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>A frequentist point estimate converges to truth (with P = 1) as N -> infty. Since we are never at infinity, our observations are always finite, all this is sa

>A frequentist point estimate converges to truth (with P = 1) as N -> infty. Since we are never at infinity, our observations are always finite, all this is saying is "we always deal with approximations of reality," i.e., statistics is never, in fact, about the "actual truth", it is about testing the set of our limited observations of reality against a model.

yummyfajitas 10y ago
This is like saying real analysis is not concerned with irrational numbers since they can only be constructed as limits of sequences of rational ones.
- astazangasta 10y ago
  No, it's not. The fact that observation is a limit on our apprehension of the real world means the "truth" is fundamentally inaccessible, and is a useless concept to the statistician. All you can hope to do is test your model against the observation, you'll never, ever get access to the truth. As a real-world example that I deal with every day, we frequently model the expected distribution of gene expression as a negative binomial. It doesn't matter how closely we observe the actual distribution of gene expression, it is never going to perfectly fit a negative binomial (it won't even get close), even though this is what we test our observation against, because the 'truth' is something different and far too complex for us to apprehend.
  - yummyfajitas 10y ago
    Irrational numbers are also "fundamentally inaccessible" in the exact same (infinite limit) sense that statistical convergence is. It's not a useless concept at all, it's actually the fundamental concept. What your real world example describes is something different entirely. That's just pragmatically choosing the wrong model due to computational or human tractability. That's not fundamental to statistics, that's just a cheat you made because it's good enough. Statistics is about acknowledging that cheat and quantifying how much it hurts you; fundamentally such a thing is not possible if truth doesn't exist.
    - astazangasta 10y ago
      Irrational numbers can be represented exactly; integrals allow us to perform exact calculation using them. In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent.
      - yummyfajitas 10y ago
        Integrals themselves are, except in very special cases (piecewise functions with rational values), only definable as limits - specifically the limit of the Riemann sum. (You can also use measure theory, but measures themselves are only definable on sigma algebras, which in the non-finite case are also not explicitly constructable.) In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent. I don't quite understand. You are arguing that statistics doesn't care about truth simply because some biologists are using a model they know to be wrong? That doesn't even make sense. In applied math in general (which includes but is not necessarily limited to statistics), the following equation holds: error = |true model - actual approximated model| We can use the triangle inequality to show: error <= |true model - best model in class X| + |best model in class X - actual approximated model| Presumably you all have decided that |true - best| is adequately small via scientific investigation. Or maybe not, maybe your workplace just doesn't care, I don't really know. Various mathematical techniques, or increasing sample size in a statistical scenario, can be used to reduce |best - actual|. Due to the triangle inequality, this brings you closer to truth. Statistics is also concerned with expanding class X in such a way as to more easily reduce the model error. I really feel like I'm missing something, because I truly can't comprehend what you are trying to argue.
        astazangasta 10y ago
        What I'm arguing is we have no idea what "true model" is, what we have is "presumed model" and "observation". In the example I gave, we can never know the true source of the data we have observed (biology), we can only test our observations against some constructed model. Biologists are using a model they know to be wrong because that is what all models are - we know them to be wrong, we just can't do otherwise, because the truth is not available to us. I feel like I've made this same point about four times already, so if you aren't getting it, let's just stop here.

yummyfajitas 10y ago

This is like saying real analysis is not concerned with irrational numbers since they can only be constructed as limits of sequences of rational ones.

astazangasta 10y ago
No, it's not. The fact that observation is a limit on our apprehension of the real world means the "truth" is fundamentally inaccessible, and is a useless concept to the statistician. All you can hope to do is test your model against the observation, you'll never, ever get access to the truth. As a real-world example that I deal with every day, we frequently model the expected distribution of gene expression as a negative binomial. It doesn't matter how closely we observe the actual distribution of gene expression, it is never going to perfectly fit a negative binomial (it won't even get close), even though this is what we test our observation against, because the 'truth' is something different and far too complex for us to apprehend.
- yummyfajitas 10y ago
  Irrational numbers are also "fundamentally inaccessible" in the exact same (infinite limit) sense that statistical convergence is. It's not a useless concept at all, it's actually the fundamental concept. What your real world example describes is something different entirely. That's just pragmatically choosing the wrong model due to computational or human tractability. That's not fundamental to statistics, that's just a cheat you made because it's good enough. Statistics is about acknowledging that cheat and quantifying how much it hurts you; fundamentally such a thing is not possible if truth doesn't exist.
  - astazangasta 10y ago
    Irrational numbers can be represented exactly; integrals allow us to perform exact calculation using them. In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent.
    - yummyfajitas 10y ago
      Integrals themselves are, except in very special cases (piecewise functions with rational values), only definable as limits - specifically the limit of the Riemann sum. (You can also use measure theory, but measures themselves are only definable on sigma algebras, which in the non-finite case are also not explicitly constructable.) In what comparable way does truth (e.g. the true process underlying gene expression) have a role in statistics? We neither measure the truth nor model it; it is absent. I don't quite understand. You are arguing that statistics doesn't care about truth simply because some biologists are using a model they know to be wrong? That doesn't even make sense. In applied math in general (which includes but is not necessarily limited to statistics), the following equation holds: error = |true model - actual approximated model| We can use the triangle inequality to show: error <= |true model - best model in class X| + |best model in class X - actual approximated model| Presumably you all have decided that |true - best| is adequately small via scientific investigation. Or maybe not, maybe your workplace just doesn't care, I don't really know. Various mathematical techniques, or increasing sample size in a statistical scenario, can be used to reduce |best - actual|. Due to the triangle inequality, this brings you closer to truth. Statistics is also concerned with expanding class X in such a way as to more easily reduce the model error. I really feel like I'm missing something, because I truly can't comprehend what you are trying to argue.
      - astazangasta 10y ago
        What I'm arguing is we have no idea what "true model" is, what we have is "presumed model" and "observation". In the example I gave, we can never know the true source of the data we have observed (biology), we can only test our observations against some constructed model. Biologists are using a model they know to be wrong because that is what all models are - we know them to be wrong, we just can't do otherwise, because the truth is not available to us. I feel like I've made this same point about four times already, so if you aren't getting it, let's just stop here.