read more)

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1Toggle
This is one of the harder problems out there, in my experience. Many extremely intelligent people spin their wheels on this one for years. Some indefinitely. Especially for a person who is talented at inferential or analytical problem-solving, looking for acceptable institutions first may be a case of putting the cart before the horse. Those places- the universities, research groups, etc.- tend to be looking for researchers that think of the institution as a tool, not as a goal. This is at least partly because it's very hard to quantify successful research, and they're looking for assurances that work will continue. If you value 'being a member of the Center for Advanced Studies', then you can succeed at that goal without actually doing any work. Imagine a dissertation for yourself, the multi-year project that you would be working on during your time in the Platonic University where everyone is accepted and everything is perfect and you study exactly what you want to study. What is that project? What can you do, right now, to pursue that project? What factors will get in your way, and what steps do you need to take to minimize or eliminate those factors? If you can't picture one just yet, that's fine. Talk with your professors if you want; not about grades but about what their own research is, and why they care about it. Ask a lot of professors about that; they almost always think that their work is important, and are happy to describe it. Get a sense of the conversation as it currently exists, and then find a niche that interests you. In other words, don't look for a pleasant societal position as its own end- just do the thing, become more skilled in thing-doing, and grab help when you need it. The more you do the thing, the more you will quite naturally build a positive reputation among a widening network, and this will grant you access to a surprising number of institutions as you need them. But at the end of the day, it is the privilege of the researcher to e
0Shmi
Seems like a place like this could be a good fit, if you are really really good. Not sure how one gets hired there.
Assessing oneself
polymer60

I agree, that I have a wealth of information to work with right now. Just trying to honestly balance it (felt like LW fit the theme somewhat).

On the one hand, both of those scores are my first time, and they were taken cold. And, I could argue I thought a lot of homework in school was unimportant and unnecessary (because of a poor philosophical attitude).

But of the 26 questions wrong or incomplete on the practice Math subject test, roughly 16 of them I had sufficient knowledge, but I simply wasn't fast enough. And the Algebra class, was really hard, and I ... (read more)

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0D_Malik
Here's some heterodox advice: Take stimulants. Before I wrote the SAT, I stayed off caffeine for a couple of weeks. Then I drank lots of coffee right before the test, and in the break between sections. Caffeine affects me very strongly, so you can use some other stimulant. It might shorten your life by a week, but it's probably worth it.
7[anonymous]
You took the GREs cold. I'm surprised you did half as well as you did. Why? Because anyone who is not mentally handicaped can pay tutor a large sum of money, do exactly what the tutor says, and get a perfect score. I'm not exagerating -- I have friends who tutor in this business, and every year they sit for the GRE as a requirement for their job, and get a perfect score. It's a teachable skill, and one which has very little to do with the subject matter. Now consider that most of the other people who took the GRE knew about this weakness. Especially internationally in places like China and India where (1) there are a lot of test takers, (2) a much larger test prep industry, and (3) massive incentives to do well (so as to get into an American or European university). Now keeping all that in mind, you still scored better than 72 / 68 percent of the competition despite having absolutely no preparation whatsoever. Why are you not congratulating yourself?
gjm330

A few disorganized remarks that may or may not be any help:

  • Different people are good at different things. In particular, the algebra/analysis dichotomy is a pretty standard one and if you're good at analysis and not so good at algebra, it probably matters how good you are at what you're best at.
  • It seems like simply not being fast enough could be largely irrelevant (if it's really just a matter of speed; the limiting factor in doing mathematical research is unlikely to be how fast you can do practice-test-level questions) or quite important (if what it r
... (read more)
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Open thread, 25-31 August 2014
polymer00

I'm not quite sure what the following means:

if you add details to a story, it becomes less plausible" is a false statement coming from human interaction.

I don't care whether it's false as a "human interaction". I care whether the idea can be modeled by probabilities.

Is my usage of the word plausible in this way really that confusing? I'd like to know why... Probable, likely, credible, plausible, are all (rough) synonyms to me.

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Open thread, 25-31 August 2014
polymer00

So plausibility isn't the only dimension for assessing how "good" a belief is.

A or not A is a certainty. I'm trying to formally understand why that statement tells me nothing about anything.

The motivating practical problem came from this question,

"guess the rule governing the following sequence" 11, 31, 41, 61, 71, 101, 131, ...

I cried, "Ah the sequence is increasing!" With pride I looked into the back of the book and found the answer "primes ending in 1".

I'm trying to zone in on what I did wrong.

If I had said inst... (read more)

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0JQuinton
I think of it in terms of making a $100 bet. So you have the sequence S: 11, 31, 41, 61, 71, 101, 131. A: is the "bet" (i.e. hypothesis) that the sequence is increasing by primes ending in 1. There are very few sequences (below the number 150) you can write where you have an increasing sequence of primes ending in 1, so your "bet" is to go all in. B: is the "bet" that the sequence is increasing. But a "sequence that's increasing" spreads more of its money around so it's not a very confident bet. Why does it spread more of its money around? If we introduced a second sequence X: 14, 32, 42, 76, 96, 110, 125 You can still see that B can account for this sequence as well, whereas A does not. So B has to at least spread its betting money between the two sequences presented A and X just in case either of those are the answer presented in the back of the book. In reality there are an untold amount of sequences that B can account for besides the two here. Meaning that B has to spread its betting money to all of those sequences if B wants to "win" by "correctly guessing" what the answer was in the back of the book. This is what makes it a bad bet; a hypothesis that is too general. This is a simple mathematical way you can compare the two "bets" via conditional probabilities: Pr(B | S) + Pr(B | X) + Pr(B | ??) = 1.00 and Pr(A | S) + Pr(A | X) + Pr(A | ??) = 1.00 Pr(A | S) is already all in because the A bet only fits something that looks like S. Pr(B | S) is less than all in because Pr(B | X) is also a possibility as well as any other increasing sequence of numbers, Pr(B | ???). This is a fancy way of saying that the strength of a hypothesis lies in what it can't explain, not what it can; ask not what your hypothesis predicts, but what it excludes. Going by what each bet excludes you can see that Pr(A | ??) < Pr(B | ??), even if we don't have any hard and fast number for them. While there is a limited amount of 7 number patterns below 150 that are increasing, this is
1gjm
"S is an increasing sequence" is a less specific hypothesis than "S consists of all prime numbers whose decimal representations end in 1, in increasing order". But "The only constraint governing the generation of S was that it had to be an increasing sequence" is not a less specific hypothesis than "The only constraint governing the generation of S was that it had to consist of primes ending in 1, in increasing order". If given a question of the form "guess the rule governing such-and-such a sequence", I would expect the intended answer to be one that uniquely identifies the sequence. So I'd give "the numbers are increasing" a much lower probability than "the numbers are the primes ending in 1, in increasing order". (Recall, again, that the propositions whose probabilities we're evaluating aren't the things in quotation marks there; they're "the rule is: the numbers are increasing" and "the rule is: the numbers are the primes (etc.)". Moving back to your question about analytic functions: Yes, more specific hypotheses may be more useful when true, and that might be a good reason to put effort into testing them rather than less specific, less useful hypotheses. But (as I think you appreciate) that doesn't make any difference to the probabilities. The subject concerned with the interplay between probabilities, preferences and actions is called decision theory; you might or might not find it worth looking up. I think there's some philosophical literature on questions like "what makes a good explanation?" (where a high probability for the alleged explanation is certainly a virtue, but not the only one); that seems directly relevant to your questions, but I'm afraid I'm not the right person to tell you who to read or what the best books or papers are. I'll hazard a guess that well over 90% of philosophical work on the topic has close to zero (or even negative) value, but I'm making that guess on general principles rather than as a result of surveying the literature i
Open thread, 25-31 August 2014
polymer30

Can someone link to a discussion, or answer a small misconception for me?

We know P(A & B) < P(A). So if you add details to a story, it becomes less plausible. Even though people are more likely to believe it.

However, If I do an experiment, and measure something which is implied by A&B, then I would think "A&B becomes more plausible then A", Because A is more vague then A&B.

But this seems to be a contradiction.

I suppose, to me, adding more details to a story makes the story more plausible if those details imply the evidence. Si... (read more)

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0IlyaShpitser
If A,B,C are binary, values of A and B are drawn from independent fair coins, and C = A XOR B, then measuring C = 1 constrains A,B to be either { 1, 1 } or { 0, 0 }, but does not constrain A alone at all. Before we conditioned on C=1, all values of the joint variable A,B had probabilities 0.25, and all values of a single variable A had probabilities 0.5. After we conditioned on C=1, values { 0, 0 } and { 1, 1 } of A,B assume probabilities 0.5, and values { 0, 1 } and { 1, 0 } of A,B assume probabilities 0, values of a single variable A remain at probability 0.5. By conditioning on C=1, you learn more about the joint variable A,B than about a single variable A (because your posterior for A,B changed, but your posterior for A did not), but that is not the same thing as the joint variable A,B being more plausible than the single variable A. In fact, it is still the case that p(A & B | C) <= p(A | C) for all values of A,B. ---------------------------------------- edit: others below said the same, and often better.
0lmm
I think we tend to intuitively "normalize" the likelihood of a complex statement. Our prior is probably Kolmogorov complexity, so if A is a 2-bit statement and B is a 3-bit statement, we would "expect" the probabilities to be P(A)=1/4, P(B)=1/8, P(A&B)=1/32. If our evidence leads us to adjust to say P(A)=1/3, P(A&B)=1/4, then while A&B is still less likely than A, there is some sense in which A&B is "higher above baseline". Coming from the other end, predictions, this sort of makes sense. Theories that are more specific are more useful. If we have a theory that this sequence consists of odd numbers, that lets us make some prediction about the next number. If our theory is that the numbers are all primes, we can make a more specific, and therefore more useful, prediction about the next number. So even though the theory that the sequence is odd is more likely than the theory that the sequence is prime, the latter is more useful. I think that's where the idea that specific theories are better than vague theories comes from.
0buybuydandavis
P(A & B) <= P(A)
0JQuinton
I'm guessing that the rule P(A & B) < P(A) is for independent variables (though it's actually more accurate to say P(A & B) <= P(A) ). If you have dependent variables, then you use Bayes Theorem to update. P(A & B) is different from P(A | B). P(A & B) <= P(A) is always true, but not so for P(A | B) viz. P(A). This is probably an incomplete or inadequate explanation, though. I think there was a thread about this a long time ago, but I can't find it. My Google-fu is not that strong.
-4Lumifer
Not so. Stories usually are considerably more complicated than can be represented as ANDing of probabilities. A simple example: Someone tells me that she read my email to Alice, let's say I think that's X% plausible. But then she adds details: she says that the email mentioned a particular cafe. This additional detail makes the plausibility of this story skyrocket (since I do know that the email did mention that cafe).
2one_forward
A&B cannot be more probable than A, but evidence may support A&B more than it supports A. For example, suppose you have independent prior probabilities of 1/2 for A and for B. The prior probability of A&B is 1/4. If you are then told "A iff B," the probability for A does not change but the probability of A&B goes up to 1/2. The reason specific theories are better is not that they are more plausible, but that they contain more useful information.
5gjm
A&B gains more evidence than A from the experiment. It doesn't (and can't) become more probable. Let's have an example. Someone is flipping a coin repeatedly. The coin is either a fair one or a weighted one that comes up heads 3x as often as tails. (A = "coin is weighted in this way".) The person doing the flipping might be honest, or might be reporting half the tails she flips (i.e., each one randomly with p=1/2) as heads. (B = "person is cheating in this way".) Let's say that ahead of time you think A and B independently have probability 1/10. Your experiment consists of getting the (alleged) results of a single coin flip, which you're told was heads. So. Beforehand the probability of A was 1/10 and that of B was 1/100. The probability of your observed results is: 1/2 under (not-A,not-B); 3/4 under (not-A,B); 3/4 under (A.not-B); and 7/8 under (A,B). So the posterior probabilities for the four possibilities are proportional to (81:9:9:1) times (4:6:6:7); that is, to (324:54:54:7). Which means the probability of A has gone up from 10% to about 14%, and the probability of A&B from 1% to about 1.6%. So you've got more evidence for A&B than for A, which translates (more or less) to a larger relative gain in probability for A&B than for A. But A&B is still less likely. If you repeat the experiment and keep getting heads, then A&B will always improve more than A alone. But the way this works is that after a long time almost all the probability of A comes from the case where A&B, so that A&B's advantage in increase-in-probability gradually goes away.
2Metus
A more specific explanation is better than a general explanation in the scientific sense exactly because it is more easily falsifiable. Your sentence is completely wrong, as the set containing the sine function is most certainly contained in the set of all analytic functions, making it more plausible that "some analytic function has roots at all multiples of pi" than to say the same of sine, assuming we do not already know a great deal of information about sine. Plain and simply no. If evidence E implies A and B, formally E -> A&B, then seperately E -> A and E -> B are true, increasing the probability of both seperately, making your conclusion invalid.
[Link] Feynman lectures on physics
polymer50

I disagree, I read the Feynman lectures in high school and learned a great deal. His presentation taught me more about how to think about these things then Giancoli did.

Giancoli better prepared me for what the standard format was for test questions, but it didn't really articulate how I was supposed to use the ideas to generate new ones. Feynman's style of connecting claims with whatever you happened to know, is extremely important. Giancoli doesn't demonstrate this style quite as well.

Of course it was my first textbook, so I could go on and on about why I like it...

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1Shmi
Actually, this was my experience, as well. But I did not truly learn the subject until I used a more standard text (not Giancoli, but similar). Feynman gives you ideas and intuition, not mastery.