Control systems win while being arational. Either explain this in terms of Bayescraft, or explain why there is no such explanation.
This (from Richard’s post) seems to me very much parallel to this (which I just made up):
Cricketers and baseball players win at ball-catching while knowing nothing about Newtonian mechanics, fluid dynamics, or solving differential equations. Either explain this in terms of physics, or explain why there is no such explanation.
Anyone who says anything close to “Cox’s theorem; therefore you must make your decisions by making Bayesian calculations” is broken. But it could still be reasonable to say “However you make your decisions, the results should be as close as you can make them to those of an ideal reasoner doing Bayesian calculations on the information you have”. I don’t see any contradiction, or even any tension, here. As for an actual specific explanation that matches the facts, that would seem to need to be done afresh for every control system that works; for some cases (like our brains) the answers might be unmanageably complicated.
Do Bayesian processes have anything to do with the mechanism of brains?
In the same sense as differential equations have something to do with the mechanism of people catching balls: when brains function well at maintaining reasonable beliefs, on some level of abstraction they have to act at least a little bit like Bayesian systems. But there needn’t be anything in the mechanisms that resembles the form (as opposed to the output) of the idealizations.
Does Bayescraft necessarily have anything to do with the task of building a machine that [...]
Since we might be able to do that by building a very big very low-level model of an entire human brain, without any understanding at all of what’s going on, obviously in some sense the answer is no. But if you want to understand what you’re doing—well, how much physics do you need to know if you want to get a space probe to Neptune? My guess is that even if you do it by making something that you launch into space at random and that then goes looking heuristically for something that might be Neptune, the chances are you’re going to want quite a lot of physics while you’re designing it.
The ball-catching example is interesting, as it’s another control problem, and has been studied as such. The fielder must get to where the ball will land. The predictive method would be to look at the ball, estimate its trajectory, then go to where you predict it will come down. This will not be very effective, because you cannot estimate the trajectory well enough. Instead, one method that will work is to move so as to maintain the direction from yourself to the ball constant in both azimuth and elevation. This is a control task, akin to the cursor-tracking task I discussed in the posting. You just have to move faster or slower and vary your direction, in whatever way will keep the direction constant. (The reason this works is that if the direction is constant, the ball is moving directly towards you in the frame of reference that moves with you. Or directly away, but in that case you won’t be able to run fast enough to catch it.)
Devise such a control model, put in some parameters, add the physics of flying balls, solve the differential equations, and compare the results to the performance of actual fielders, and you have explained it in terms of physics.
The ball-catching example is interesting, as it’s another control problem [...]
That’s why I chose it.
How would Jeffreyssai analyse a PID loop?
I decline to speculate on the internal workings of someone who is (1) fictional and (2) much cleverer, or at least better-trained in relevant areas, than me. But a generic Bayesian rationalist might say something like this:
“My goal is to have my beliefs track the range of possible futures. The mathematics of probabilistic inference founded by Bayes, Lagrange, Jaynes, etc., tells me that if I do this then the dynamics of my belief-updating must satisfy certain equations. Unfortunately, my brain is not fast enough, precise enough, or reliable enough to do that in real time, so I’d better look for some more tractable approximation that will produce results similar enough to those of updating according to the equations. Hmm, let’s see … scribble scribble … hack hack … scribble hack think scribble think scribble … OK, it turns out that in this special case, I can do well enough by just keeping track of the expected value minus half the standard deviation, and (provided things change in roughly the way I expect them to) that quantity satisfies this nice simple differential equation, which I can approximate with a finite-difference equation; so, simplifying a bit, it turns out that I can do a decent job by updating my estimate like so. [At which point he has written down the guts of a PID controller.] Unfortunately, that only gives me a point estimate; fortunately, if all goes according to plan my optimal posteriors are all of roughly the same shape and if I really have to I can get a decent approximation to the other parameter I need by doing this… [He writes down the coefficients for another PID controller.] I’ve had to make some assumptions that amount to having a prior with zero probabilities all over the place, which is ugly. Perhaps there’s some quantity I can keep track of that will stay close to zero as long as my model holds, but that has no reason to do so if the model’s wrong. … scribble scribble think scribble hack … Well, it’s not great, but if I also compute this and this, then while my underlying assumptions hold they should be very close to equal, so a lower bound on Pr(the model is broken) is such-and-such, so let’s watch that as well.”
Of course all the actual analysis is missing here. That would be because “a PID loop” can describe a vast range of different systems. And I’m assuming that our hypothetical rationalist knows enough about the relevant domain to be able to do the analysis, because otherwise your question seems a bit like asking how Jeffreyssai would do the biological research to know that he could take Pr(evolution) to be very close to 1. (Answer: He doesn’t need to; other people have already done it.)
(I have the feeling that one of us is missing the other’s point.)
RichardKennaway, very interesting post. I actually specialized in control theory in graduate school, but didn’t finish the program. I must object to what you’ve said here, in that control theory most certainly does make extensive use of Bayesian inferenence, under the name of the Kalman filter.
The Kalman filter is a way of estimating the paramaters of a system, given your observations and your knowledge of the system’s dynamics. While it may not help you pick a good control input algorithm, and while the problems you listed there may not need such accurate estimation of the data, it is an integral part of finding out how much the system deviates from where you want it to be, and is used extensively in controls.
I like this perspective.
This (from Richard’s post) seems to me very much parallel to this (which I just made up):
Anyone who says anything close to “Cox’s theorem; therefore you must make your decisions by making Bayesian calculations” is broken. But it could still be reasonable to say “However you make your decisions, the results should be as close as you can make them to those of an ideal reasoner doing Bayesian calculations on the information you have”. I don’t see any contradiction, or even any tension, here. As for an actual specific explanation that matches the facts, that would seem to need to be done afresh for every control system that works; for some cases (like our brains) the answers might be unmanageably complicated.
In the same sense as differential equations have something to do with the mechanism of people catching balls: when brains function well at maintaining reasonable beliefs, on some level of abstraction they have to act at least a little bit like Bayesian systems. But there needn’t be anything in the mechanisms that resembles the form (as opposed to the output) of the idealizations.
Since we might be able to do that by building a very big very low-level model of an entire human brain, without any understanding at all of what’s going on, obviously in some sense the answer is no. But if you want to understand what you’re doing—well, how much physics do you need to know if you want to get a space probe to Neptune? My guess is that even if you do it by making something that you launch into space at random and that then goes looking heuristically for something that might be Neptune, the chances are you’re going to want quite a lot of physics while you’re designing it.
The ball-catching example is interesting, as it’s another control problem, and has been studied as such. The fielder must get to where the ball will land. The predictive method would be to look at the ball, estimate its trajectory, then go to where you predict it will come down. This will not be very effective, because you cannot estimate the trajectory well enough. Instead, one method that will work is to move so as to maintain the direction from yourself to the ball constant in both azimuth and elevation. This is a control task, akin to the cursor-tracking task I discussed in the posting. You just have to move faster or slower and vary your direction, in whatever way will keep the direction constant. (The reason this works is that if the direction is constant, the ball is moving directly towards you in the frame of reference that moves with you. Or directly away, but in that case you won’t be able to run fast enough to catch it.)
Devise such a control model, put in some parameters, add the physics of flying balls, solve the differential equations, and compare the results to the performance of actual fielders, and you have explained it in terms of physics.
How would Jeffreyssai analyse a PID loop?
That’s why I chose it.
I decline to speculate on the internal workings of someone who is (1) fictional and (2) much cleverer, or at least better-trained in relevant areas, than me. But a generic Bayesian rationalist might say something like this:
“My goal is to have my beliefs track the range of possible futures. The mathematics of probabilistic inference founded by Bayes, Lagrange, Jaynes, etc., tells me that if I do this then the dynamics of my belief-updating must satisfy certain equations. Unfortunately, my brain is not fast enough, precise enough, or reliable enough to do that in real time, so I’d better look for some more tractable approximation that will produce results similar enough to those of updating according to the equations. Hmm, let’s see … scribble scribble … hack hack … scribble hack think scribble think scribble … OK, it turns out that in this special case, I can do well enough by just keeping track of the expected value minus half the standard deviation, and (provided things change in roughly the way I expect them to) that quantity satisfies this nice simple differential equation, which I can approximate with a finite-difference equation; so, simplifying a bit, it turns out that I can do a decent job by updating my estimate like so. [At which point he has written down the guts of a PID controller.] Unfortunately, that only gives me a point estimate; fortunately, if all goes according to plan my optimal posteriors are all of roughly the same shape and if I really have to I can get a decent approximation to the other parameter I need by doing this… [He writes down the coefficients for another PID controller.] I’ve had to make some assumptions that amount to having a prior with zero probabilities all over the place, which is ugly. Perhaps there’s some quantity I can keep track of that will stay close to zero as long as my model holds, but that has no reason to do so if the model’s wrong. … scribble scribble think scribble hack … Well, it’s not great, but if I also compute this and this, then while my underlying assumptions hold they should be very close to equal, so a lower bound on Pr(the model is broken) is such-and-such, so let’s watch that as well.”
Of course all the actual analysis is missing here. That would be because “a PID loop” can describe a vast range of different systems. And I’m assuming that our hypothetical rationalist knows enough about the relevant domain to be able to do the analysis, because otherwise your question seems a bit like asking how Jeffreyssai would do the biological research to know that he could take Pr(evolution) to be very close to 1. (Answer: He doesn’t need to; other people have already done it.)
(I have the feeling that one of us is missing the other’s point.)
RichardKennaway, very interesting post. I actually specialized in control theory in graduate school, but didn’t finish the program. I must object to what you’ve said here, in that control theory most certainly does make extensive use of Bayesian inferenence, under the name of the Kalman filter.
The Kalman filter is a way of estimating the paramaters of a system, given your observations and your knowledge of the system’s dynamics. While it may not help you pick a good control input algorithm, and while the problems you listed there may not need such accurate estimation of the data, it is an integral part of finding out how much the system deviates from where you want it to be, and is used extensively in controls.