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Eliezer_Yudkowsky comments on Why (and why not) Bayesian Updating? - Less Wrong
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Comments (26)
1) Because over a lifetime of this, we would rapidly fall off the precision that can be directly represented by analogue neurons. This takes floating-point math with large exponents.
2) Because it is convenient to cache some mental quantities around the standard probability - e.g., having emotional strength go as the absolute probability avoids the need to do lots of comparisons each time.
3) Because evolution isn't that clever, of course!
4) With Bayesian updating, you only have to update the beliefs that are correlated with the evidence you observe. Beliefs that are independent of the evidence can stay constant. Trying to "save" on the division means you have to update just about every belief upon every new observation, which ends up being much more costly.
Oh well, it was fun trying to imagine being a mind whose beliefs only get weaker with time. :)
What does "correlated" mean when talking about alternative kinds of updating?
ETA: And how would you know whether the belief and the evidence are correlated without performing the updating to check?
A and B are correlated if P(A ∩ B) != P(A) * P(B).
The idea is that you'd represent the prior using a data structure which allows you to easily determine which beliefs are correlated with a given evidence. I'm not an expert here, but I think this is what Bayesian networks are all about.
Careful, though. That's the definition of when there's mutual information, and the term "correlated" can also be used to mean a "linear statistical correlation", which is not the same thing.
And before you roll your eyes, note that this entire LW article is based on equivocating between the two meanings! (See my comment. )
No. The data in the scatter-plot in that article contains no mutual information between the variables A and B, not merely zero product-moment correlation. I linked there to the data that are plotted; anyone is welcome to have a go at finding mutual information in them.
I challenge anyone to analyse these data and demonstrate substantial mutual information between A and B. If the data are insufficient for your favorite method of analysis, I can generate arbitrarily large quantities of it, and if I were using a quantum RNG instead of a PRNG, there would be absolutely no way to determine any connection between the two variables.
Despite that, there is one. It only shows up when the process from which these data are taken is sampled on a sufficiently short timescale, as in the other data file I linked to in that post.
Correct me if I'm wrong, but would the actual measure of the connection between A and B be more accurately summarized as K(A + B) < K(A) + K(B), then?
I believe that's an equivalent way to express "H(X) - H(X|Y) > 0" and "P(A ∩ B) != P(A) * P(B)". Or at least, any one of the three can be derived from any of the others.
Note that the Kullback-Leibler divergence (a generalization of entropy) between X and Y is equivalent to the number of extra bits required to code data sampled from X when your compression algorithm is optimized for Y, which shows how these all relate.
If you separate out the variables into simultaneous pairs, then yes, you've destroyed the mutual information.
But if someone is allowed to look at the timewise development of each variable, they would see the mutual information, which necessarily results from one causing the other! If A causes B, then, by knowing A, you require less data to describe B (than if you did not know or could not reference A). That's the very definition of mutual information.
You can't just say that because the simultaneous pairs are uncorrelated, there is no mutual information between the variables. You showed as much when you later demonstrated that the simultaneous pairs between a function and its derivative are uncorrelated. But who denies that learning a function tells you something about its derivative? (which would mean there's mutual information between the two...)
You're heading towards redefining correlation to mean causal connection.
When people actually do causal analysis (example, example, example) they perform specific calculations to detect various relationships among the variables. There are many different calculations they may do, which is the point of the first of those references, but we are not talking -- at least, I'm not -- about uncomputable Kolmogorov-based concepts (and even by that standard, the first data file, were it to use a QRNG, contains no mutual information). The moral that I was drawing was a practical one: certain very simple relationships between physical variables can generate time series containing no mutual information detectable by any of these methods. This suggests a substantial limitation of their practical applicability.
Specifics, please. Given the actual dynamical process generating those data (which is that B is the derivative of A, and A is a smoothly varying random variable), show me a mathematical definition of the mutual information between A and B, and a method of calculating it.
Nope. I'm pointing out that "correlated" can mean "there exists a linear statistical correlation" or "there exists mutual information" -- but whichever you use, you need to be consistent. And at no point did I say it meant causal connection -- I just noted that that's one way mutual information can develop.
What you showed is that there is more than one way for two variables to be mutually informative, and if you limit yourself to a linear statistical regression on the simultaneous pairs, you might not find the mutual information. So what? If you know more than just the unordered simultaneous pairs, use that knowledge!
Sure. Let's use your point about derivatives. I tell you sin(x) = 4/5. Have I told you something about cos(x)? (And no it doen't matter that the cosine can have two values; you've still learned something.)
I tell you f(x) = sin(x) + cos(x). Have I told you something about f ' (x)?
Yes.
Yes.
But in real experiments, you're not given the underlying function, only observations of some of its values.
So, I tell you a time series for an unknown function f.
What have I told you about f'? What further information would you need to make a numerical calculation of the amount of information you now have about f'?
In the data file I originally linked to, there is not merely no linear relationship, but virtually no relationship whatsoever, discoverable by any means whatever, between the two columns, which tabulate f and f' for a certain stochastic function f. Mutual information, even in Kolmogorov heaven, is not present.
Mutual information is the difference of marginal and conditional entropy (eq 4 of this): I(X,Y) = H(X) - H(X|Y)
Suppose X is a deterministic function of Y (e.g., Y is a function sampled from a stochastic process and X is its derivative). Then P(X|Y) is a degenerate distribution and the conditional entropy H(X|Y) is 0. Hence Y is maximally informative about X.
I think the words Richard used in his question denoted the mutual information between the functions A and B, but I think he meant to ask about the mutual information between two time series datasets sampled from A and B over the same interval.
No need to bring up causality. It's enough that knowledge of A specifies B too.
Yes, that's correct. I only mentioned causality to make my comment relevant to the context Kennaway brought up.
Considering all the different combinations of things you might condition on, the task does not sound trivial.