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.
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.