Didn't we already have this exact post?

You had "Optimization", and you may have read a draft elsewhere with some of this material in it. I'm posting it now by way of finally finishing it after years, and by way of taking a vacation (feeling a bit burned-out lately).

On a whimsical note, it is reminiscent of the unpredictability of the Infinite Improbability Drive :-)

"If we walk without rhythm, we won't attract the worm."

Set up a pattern-recognition system directed at your own actions, and when you fall into a predictable rut, do something differently.

Could you give a more precise definition of "calibrated"? Your example of 1/37 for each of 37 different possibilities, justified by saying that indeed one of the 37 will happen, seems facile. Do you mean that the "correct" distribution, relative to your guess, has low relative entropy?

I think what he means by "calibrated" is something like it not being possible for someone else to systematically improve the probabilities you give for the possible answers to a question just from knowing what values you've assigned (and your biases), without looking at what the question is.

I suppose the improvement would indeed be measured in terms of relative entropy of the "correct" guess with respect to the guess given.

anon: The short definition of "well-calibrated" is that if things you assign X% probability to actually do happen X% of the time, you are well-calibrated. If you go around saying that you are 90% certain of your answers on an exam, and you get a 90, you are well-calibrated. If you say that you are 90% certain and you get a 50, you are not well-calibrated.

For the long version, see A Technical Explanation of Technical Explanation". (Highly recommended!)

Anon: well-calibrated means roughly that in the class of all events you think have probability p to being true, the proportion of them that turn out to be true is p.

More formally, suppose you have a probability distribution over something you are going to observe. If the log probability of the event which actually occurs is equal to the entropy of your distribution, you are well calibrated. If it is above you are over confident, if it is below you are under confident. By this measure, assigning every possibility equal probability will always be calibrated.

This is related to relative entropy.

As with jsalvati's comment, upon reading this post I was convinced that either I was hallucinating, that almost the entirety of this post had appeared elsewhere before, or that I've read so much of Eliezer's writing that I'm anticipating his arguments in detail before he makes them.

Fortunately, it turns out that the second (and least disturbing) option was correct, in that a substantial amount of this post already existed at: [link deleted]

Not that I'm complaining, but it was seriously bothering me that I knew I'd heard this stuff before, but Google said it hadn't previously been posted on Overcoming Bias, and I was starting to doubt my sanity...

This notion of calibratedness seems to have bad properties to me. Consider a situation where I'm trying to guess a distribution for the outcomes of a coin flip with a coin which, my information tells me, lands "heads" 99% of the time. Then a guess of 50% and 50% is "calibrated" because of the 50% predictions I make, exactly half come out right. But a guess 49.9% heads and 50.1% tails is horribly calibrated; the "49.9%" predictions come out 99% correct, and the "50.1%" predictions come out 1% correct. So the concept, as defined like this, seems hypersensitive, and therefore not very useful. I think a proper definition must necessarily be in terms of relative entropy, or perhaps considering Bayesian posteriors from subsets of your information, but I still don't see how it should work. Sorry if someone already gave a robust definition that I missed.

Nick: If you don't mean expected log probability, then I don't know what you're talking about. And if you do, it seems to me that you're saying that well-calibratedness means that relative entropy of the "correct" distribution relative to yours is equal to your entropy. But then the uniform prior doesn't seem well-calibrated; again, consider a coin that lands "heads" 99% of the time. Then your entropy is 1, while the relative entropy of the "correct" distribution is (-log(99%)-log(1%))/2, which is >2.

Nick: Sorry, I got it backwards. What you seem to be saying is that well-calibratedness means that relative entropy of your distribution relative to the "correct" one is equal to your entropy. This does hold for the uniform guess. But once again, considering a situation where your information tells you the coin will land "heads" with 99% probability, it would seem that the only well-calibrated guesses are 99%-1% and 50%-50%. I don't yet have an intuition for why both of these guesses are strictly "better" in any way than an 80%-20% guess, but I'll think about it. It definitely avoids the sensitivity that seemed to come out of the "rough" definition, where 50% is great but 49.9% is horrible.

Please don't post a link to the draft version of this. This just means that lots of Overcoming Bias readers will be bored for the next couple of weeks, because they'll already have read the old version of something.

Anon: no, I mean the log probability. In your example, the calibratedness will generally be high: - \log 0.499 - H(p) ~= 0.00289 each time you see tails, and - log 0.501 - H(p) ~= - 0.00289 each time you come up tails. It's continuous.

Let's be specific. We have H(p) = - \sum_x p(x) \log p(x), where p is some probability distribution over a finite set. If we observe x0, the say the predictor's calibration is

C(x0) = \sum_x p(x) \log p(x) - \log p(x0) = - \log p(x0) - H(p)

so the expected calibration is 0 by the definition of H(p). The calibration is continuous in p. If \log p(x0) is higher then the expected value of \log p(x) then we are underconfident and C(x0) < 0; if \log p(x0) is lower than expected we are overconfident, and C>0.

With q = p(x) d(x,x0) the non-normalised probability distribution that assigns value only x0, we have

C = D(p||q)

so this is a relative entropy of sorts.

If the possibility of "creative surprises" depends on ignorance about the logical consequences of a game move, it seems that this would be best modeled as an asymmetric information problem.

It's interesting to note that the usual "Dutch-book" arguments for subjective probability break down under asymmetric information - the only way to avoid losing money to a possibly better-informed opponent is refusing to enter some bets, i.e. adopting an upper-lower probability interval.

Of course, such upper-lower spreads show up routinely in illiquid financial markets; I wonder whether any connections have been made between rational pricing conditions in such markets and imprecise probability theories like Dempster-Shafer, etc.

I think there's a sign error in my post -- C(x0) = \log p(x0) + H(p) it should be.

Nick Hay - IIRC the minus-log probability of an outcome is usually called "surprisal" or "self-information". The Shannon entropy of a random variable is just the expected value of its surprisal.

guest: right, so with those definitions you are overconfident if you are suprised more than you expected, underconfident if you are suprised less, calibration being how close your suprisal is to your expectation of it.

If you use upper-lower then you will forego combinations of bets that are sure gains. Dutch Book is inescapable, and only your presumption that the environment will contain agents who try to offer you losing bet combinations but no agents who are nice enough to take them, would lead you think that upper-lower "solves" the problem. No real agent can afford the computing power to be an ideal Bayesian, but the laws are ineluctable: whoever is not Bayesian - regardless of the reasons, regardless of whether it is unavoidable - is losing something.

whoever is not Bayesian - regardless of the reasons, regardless of whether it is unavoidable - is losing something.

And whoever is bayesian is also losing something, a quantity of easily acquirable energy. It is a loss one way or the other.

Nick: It seems like a bad idea to me to call a prediction underconfident or overconfident depending on the particular outcome. Shouldn't it depend rather on the "correct" distribution of outcomes, i.e. the Bayesian posterior taking all your information into account? I mean, with your definition, if we do the coin flip again, with 99% heads and 1% tails, and our prediction is 99% heads and 1% tails, then if it comes up heads we're slightly underconfident, and if it comes up tails we're strongly overconfident. Hence there's no such thing as an actually well-calibrated prediction for this (?). If we take into account the existence of a correct Bayesian posterior then it's clear that "expected calibration" is not at all 0. For instance if p is the "correct" probability of heads and q is your prediction then the "expected calibration" would seem to be -plog(q)-(1-p)log(1-q)+qlog(q)+(1-q)log(1-q). And, for instance, if you know for a fact that a certain experiment can go one of 3 ways, and over a long period of time the proportion has been 60%-30%-10%, then not only 33.3%-33.3%-33.3%, but also 45%-45%-10% and 57%-19%-24% have "expected calibration" ~0 by this definition.

Will RYK be as good a player as Kasparov? Of course not. Sometimes the RYK system will randomly make dreadful moves which the real-life Kasparov would never make - start the game with P-KN4. I assign such moves a low probability, but sometimes the computer makes them anyway, by sheer random chance.

Don't think this is right. RYK would not make dreadful moves that you would not think Kasparov would make under any circumstances, unless you were assigning odds too high to those moves. So unless you think there's a chance that Kasparov would make that move more often than you would, you would assign a log-odd of -200 dB or something, effective 0 chance.

To clarify: what I'm specifically objecting to, I think, is "by sheer random chance". If it's the chance that's causing those bad moves, then you're not well-calibrated. If it's your uncertainty about Kasparov that's causing the moves, then my objection doesn't apply.

I swear to god I've read these Kasparov posts before...

I think the weakness of R[xx] systems is that they're only working one move at a time. Any position suggests a number of strategies to a good player. Being stuck with a randomized system that tends towards plausible moves means that a move that builds in one direction could be weakened in the next round by a move that's tending towards another good strategy.

Could R[xx] be improved (in theory--I think the calculations would be unmanageable) by considering more than one move at a time? Is there some number of moves where you don't exactly have an R[xx] anymore because the program is more like simulating the way the real chess player thinks?

re: calibration, it seems like what we want to do is ask ourselves what happens if an agent is asked lots of different probability questions, consider the the true probability as a function of probability stated by the agent, use some prior distribution (describing our uncertainty) on all such functions that the agent could have, update this prior using a finite set of answers we have seen the agent give and their correctness, and end up with a posterior distribution on functions (agent's probability -> true probability) from which we can get estimates of how over/underconfident the agent is at each probability level, and use those to determine what the agent "really means" when it says 90%; if the agent is overconfident at all probabilities then it's "overconfident" period, if it's underconfident at all probabilities then it's "underconfident" period, if it's over at some and under at some then I guess it's just "misconfident"? (an agent could be usually overconfident in an environment that usually asked it difficult questions and usually underconfident in an environment that usually asked it easy questions, or vice versa) If we keep asking an agent that doesn't learn the same question like in anon's comment then that seems like a degenerate case. On first think it doesn't seem like an agent's calibration function necessarily depends on what questions you ask it; Solomonoff induction is well-calibrated in the long run in all environments (right?), and you could imagine an agent that was like SI but with all probability outputs twice as close to 1, say. Hope this makes any sense.

pdf23ds, thanks. I wasn't sure whether the whole strategic situation could be implied in each position. Have people been able to learn anything about chess strategy by studying how programs play? Do program vs. program games make sense to people?

pdf23ds, most chess engines store analyzed subtrees into a transposition table. This is not the same as having a persistent strategy, but it does mean that much evaluation work can be saved by keeping a persistent state between moves.

This is similar to how iterative deepening works: in theory you could obtain the same result by performing a depth-first search directly, but iterative deepening provides a hint at the "best" moves which makes search more efficient.

I wasn't under the impression that transposition tables were kept between moves, but now that I think about it, they probably are. On the other hand, I'd wager that the reuse isn't very high, depending on the caching criteria. Maybe around 30%.

Have people been able to learn anything about chess strategy by studying how programs play?

I'm don't know if they've learned anything. Computer style is heavy on tactics. I'm not sure how their strategy is characterized. But it's notable that most new young grandmasters nowadays rely heavily on chess engines to help them train.

Do program vs. program games make sense to people?

Yes. Only chess engine fans are really too interested in them, though.

It'd be interesting to know if perfect checkers games (checkers has been solved) were amenable to human analysis.

A bit more on chess engines: many engines have a number of parameters you can set that determines their playing style, and some might say their "strategy". Things like aggressiveness, willingness to sacrifice, valuation of pieces, willingness to trade, etc. What many of these parameters actually do is change the static board evaluation function. A chess engine looks ahead many moves, and due to time constraints, picks a board position after X moves and runs the static evaluation function, which is a very naive estimate of the quality of the engine's position. A human's static evaluation function is much higher quality, but humans only look ahead a few moves at most. (Well, grandmasters look ahead a dozen or two moves at most but they're still evaluating many, many fewer positions (their pruning factor is higher).)

When I play against a superior player, I can't predict exactly where my opponent will move against me. [...] But I can predict the consequence of the unknown move, which is a win for the other player; and the more the player's actual action surprises me, the more confident I become of this final outcome."

Interestingly, playing an opponent which selects completely randomly from the range of possible moves gives a similar result: you never know what your opponent will do, but you can predict with a fair amount of confidence that you will win. And the more their actions surprise you (because you didn't spend any time thinking about such a remarkably dumb move as that), the more confident you become in your prediction of the result.

bah. moving too fast, forget to close an em tag. in the preceding comment, only the first paragraph should be em'd as a quote.

The quantity called Bayesian surprise expresses what you're trying to say. It is the KL divergence between the guessed probability distribution and the 'actual' distribution (distance between your probability distribution computed from some model (RYK) and Kasparov's actual distribution over moves (perhaps uniforms over the set of the 'top 5' moves according to some scoring heuristic)). When these are far apart in terms of KL divergence, your model will be greatly surprised (I recall a quote of yours somewhere in a post where you say, "There are no surprising facts, only models that are surprised by facts. And if a model is surprised by facts then it is no credit to that model.")

Bayesian surprise is experimentally linked to focus of attention mechanisms in human vision. If I have to have a good model of my world to avoid being hit by cars or eaten by bears, then I had better divert precious resources to allow me to scrutinize events of high Bayesian surprise (high shock to my current model) more intensely. This will almost certainly play a large role in understanding the specific biological mechanisms that lead to cognitive biases.

I also think that playing illustrates a use of Bayesian surprise. You are able to modify your map and discover new, perhaps unanticipated components of the territory, and (and this is what distinguishes 'playing' from other types of map updates) you feel safe while you do it. When a child learns to swim, they usually have adults around to protect them and though there is an element of danger, one feels reasonably safe while having your model of buoyancy, gravity, and movement totally surprised (in the Bayesian sense) by the experience of being in the water. The safety and map updating at the same time are what we refer to as fun, at least in many cases.

At any rate, I have noticed a number of places where this appeal to surprise is made and can benefit from rigorously linking to actual Bayesian surprise. For example, if I have two different probability distributions P and Q, then if they fail to have the same support over the domain, then D(P || Q) = \infty. Perhaps Engelbart invented the mouse and was able to extrapolate and anticipate a space of computer tools that could possibly be invented and he had some internal probability distribution over this space. But it's not those aspects of modern computing that are easily formed from this probability model of his that cause surprise or problems. If he understood mice conceptually, then trackballs clearly aren't surprising. Probably pen-based computing and even tablets are not surprising. But natural language processing allowing voice control might have been. Or brain-computer interface might have been... etc. Something in device space currently has non-zero probability that did not have non-zero probability in his model, and so there's infinite surprise between his model and the current situation. But maybe in concept space he did better. But by comparison, device space is to us as concept space is to FOOM'd AI. I certainly believe that a FOOM'd AI's probability model about various issues will have a strictly larger support set than my own would. There are concepts, heuristics, and metaheuristics that it will be able to assign meaningful probability to and of which I won't even be able to conceive. There will be infinite surprise.

All cases where humans can abstract problems (of economics, physics, mathematics, etc.) and accurately predict things in solution space indicate domains where the surprise isn't infinite. There are salient search directions to investigate that produce useful results. This is Robin's FOOM argument as I understand it from the debate. But it fails to consider cases where there literally are no salient search directions... at least not to us because we can't model them. Or the ones we can model are pitifully inadequate by comparison.

Consider RYY : your best probabilistic guess of your next move. Assuming you know yourself perfectly (or good enough to predict your own moves reliably at least), it will turn out RYY is very similar to you (RYY is not deterministic if you are not, but then it is still an opponent as skilled as you for all chess-related purposes).

Then since you have shown you win against RYK, I can guess that RYY will reliably win against RYK, which I find very surprising.

Why is emulating a stronger player less efficient than emulating yourself ? (It sounds more surprising to me formulated like that than how you said it.)

The explanation I see is that you don't know Kasparov well enough to emulate him correctly (what you already pointed out) whereas you know yourself very well. Then, the question that comes to my mind is : how can you use this knowledge to improve your play ?

I have received the advice "what would X do in your stead" by a number of people in a number of circumstances, including here by rationalists. How can it be useful ? If it is helpful, then it means that your cognition algorithm can be optimized and you know a specific way to improve it. If you find yourself frequently finding that wondering about what someone else would do is helpful, then there is additional computation that could be saved if you knew how you know how that someone else behaves (by doing it directly instead of imitating) . So, it's purely a matter of knowing yourself : the advice I had received was no better than "think about yourself".

The other question I wonder about is how that applies to artificial intelligence. I don't know more much about it. Is the "know yourself" important in that case ? How does an AI see its own source code ? I guess the first step toward making this question meaningful would be to design precisely how the emulator works. A naive approach (that doesn't account for styla and psychology) would be to use statistics for every position, and be random if an unprecedented position occurs. Then it becomes clear that there is no "default player" to emulate. There is no such thing as the emulator emulating itself because the emulator is not a player. Bummer.