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No, Chomsky's problem with statistical models is that you can't learn much about the underlying system from them, and so it isn't science, which has as a fundam

No, Chomsky's problem with statistical models is that you can't learn much about the underlying system from them, and so it isn't science, which has as a fundamental aim to understand the world. Statistical models can be useful but they generally aren't meaningful to scientific progress.

dragonwriter 10y ago
> No, Chomsky's problem with statistical models is that you can't learn much about the underlying system from them, and so it isn't science, which has as a fundamental aim to understand the world. Science has the fundamental aim to make testable predictive models of the world; any "understanding" other than that represented by a testable predictive model is irrelevant to science, except insofar as it might provide intuition on which to found hypotheses of better predictive models. Statistical models, to the extent that they are testable and predictive, are exactly the kind of thing that science is about.
- foobarqux 10y ago
  Putting aside whether how we want to define "science" there are models which attempt to represent the underlying system (e.g. generative grammars) and other "models" which merely try to predict the input/output behavior of those systems (e.g. neural nets). The latter typically doesn't lend any understanding of the underlying system (e.g. we don't understand much about language from a deep neural net translation system), which isn't surprising because they are designed for prediction not for representing the internal behavior of the system. To use one of Chomsky's examples, if you built a deep neural net that could predict the weather with 100% accuracy then you have performed an amazing feat of engineering that is incredibly useful to the world. But you haven't necessarily learned anything about the weather.
  - dragonwriter 10y ago
    The problem there seems to be not one with statistical models, but with the fact that extracting a parisomonious mathematical model from a neural net is nontrivial (presumably, you can extract a mathematical model from the internal weights, etc., but for even known simple relationships neural net models produced without an a priori prediction of the general shape of the relationship tend to be very complex.) Of course, were the model really 100% accurate, then converting it to a the most parsimonious perfectly accurate model would be simply a matter of reduction; the real problem is that real neural net models are not 100% accurate, and are often far more complex than more accurate models, and often aren't convenient to analyze to deduce the more-accurate and simpler model.
    - arcanus 10y ago
      > extracting a parisomonious mathematical model from a neural net is nontrivial Has this ever been done? It strikes me as impossible. > Of course, were the model really 100% accurate, then converting it to a the most parsimonious perfectly accurate model would be simple a matter of reduction There are entire sub-fields of physics focused on reduced order models and phenomenalogical prediction. It is not 'simply a matter of reduction', it is non-trivial and the resulting models are almost always highly imperfect.
      - dragonwriter 10y ago
        > Has this ever been done? Probably not. > It strikes me as impossible. "Impossible" is probably, in principle, not quite right, but when I said "nontrivial", I really did mean "almost certainly impractical in virtually all real cases". > There are entire sub-fields of physics focused on reduced order models and phenomenalogical prediction. It is not 'simply a matter of reduction', it is non-trivial and the resulting models are almost always highly imperfect. "Simply" there meant that it was just reduction, not that reduction is necessarily simple. The point was that in real-world cases, reduction to an equivalent model isn't the only problem.
    - foobarqux 10y ago
      The problem is not one of minimality, but of meaning and exposition. If you model a system using the smallest possible mathematical model you don't, from that act alone, understand how the system works.
mrow84 10y ago
For the purposes of this discussion, Norvig has defined a statistical model as: "a mathematical model which is modified or trained by the input of data points." He then illustrates how what would be considered a "scientific" model, Newton's law of gravitation, is a statistical model under his definition, but simple one with not many parameters. He contrasts this with a Markov model of a communication channel with a large vocabulary, which has many parameters. His argument is, then, that Chomsky dislikes statistical models with large numbers of parameters, as stated in the passage I quoted before. My point was that Chomsky's concerns are, with reference to Norvig's argument, equivalent to concerns about model fitting, namely that to fit models with many more parameters than you have data, you require additional assumptions about your model structure. It is difficult (though not necessarily impossible) to learn about the system from your model, because in order to construct your model you have had to assume things about reality that you will not be verifying against observations. In the opposite case, where you have many more data than parameters, you can fit your model with confidence, given only assumptions about your sampling (which you address by being a good experimentalist). This is what allows you to "learn about the underlying system" - you have a model that describes reality well by itself, without requiring additional assumptions about the nature of reality, so the structure of your model reflects something about the structure of reality, and you can explore your model as though you were exploring reality. Of course, sometimes it turns out the equivalence wasn't as good as we thought, but often it provides us with new directions of investigation. Hopefully that clarifies the equivalence between the two statements - I apologise for not making it more obvious earlier.
- xixi77 10y ago
  It's more than just overfitting: e.g. if a neural net model with several hundreds or thousands of parameters can be trained on a dataset of billions of observations, and may even be OK at predicting behavior both in and out of sample, but it still remains a black box, since we generally do not know what phenomena particular parameters (or their combinations) represent. On the other hand, IMO there is nothing wrong or unscientific with having empirically estimated relationships as part of the model -- I just see them as shortcuts whose purpose is to parcel the problem so as to allow other analysis, and as something to potentially investigate further to see why the relationship takes a particular form. Some ML methods are more amenable to this type of analysis than others though.
Barrin92 10y ago
I get Chomsky's point about the importance of fundamental models and understanding in principle, but I'm confused why it should apply to machine learning. The problem with Chomsky's knowledge based approach is that it seems to be domain specific. You can learn the fundamentals of music but you can't apply it to physics. You can learn the fundamental mechanics of weather but you can't apply it to psychology etc.. General learning though requires exactly that and there the statistical model seems to be much better because it doesn't run into the domain problem.
- foldr 10y ago
  Chomsky's not interested in solving engineering problems, so that's not really a problem from his point of view.
mcguire 10y ago
Well, no, Chomsky's problem with statistical models is that they are based on data. Chomsky is a Grand System Builder in the style of the medieval and renaissance thinkers. Data is, at best, irrelevant and, at worst, a confusing distraction that gets in the way of his program, which is elaborating on his system, basing it entirely on intuition. One advantage of such grand systems is that they can be made to look like the mathematical model, based on "intuitively obvious" axioms and built by thinking deeply in the comfort of one's armchair.
- foldr 10y ago
  It's sad to see HN descend to this level. You clearly aren't very familiar with anything that Chomsky's written. Maybe just refrain from commenting instead of making up nonsense?
  - mcguire 10y ago
    Excellent argument.
    - foldr 10y ago
      Your post says lots of untrue things about Chomsky without any references, so what kind of response did you expect?

dragonwriter 10y ago

> No, Chomsky's problem with statistical models is that you can't learn much about the underlying system from them, and so it isn't science, which has as a fundamental aim to understand the world. Science has the fundamental aim to make testable predictive models of the world; any "understanding" other than that represented by a testable predictive model is irrelevant to science, except insofar as it might provide intuition on which to found hypotheses of better predictive models. Statistical models, to the extent that they are testable and predictive, are exactly the kind of thing that science is about.

foobarqux 10y ago
Putting aside whether how we want to define "science" there are models which attempt to represent the underlying system (e.g. generative grammars) and other "models" which merely try to predict the input/output behavior of those systems (e.g. neural nets). The latter typically doesn't lend any understanding of the underlying system (e.g. we don't understand much about language from a deep neural net translation system), which isn't surprising because they are designed for prediction not for representing the internal behavior of the system. To use one of Chomsky's examples, if you built a deep neural net that could predict the weather with 100% accuracy then you have performed an amazing feat of engineering that is incredibly useful to the world. But you haven't necessarily learned anything about the weather.
- dragonwriter 10y ago
  The problem there seems to be not one with statistical models, but with the fact that extracting a parisomonious mathematical model from a neural net is nontrivial (presumably, you can extract a mathematical model from the internal weights, etc., but for even known simple relationships neural net models produced without an a priori prediction of the general shape of the relationship tend to be very complex.) Of course, were the model really 100% accurate, then converting it to a the most parsimonious perfectly accurate model would be simply a matter of reduction; the real problem is that real neural net models are not 100% accurate, and are often far more complex than more accurate models, and often aren't convenient to analyze to deduce the more-accurate and simpler model.
  - arcanus 10y ago
    > extracting a parisomonious mathematical model from a neural net is nontrivial Has this ever been done? It strikes me as impossible. > Of course, were the model really 100% accurate, then converting it to a the most parsimonious perfectly accurate model would be simple a matter of reduction There are entire sub-fields of physics focused on reduced order models and phenomenalogical prediction. It is not 'simply a matter of reduction', it is non-trivial and the resulting models are almost always highly imperfect.
    - dragonwriter 10y ago
      > Has this ever been done? Probably not. > It strikes me as impossible. "Impossible" is probably, in principle, not quite right, but when I said "nontrivial", I really did mean "almost certainly impractical in virtually all real cases". > There are entire sub-fields of physics focused on reduced order models and phenomenalogical prediction. It is not 'simply a matter of reduction', it is non-trivial and the resulting models are almost always highly imperfect. "Simply" there meant that it was just reduction, not that reduction is necessarily simple. The point was that in real-world cases, reduction to an equivalent model isn't the only problem.
  - foobarqux 10y ago
    The problem is not one of minimality, but of meaning and exposition. If you model a system using the smallest possible mathematical model you don't, from that act alone, understand how the system works.

mrow84 10y ago

For the purposes of this discussion, Norvig has defined a statistical model as: "a mathematical model which is modified or trained by the input of data points." He then illustrates how what would be considered a "scientific" model, Newton's law of gravitation, is a statistical model under his definition, but simple one with not many parameters. He contrasts this with a Markov model of a communication channel with a large vocabulary, which has many parameters. His argument is, then, that Chomsky dislikes statistical models with large numbers of parameters, as stated in the passage I quoted before. My point was that Chomsky's concerns are, with reference to Norvig's argument, equivalent to concerns about model fitting, namely that to fit models with many more parameters than you have data, you require additional assumptions about your model structure. It is difficult (though not necessarily impossible) to learn about the system from your model, because in order to construct your model you have had to assume things about reality that you will not be verifying against observations. In the opposite case, where you have many more data than parameters, you can fit your model with confidence, given only assumptions about your sampling (which you address by being a good experimentalist). This is what allows you to "learn about the underlying system" - you have a model that describes reality well by itself, without requiring additional assumptions about the nature of reality, so the structure of your model reflects something about the structure of reality, and you can explore your model as though you were exploring reality. Of course, sometimes it turns out the equivalence wasn't as good as we thought, but often it provides us with new directions of investigation. Hopefully that clarifies the equivalence between the two statements - I apologise for not making it more obvious earlier.

xixi77 10y ago
It's more than just overfitting: e.g. if a neural net model with several hundreds or thousands of parameters can be trained on a dataset of billions of observations, and may even be OK at predicting behavior both in and out of sample, but it still remains a black box, since we generally do not know what phenomena particular parameters (or their combinations) represent. On the other hand, IMO there is nothing wrong or unscientific with having empirically estimated relationships as part of the model -- I just see them as shortcuts whose purpose is to parcel the problem so as to allow other analysis, and as something to potentially investigate further to see why the relationship takes a particular form. Some ML methods are more amenable to this type of analysis than others though.

Barrin92 10y ago

I get Chomsky's point about the importance of fundamental models and understanding in principle, but I'm confused why it should apply to machine learning. The problem with Chomsky's knowledge based approach is that it seems to be domain specific. You can learn the fundamentals of music but you can't apply it to physics. You can learn the fundamental mechanics of weather but you can't apply it to psychology etc.. General learning though requires exactly that and there the statistical model seems to be much better because it doesn't run into the domain problem.

foldr 10y ago
Chomsky's not interested in solving engineering problems, so that's not really a problem from his point of view.

mcguire 10y ago

Well, no, Chomsky's problem with statistical models is that they are based on data. Chomsky is a Grand System Builder in the style of the medieval and renaissance thinkers. Data is, at best, irrelevant and, at worst, a confusing distraction that gets in the way of his program, which is elaborating on his system, basing it entirely on intuition. One advantage of such grand systems is that they can be made to look like the mathematical model, based on "intuitively obvious" axioms and built by thinking deeply in the comfort of one's armchair.

foldr 10y ago
It's sad to see HN descend to this level. You clearly aren't very familiar with anything that Chomsky's written. Maybe just refrain from commenting instead of making up nonsense?
- mcguire 10y ago
  Excellent argument.
  - foldr 10y ago
    Your post says lots of untrue things about Chomsky without any references, so what kind of response did you expect?