You need much more than limiting behavior to say anything about whether or not the processes are ‘similar’ in a useful way before that.
Perhaps the synthesis here is that while looking at asymptotic behaviour of a simpler system can be supremely useful, we should be surprised that it works so well. To rely on this technique in a new domain we should, every time, demonstrate that it actually works in practice.
Also, it’s interesting that many of these examples do have ‘pathological cases’ where the limit doesn’t match practice. And this isn’t necessarily restricted to toy domains or weird setups: for example, the most asymptotically efficient matrix multiplication algorithms are impractical (although in fairness that’s the most compelling example on that page).
In general I very much appreciate people reasoning from examples like these. The sarcasm does make me less motivated to engage with this thoroughly, though. Anyway, idk how to come up with general rules for which abstractions are useful and which aren’t. Seems very hard. But when we have no abstractions which are empirically verified to work well in modelling a phenomenon (like intelligence), it’s easy to overestimate how relevant our best mathematics is, because proofs are the only things that look like concrete progress.
On Big-O analysis in particular: this is a pretty interesting example actually, since I don’t think it was obvious in advance that it’d work as well as it has (i.e. that the constants would be fairly unimportant in practice). Need to think more about this one.
But when we have no empirical verifications for abstractions which work well to model a phenomenon (like intelligence)...
We have tons of empirical evidence on this. We may not have a fully general model of intelligence yet, but we don’t have a fully general model of physics yet either. We do know of some reasonably general approaches which work well to model intelligence in practice in a very wide variety of situations, and Bayesianism is one of those.
The lack of a fully-general model of all of physics is not a very good argument against quantum mechanics. Likewise, the lack of a fully-general model of all of intelligence is not a very good argument against Bayesianism. In particular, we expect new theories of physics to “add up to normality”—they should reduce to the old theory in places where the old theory worked. The same applies to models of intelligence: they should reduce to Bayesianism in places where Bayesianism works. Like quantum mechanics, that’s an awful lot of places.
All of the applications in which Bayesian statistics/ML methods work so well. All of the psychology/neuroscience research on human intelligence approximating Bayesianism. All the robotics/AI/control theory applications where Bayesian methods are used in practice.
All of the applications in which Bayesian statistics/ML methods work so well. All the robotics/AI/control theory applications where Bayesian methods are used in practice.
This does not really seem like much evidence to me, because for most of these cases non-bayesian methods work much better. I confess I personally am in the “throw a massive neural network at it” camp of machine learning; and certainly if something with so little theoretical validation works so well, it makes one question whether the sort of success you cite really tells us much about bayesianism in general.
All of the psychology/neuroscience research on human intelligence approximating Bayesianism.
I’m less familiar with this literature. Surely human intelligence *as a whole* is not a very good approximation to bayesianism (whatever that means). And it seems like most of the heuristics and biases literature is specifically about how we don’t update very rationally. But at a lower level, I defer to your claim that modules in our brain approximate bayesianism.
Then I guess the question is how to interpret this. It certainly feels like a point in favour of some interpretation of bayesianism as a general framework. But insofar as you’re thinking about an interpretation which is being supported by empirical evidence, it seems important for someone to formulate it in such a way that it could be falsified. I claim that the way bayesianism has been presented around here (as an ideal of rationality) is not a falsifiable framework, and so at the very least we need someone else to make the case for what they’re standing for.
Problem is, “throw a massive neural network at it” fails completely for the vast majority of practical applications. We need astronomical amounts of data to make neural networks work. Try using them at a small company on a problem with a few thousand data points; it won’t work.
I see the moral of that story as: if you have enough data, any stupid algorithm will work. It’s when data is not superabundant that we need Bayesian methods, because nothing else reliably works. (BTW, this is something we could guess based on Bayesian foundations: Cox’ Theorem or an information-theoretic foundation of Bayesian probability do not depend on infinite data for any particular problem, whereas things like frequentist statistics or brute-force neural nets do.)
(Side note for people confused about how that plays with the comment at the top of this thread: the relevant limit there was not the limit of infinite data, but the limit of reasoning over all possible models.)
I claim that the way bayesianism has been presented around here (as an ideal of rationality) is not a falsifiable framework, and so at the very least we need someone else to make the case for what they’re standing for.
Around here, rationality is about winning. To the extent that we consider Bayesianism an ideal of rationality, that can be falsified by outperforming Bayesianism, in places where behavior of that ideal can be calculated or at least characterized enough to prove that something else outperforms the supposed ideal.
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Perhaps the synthesis here is that while looking at asymptotic behaviour of a simpler system can be supremely useful, we should be surprised that it works so well. To rely on this technique in a new domain we should, every time, demonstrate that it actually works in practice.
Also, it’s interesting that many of these examples do have ‘pathological cases’ where the limit doesn’t match practice. And this isn’t necessarily restricted to toy domains or weird setups: for example, the most asymptotically efficient matrix multiplication algorithms are impractical (although in fairness that’s the most compelling example on that page).
Most of the arguments in the OP apply just as well to all those other limit use-cases as well.
In general I very much appreciate people reasoning from examples like these. The sarcasm does make me less motivated to engage with this thoroughly, though. Anyway, idk how to come up with general rules for which abstractions are useful and which aren’t. Seems very hard. But when we have no abstractions which are empirically verified to work well in modelling a phenomenon (like intelligence), it’s easy to overestimate how relevant our best mathematics is, because proofs are the only things that look like concrete progress.
On Big-O analysis in particular: this is a pretty interesting example actually, since I don’t think it was obvious in advance that it’d work as well as it has (i.e. that the constants would be fairly unimportant in practice). Need to think more about this one.
We have tons of empirical evidence on this. We may not have a fully general model of intelligence yet, but we don’t have a fully general model of physics yet either. We do know of some reasonably general approaches which work well to model intelligence in practice in a very wide variety of situations, and Bayesianism is one of those.
The lack of a fully-general model of all of physics is not a very good argument against quantum mechanics. Likewise, the lack of a fully-general model of all of intelligence is not a very good argument against Bayesianism. In particular, we expect new theories of physics to “add up to normality”—they should reduce to the old theory in places where the old theory worked. The same applies to models of intelligence: they should reduce to Bayesianism in places where Bayesianism works. Like quantum mechanics, that’s an awful lot of places.
What sort of evidence are you referring to; can you list a few examples?
All of the applications in which Bayesian statistics/ML methods work so well. All of the psychology/neuroscience research on human intelligence approximating Bayesianism. All the robotics/AI/control theory applications where Bayesian methods are used in practice.
This does not really seem like much evidence to me, because for most of these cases non-bayesian methods work much better. I confess I personally am in the “throw a massive neural network at it” camp of machine learning; and certainly if something with so little theoretical validation works so well, it makes one question whether the sort of success you cite really tells us much about bayesianism in general.
I’m less familiar with this literature. Surely human intelligence *as a whole* is not a very good approximation to bayesianism (whatever that means). And it seems like most of the heuristics and biases literature is specifically about how we don’t update very rationally. But at a lower level, I defer to your claim that modules in our brain approximate bayesianism.
Then I guess the question is how to interpret this. It certainly feels like a point in favour of some interpretation of bayesianism as a general framework. But insofar as you’re thinking about an interpretation which is being supported by empirical evidence, it seems important for someone to formulate it in such a way that it could be falsified. I claim that the way bayesianism has been presented around here (as an ideal of rationality) is not a falsifiable framework, and so at the very least we need someone else to make the case for what they’re standing for.
Problem is, “throw a massive neural network at it” fails completely for the vast majority of practical applications. We need astronomical amounts of data to make neural networks work. Try using them at a small company on a problem with a few thousand data points; it won’t work.
I see the moral of that story as: if you have enough data, any stupid algorithm will work. It’s when data is not superabundant that we need Bayesian methods, because nothing else reliably works. (BTW, this is something we could guess based on Bayesian foundations: Cox’ Theorem or an information-theoretic foundation of Bayesian probability do not depend on infinite data for any particular problem, whereas things like frequentist statistics or brute-force neural nets do.)
(Side note for people confused about how that plays with the comment at the top of this thread: the relevant limit there was not the limit of infinite data, but the limit of reasoning over all possible models.)
Around here, rationality is about winning. To the extent that we consider Bayesianism an ideal of rationality, that can be falsified by outperforming Bayesianism, in places where behavior of that ideal can be calculated or at least characterized enough to prove that something else outperforms the supposed ideal.