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wedrifid comments on Fake Explanations - Less Wrong
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I disagree. The proper response to not knowing the answer is to admit to not knowing and then give your best guess, not to try to hide your ignorance, because if you succeed then the teacher doesn't know you need help. A student who is more concerned with not displaying ignorance than with not being ignorant is not trying to learn, which is not rational. That which can be destroyed by the truth should be, and it probably won't be if you try to avoid finding out what the truth is.
The key phrase here is "on average". If you guess at random from all possible explanations of a given phenomenon, you will, on average, die before guessing the correct answer. There is a reason the monkeys with typewriters are given infinite time to reproduce Hamlet.
Moreover, as the set of answers considered increases in size, the expected utility from giving any one answer tends towards the expected utility of a wrong answer. As long as giving the wrong answer gives less utility than admitting ignorance, admitting ignorance is almost always the utility maximising option if you don't know.
If I write down a number and then take a number from a table of random numbers, and the numbers are the same, does this mean that I'm psychic? Because if getting the correct answer means that I have useful anticipation controllers then I must be.
"I don't know" is not an excuse for not knowing. That makes no sense whatsoever. "I don't know" is a statement about whether I know something or not, not a statement about whether I ought to know. If you can't admit fallibility then you will never learn anything.
The points you make about the benefits of testing students' knowledge are true. Unfortunately, they miss the point - while it is important not to penalise guessing incorrectly, so as not to dissuade admitting ignorance, it is much better to actively reward admitting that you have tried and failed. If a confused student does not always seek an explanation, the reward for seeking explanations isn't large enough yet. If students are content to remain ignorant, something is seriously wrong with your system for making students less ignorant.
If students could always get away with an "I don't know" they wouldn't have much incentive to learn anything.
More importantly, the school system main purpose is not to teach you just a collection of facts. It has to teach you how to behave in the world, where you often have to make choices based on incomplete information.
0 marks for "I don't know". 1 mark for a correct answer. -1 mark for an incorrect answer.
Not only is it a simple incentive system I've done exams that implemented similar systems. (Westpac math competition for example.)
That is a sensible scoring system which is in fact widely used.
Allow both an answer and a certainty.
-x points for an incorrect answer with certainty x
+2x points for the correct answer with certainty x
Alternately, +10^x points for a correct answer with certainty x, and +Log(1-x) points for the incorrect answer. This encourages an attempt to answer every question, even if the certainty is rated as 0.
Yes, I know, old post.
If you give the student -X points for an incorrect answer with certainty X, and +2X points for a correct answer with certainty X, the expected value of giving an answer and lying about its certainty as Y is (1-X)(-Y) + (X)(2Y) = 3XY - Y. If X is less than 1/3, the student should lie and claim that his certainty is 0, while if X is greater than 1/3, he should lie and claim that his certainty is 1.
I'm not going to try to find the maximum for the second version, but it should be obvious that the student is still better off lying about his true certainty. Of course, you could just avoid telling the student how you're going to grade, but the score will then just depend on how well the student guesses your grading criteria.
Neither of my described systems are ideal. Squared error works for binary questions, but it would reward "Pi is exactly 3, with 0 confidence".
Rather than allow continuous estimates of accuracy, I think that the ideal system would ask the student to provide a range of confidence, (five choices from "guessing" to "Certain", with equivalent probabilities), and an appropriate scoring rule; a guess would be penalized 0 for being wrong but gain little for being right, and going from "almost certain" to "certain" would add a small value to a correct answer but a large penalty to a wrong answer.
Having established the +points for correct and -points for wrong for each confidence description, do the math to determine what the actual ranges of confidence are, sanity check them against the descriptions, and then tell the student the confidence intervals. (Alternately, pick the intervals and terms and do the math to figure out the + for correct answer and -for incorrect answer for those intervals.)
It's hard to come up with a system where the student doesn't benefit from lying about his certainty. What you describe would fix the case from 4 (almost certain) to 5 (certain), but you need to get all the cases to work and it's plausible that fixing the 4 to 5 case (and, in general, increasing the incentive to pick 4) breaks the 3 to 4 case.
After all, you can't have all the transitions between certainty values add a small value to a correct answer. You must have a transition where a large value is added for a correct answer and your system may break down around such transitions.
The largest value would be added for the first confidence interval, which would also add the smallest cost to being wrong with that confidence.
That would mean a large value would be added when going from "guess" to "almost guess", which would mean that it would be beneficial for a student to lie and claim to almost guess when he's really completely guessing.
Suppose the student thinks that there is a 10% chance that he is right, and the reward structure is +5/-1 for confidence interval 1.
In fact, make the reward structure:(right/wrong) 1/0, 6/-1, 10/-3, 13/-6, 15/-10, 16/-15
That puts the breakpoints at roughly even intervals, keeps the math easy, and with a little bit of clarifying exactly where the breakpoints are, doesn't reward someone who accurately determines their accuracy and then lies about it.