Comment author:Matt_Simpson
21 February 2013 06:48:37AM
*
3 points
[-]

Nope. This was a good point by Jaynes. The truth may not exist in your hypothesis space. It may be (and often is) something you haven't conceived of.

Yes, the implicit assumption here is that the model is true.

Low likelihood of data under a hypothesis in no way implies rejection of that hypothesis.

\6. Therefore the alternative hypothesis is true.

Without also calculating the likelihood under the alternative hypothesis (it may be less), this is unjustified as well.

I don't think you understood my point. I'm avoiding claiming any inductive theory is correct - including Bayes' - and trying to show how hypothesis testing may be a way to do induction while simultaneously being agnostic about the correct theory. That Bayesian theory rejects certain steps of the hypothesis testing process is irrelevant to my point (and if you read closely, you'll see that I acknowledge it anyway).

Comment author:Matt_Simpson
21 February 2013 07:45:28AM
4 points
[-]

I think that's a bad assumption, and if you're trying to steelman, you should avoid relying on bad assumptions.

In any given problem the model is almost certainly false, but whether you use frequentist or Bayesian inference you have to implicitly assume that it's (approximately) true in order to actually conduct inference. Saying "don't assume the model is true because it isn't" is unhelpful and a nonstarter. If you actually want to get an answer, you have to assume something even if you know it isn't quite right.

Going from 4 to 5 looks dependent on an inductive theory to me.

Why yes it does. Did you read what I wrote about that?

Saying "don't assume the model is true because it isn't" is unhelpful and a nonstarter.

It starts fine for me.

Testing just the Null hypothesis is the least one can do. Then one can test the alternative, That way you at least get a likelihood ratio. You can add priors or not. Then one can build in terms modeling your ignorance.

Testing just the Null hypothesis is the least one can do. Then one can test the alternative, That way you at least get a likelihood ratio. You can add priors or not. Then one can build in terms modeling your ignorance.

This doesn't address the problem that the truth isn't in your hypothesis space (which is what I thought you were criticizing me for). If your model assumes constant variance, for example, when in truth there's nonconstant variance, the truth is outside your hypothesis space. You're not even considering it as a possibility. What does considering likelihood ratios of the hypotheses in your hypothesis space do to help you out here?

Reading that thread, I think jsteinart is right - if the truth is outside of your hypothesis space, you're screwed no matter if you're a Bayesian or a frequentist (which is a much more succinct way of putting my response to you). Setting up a "everything else" hypothesis doesn't really help because you can't compute a likelihood without some assumptions that, in all probability, expose you to the problem you're trying to avoid.

Yes. It conflicted with what you subsequently wrote:

Are you happier if I say that Bayes is a "thick" inductive theory and that NHST can be viewed as induction with a "thin" theory which therefore keeps you from committing yourself to as much? (I do acknowledge that others treat NHST as a "thick" theory and that this difference seems like it should result in differences in the details of actually doing hypothesis tests.)

What does considering likelihood ratios of the hypotheses in your hypothesis space do to help you out here?

The likelihood ratio was for comparing the hypotheses under consideration, the Null and the alternative. My point is that the likelihood of the alternative isn't taken into consideration at all. Prior to anything Bayesian, hypothesis testing moved from only modeling the likelihood of the null to also modeling the likelihood of a specified alternative, and comparing the two.

if the truth is outside of your hypothesis space, you're screwed no matter if you're a Bayesian or a frequentist

Therefore, you put an error placeholder of appropriate magnitude onto "it's out of my hypothesis space" so that unreasonable results have some systematic check.

And the difference between Bayesian and NHST isn't primarily how many assumptions you've committed too, which is enormous, but how many of those assumptions you've identified, and how you've specified them.

Comment author:Viliam_Bur
21 February 2013 08:23:17AM
1 point
[-]

Going from 4 to 5 seems to me like silently changing "if A then B" to "if B then A". Which is a logical mistake that many people do.

More precisely, it is a silent change from "if NULL, then DATA with very low proability" to "if DATA, then NULL with very low probability".

Specific example: Imagine a box containing 1 green circle, 10 red circles, and 100 red squares; you choose a random item. It is true that "if you choose a red item, it is unlikely to be a circle". But it is not true that "if you choose a circle, it is unlikely to be red".

## Comments (37)

Best*7 points [-]Nope. This was a good point by Jaynes. The truth may not exist in your hypothesis space. It may be (and often is) something you haven't conceived of.

Low likelihood of data under a hypothesis in no way implies rejection of that hypothesis.

Without also calculating the likelihood under the alternative hypothesis (it may be less), this is unjustified as well.

*3 points [-]Yes, the implicit assumption here is that the model is true.

I don't think you understood my point. I'm avoiding claiming any inductive theory is correct - including Bayes' - and trying to show how hypothesis testing may be a way to do induction while simultaneously being agnostic about the correct theory. That Bayesian theory rejects certain steps of the hypothesis testing process is irrelevant to my point (and if you read closely, you'll see that I acknowledge it anyway).

I think that's a bad assumption, and if you're trying to steelman, you should avoid relying on bad assumptions.

Going from 4 to 5 looks dependent on an inductive theory to me.

In any given problem the model is almost certainly false, but whether you use frequentist or Bayesian inference you have to implicitly assume that it's (approximately) true in order to actually conduct inference. Saying "don't assume the model is true because it isn't" is unhelpful and a nonstarter. If you actually want to get an answer, you have to assume something even if you know it isn't quite right.

Why yes it does. Did you read what I wrote about that?

It starts fine for me.

Testing just the Null hypothesis is the least one can do. Then one can test the alternative, That way you at least get a likelihood ratio. You can add priors or not. Then one can build in terms modeling your ignorance.

See previous comment: http://lesswrong.com/lw/gqt/the_logic_of_the_hypothesis_test_a_steel_man/8ioc

One could keep going and going on modeling ignorance, but few even get that far, and I suspect it isn't helpful to go further.

Yes. It conflicted with what you subsequently wrote:

This doesn't address the problem that the truth isn't in your hypothesis space (which is what I thought you were criticizing me for). If your model assumes constant variance, for example, when in truth there's nonconstant variance, the truth is outside your hypothesis space. You're not even considering it as a possibility. What does considering likelihood ratios of the hypotheses in your hypothesis space do to help you out here?

Reading that thread, I think jsteinart is right - if the truth is outside of your hypothesis space, you're screwed no matter if you're a Bayesian or a frequentist (which is a much more succinct way of putting my response to you). Setting up a "everything else" hypothesis doesn't really help because you can't compute a likelihood without some assumptions that, in all probability, expose you to the problem you're trying to avoid.

Are you happier if I say that Bayes is a "thick" inductive theory and that NHST can be viewed as induction with a "thin" theory which therefore keeps you from committing yourself to as much? (I do acknowledge that others treat NHST as a "thick" theory and that this difference seems like it should result in differences in the details of actually doing hypothesis tests.)

The likelihood ratio was for comparing the hypotheses under consideration, the Null and the alternative. My point is that the likelihood of the alternative isn't taken into consideration at all. Prior to anything Bayesian, hypothesis testing moved from only modeling the likelihood of the null to also modeling the likelihood of a specified alternative, and comparing the two.

Therefore, you put an error placeholder of appropriate magnitude onto "it's out of my hypothesis space" so that unreasonable results have some systematic check.

And the difference between Bayesian and NHST isn't primarily how many assumptions you've committed too, which is enormous, but how many of those assumptions you've identified, and how you've specified them.

Going from 4 to 5 seems to me like silently changing "if A then B" to "if B then A". Which is a logical mistake that many people do.

More precisely, it is a silent change from "if NULL, then DATA with very low proability" to "if DATA, then NULL with very low probability".

Specific example: Imagine a box containing 1 green circle, 10 red circles, and 100 red squares; you choose a random item. It is true that "if you choose a red item, it is unlikely to be a circle". But it is not true that "if you choose a circle, it is unlikely to be red".