Comment author:Rune
26 July 2009 05:55:03PM
10 points
[-]

Can you give a detailed numerical examples of some problem where the Bayesian and Frequentist give different answers, and you feel strongly that the Frequentist's answer is better somehow?

I think you've tried to do that, but I don't fully understand most of your examples. Perhaps if you used numbers and equations, that would help a lot of people understand your point. Maybe expand on your "And here's an ultra-short example of what frequentists can do" idea?

A Bayesian will have a probability distribution over possible outcomes, some of which give her lower scores than her probabilistic expectation of average score, and some of which give her higher scores than this expectation.

I am unable to parse your above claim, and ask for specific math on a specific example. If you know your score will be lower than you expect, you should lower your expectation. If you know something will happen less often than the probability you assign, you should assign a lower probability. This sounds like an inconsistent epistemic state for a Bayesian to be in.

Comment author:Cyan
29 July 2009 02:32:24AM
*
2 points
[-]

I spent some time looking up papers, trying to find accessible ones. The main paper that kicked off the matching prior program is Welch and Peers, 1963, but you need access to JSTOR.

The best I can offer is the following example. I am estimating a large number of positive estimands. I have one noisy observation for each one; the noise is Gaussian with standard deviation equal to one. I have no information relating the estimands; per Jaynes, I give them independent priors, resulting in independent posteriors*. I do not have information justifying a proper prior. Let's say I use a flat prior over the positive real line. No matter the true value of each estimand, the sampling probability of the event "my posterior 90% quantile is greater than the estimand" is less than 0.9 (see Figure 6 of this working paper by D.A.S. Fraser). So the more estimands I analyze, the more sure I am that the intervals from 0 to my posterior 90% quantiles will contain less than 90% of the estimands.

I don't know if there's an exact matching prior in this problem, but I suspect it lacks the correct structure.

* This is a place I think Jaynes goes wrong: the quantities are best modeled as exchangeable, not independent. Equivalently, I put them in a hierarchical model. But this only kicks the problem of priors guaranteeing calibration up a level.

I'm sorry, but the level of frequentist gibberish in this paper is larger than I would really like to work through.

If you could be so kind, please state:

What the Bayesian is using as a prior and likelihood function;

and what distribution the paper assumes the actual parameters are being drawn from, and what the real causal process is governing the appearance of evidence.

If the two don't match, then of course the Bayesian posterior distributions, relative to the experimenter's higher knowledge, can appear poorly calibrated.

If the two do match, then the Bayesian should be well-calibrated. Sure looks QED-ish to me.

Comment author:Cyan
29 July 2009 05:08:56AM
*
6 points
[-]

The example doesn't come from the paper; I made it myself. You only need to believe the figure I cited -- don't bother with the rest of the paper.

Call the estimands mu_1 to mu_n; the data are x_1 to x_n. The prior over the mu parameters is flat in the positive subset of R^n, zero elsewhere. The sampling distribution for x_i is Normal(mu_i,1). I don't know the distribution the parameters actually follow. The causal process is irrelevant -- I'll stipulate that the sampling distribution is known exactly.

Call the 90% quantiles of my posterior distributions q_i. From the sampling perspective, these are random quantities, being monotonic functions of the data. Their sampling distributions satisfy the inequality Pr(q_i > mu_i | mu_i) < 0.9. (This is what the figure I cited shows.) As n goes to infinity, I become more and more sure that my posterior intervals of the form (0, q_i] are undercalibrated.

You might cite the improper prior as the source of the problem. However, if the parameter space were unrestricted and the prior flat over all of R^n, the posterior intervals would by correctly calibrated.

But it really is fair to demand a proper prior. How could we determine that prior? Only by Bayesian updating from some pre-prior state of information to the prior state of information (or equivalently, by logical deduction, provided that the knowledge we update on is certain). Right away we run into the problem that Bayesian updating does not have calibration guarantees in general (and for this, you really ought to read the literature), so it's likely that any proper prior we might justify does not have a calibration guarantee.

Comment author:wedrifid
27 July 2009 01:04:01PM
1 point
[-]

How about this: a Bayesian will always predict that she is perfectly calibrated, even though she knows the theorems proving she isn't.

Wanna bet? Literally. Have a Bayesian to make and a whole bunch of predictions and then offer her bets with payoffs based on what apparent calibration the results will reflect. See which bets she accepts and which she refuses.

Comment author:wedrifid
27 July 2009 03:23:27PM
0 points
[-]

Find a candidate.

I was about to suggest we could just bet raw ego points by publicly posting here... but then I realised I prove my point just by playing.

It should be obvious, by the way, that if the predictions you have me make pertain to black boxes that you construct then I would only bet if the odds gave a money pump. There are few cases in which I would expect my calibration to be superior to what you could predict with complete knowledge of the distribution.

Comment author:Cyan
27 July 2009 03:33:34PM
*
1 point
[-]

It should be obvious, by the way, that if the predictions you have me make pertain to black boxes that you construct then I would only bet if the odds gave a money pump.

Comment author:wedrifid
26 July 2009 09:22:19PM
6 points
[-]

I didn't like the "by Bayesian lights" phrase either. What I take as the relevant part of the question is this:

Can you provide an example of a frequentist concept that can be used to make predictions in the real world for which a bayesian prediction will fail?

"Bayesian answers don't give coverage guarantees" doesn't demonstrate anything by itself. The question is could the application of Bayes give a prediction equal to or superior to the prediction about the real world implicit in a coverage guarantee?

If you can provide such an example then you will have proved many people to be wrong in a significant, fundamental way. But I haven't seen anything in this thread or in either of Cyan's which fits that category.

Comment author:cousin_it
26 July 2009 09:32:16PM
*
2 points
[-]

Once again: the real-world performance (as opposed to internal coherence) of the Bayesian method on any given problem depends on the prior you choose for that problem. If you have a well-calibrated prior, Bayes gives well-calibrated results equal or superior to any frequentist methods. If you don't, science knows no general way to invent a prior that will reliably yield results superior to anything at all, not just frequentist methods. For example, Jaynes spent a large part of his life searching for a method to create uninformative priors with maxent, but maxent still doesn't guarantee you anything beyond "cross your fingers".

If your prior is screwed up enough, you'll also misunderstand the experimental setup and the likelihood ratios. Frequentism depends on prior knowledge just as much as Bayesianism, it just doesn't have a good formal way of treating it.

Comment author:cousin_it
27 July 2009 06:34:02AM
*
3 points
[-]

I give you some numbers taken from a normal distribution with unknown mean and variance. If you're a frequentist, your honest estimate of the mean will be the sample mean. If you're a Bayesian, it will be some number off to the side, depending on whatever bullshit prior you managed to glean from my words above - and you don't have the option of skipping that step, and don't have the option of devising a prior that will always exactly match the frequentist conclusion because math doesn't allow it in the general case . (I kinda equivocate on "honest estimate", but refusing to ever give point estimates doesn't speak well of a mathematician anyway.) So nah, Bayesianism depends on priors more, not "just as much".

If tomorrow Bayesians find a good formalization of "uninformative prior" and a general formula to devise them, you'll happily discard your old bullshit prior and go with the flow, thus admitting that your careful analysis of my words about "unknown normal distribution" today wasn't relevant at all. This is the most fishy part IMO.

(Disclaimer: I am not a crazy-convinced frequentist. I'm a newbie trying to get good answers out of Bayesians, and some of the answers already given in these threads satisfy me perfectly well.)

Comment author:Cyan
27 July 2009 06:57:19AM
9 points
[-]

The normal distribution with unknown mean and variance was a bad choice for this example. It's the one case where everyone agrees what the uninformative prior is. (It's flat with respect to the mean and the log-variance.) This uninformative prior is also a matching prior -- posterior intervals are confidence intervals.

Comment author:cousin_it
27 July 2009 07:27:33AM
*
2 points
[-]

I didn't know that was possible, thanks. (Wow, a prior with integral=infinity! One that can't be reached as a posterior after any observation! How'd a Bayesian come by that? But seems to work regardless.) What would be a better example?

ETA: I believe the point raised in that comment still deserves an answer from Bayesians.

Comment author:wedrifid
27 July 2009 12:48:28PM
*
1 point
[-]

I give you some numbers taken from a normal distribution with unknown mean and variance. If you're a frequentist, your honest estimate of the mean will be the sample mean. If you're a Bayesian, it will be some number off to the side, depending on whatever bullshit prior you managed to glean from my words above - and you don't have the option of skipping that step, and don't have the option of devising a prior that will always exactly match the frequentist conclusion because math doesn't allow it in the general case . (I kinda equivocate on "honest estimate", but refusing to ever give point estimates doesn't speak well of a mathematician anyway.) So nah, Bayesianism depends on priors more, not "just as much".

A Bayesian does not have the option of 'just skipping that step' and choosing to accept whichever prior was mandated by Fisher (or whichever other statistitian created or insisted upon the use of the particular tool in question). It does not follow that the Bayesian is relying on 'Bullshit' more than the frequentist. In fact, when I use the label 'bullshit' I usually mean 'the use of authority or social power mechanisms in lieu of or in direct defiance of reason'. I obviously apply 'bullshit prior' to the frequentist option in this case.

Comment author:cousin_it
27 July 2009 02:25:13PM
*
2 points
[-]

A Bayesian does not have the option of 'just skipping that step' and choosing to accept whichever prior was mandated by Fisher

Why in the world doesn't a Bayesian have that option? I thought you were a free people. :-) How'd you decide to reject those priors in favor of other ones, anyway? As far as I currently understand, there's no universally accepted mathematical way to pick the best prior for every given problem and no psychologically coherent way to pick it of your head either, because it ain't there. In addition to that, here's some anecdotal evidence: I never ever heard of a Bayesian agent accepting or rejecting a prior.

## Comments (155)

BestCan you give a detailed numerical examples of some problem where the Bayesian and Frequentist give different answers, and you feel strongly that the Frequentist's answer is better somehow?

I think you've tried to do that, but I don't fully understand most of your examples. Perhaps if you used numbers and equations, that would help a lot of people understand your point. Maybe expand on your "And here's an ultra-short example of what frequentists can do" idea?

*0 points [-]Short answer: Bayesian answers don't give coverage guarantees.

Long answer: see the comments to Cyan's post.

"Coverage guarantees" is a frequentist concept. Can you explain where Bayesians fail by Bayesian lights? In the real world, somewhere?

How about this: a Bayesian will always predict that she is perfectly calibrated, even though she knows the theorems proving she isn't.

A Bayesian will have a probability distribution over possible outcomes, some of which give her lower scores than her probabilistic expectation of average score, and some of which give her higher scores than this expectation.

I am unable to parse your above claim, and ask for specific math on a specific example. If you

knowyour score will be lower than you expect, you should lower your expectation. If you know something will happen less often than the probability you assign, you should assign a lower probability. This sounds like an inconsistent epistemic state for a Bayesian to be in.*2 points [-]I spent some time looking up papers, trying to find accessible ones. The main paper that kicked off the matching prior program is Welch and Peers, 1963, but you need access to JSTOR.

The best I can offer is the following example. I am estimating a large number of positive estimands. I have one noisy observation for each one; the noise is Gaussian with standard deviation equal to one. I have no information relating the estimands; per Jaynes, I give them independent priors, resulting in independent posteriors*. I do not have information justifying a proper prior. Let's say I use a flat prior over the positive real line. No matter the true value of each estimand, the sampling probability of the event "my posterior 90% quantile is greater than the estimand" is less than 0.9 (see Figure 6 of this working paper by D.A.S. Fraser). So the more estimands I analyze, the more sure I am that the intervals from 0 to my posterior 90% quantiles will contain less than 90% of the estimands.

I don't know if there's an exact matching prior in this problem, but I suspect it lacks the correct structure.

* This is a place I think Jaynes goes wrong: the quantities are best modeled as exchangeable, not independent. Equivalently, I put them in a hierarchical model. But this only kicks the problem of priors guaranteeing calibration up a level.

I'm sorry, but the level of frequentist gibberish in this paper is larger than I would really like to work through.

If you could be so kind, please state:

What the Bayesian is using as a prior and likelihood function;

and what distribution the paper assumes the actual parameters are being drawn from, and what the real causal process is governing the appearance of evidence.

If the two don't match, then of course the Bayesian posterior distributions, relative to the experimenter's higher knowledge, can appear poorly calibrated.

If the two do match, then the Bayesian should be well-calibrated. Sure looks QED-ish to me.

*6 points [-]The example doesn't come from the paper; I made it myself. You only need to believe the figure I cited -- don't bother with the rest of the paper.

Call the estimands mu_1 to mu_n; the data are x_1 to x_n. The prior over the mu parameters is flat in the positive subset of R^n, zero elsewhere. The sampling distribution for x_i is Normal(mu_i,1). I don't know the distribution the parameters actually follow. The causal process is irrelevant -- I'll stipulate that the sampling distribution is known exactly.

Call the 90% quantiles of my posterior distributions q_i. From the sampling perspective, these are random quantities, being monotonic functions of the data. Their sampling distributions satisfy the inequality Pr(q_i > mu_i | mu_i) < 0.9. (This is what the figure I cited shows.) As n goes to infinity, I become more and more sure that my posterior intervals of the form (0, q_i] are undercalibrated.

You might cite the improper prior as the source of the problem. However, if the parameter space were unrestricted and the prior flat over all of R^n, the posterior intervals would by correctly calibrated.

But it really is fair to demand a proper prior. How could we determine that prior? Only by Bayesian updating from some pre-prior state of information to the prior state of information (or equivalently, by logical deduction, provided that the knowledge we update on is certain). Right away we run into the problem that Bayesian updating does not have calibration guarantees in general (and for this, you really ought to read the literature), so it's likely that any proper prior we might justify does not have a calibration guarantee.

Wanna bet? Literally. Have a Bayesian to make and a whole bunch of predictions and then offer her bets with payoffs based on what apparent calibration the results will reflect. See which bets she accepts and which she refuses.

Are you volunteering?

Sure. :)

But let me warn you... I actually predict my calibration to be pretty darn awful.

We need a trusted third party.

Find a candidate.

I was about to suggest we could just bet raw ego points by publicly posting here... but then I realised I prove my point just by playing.

It should be obvious, by the way, that if the predictions you have me make pertain to black boxes that you construct then I would only bet if the odds gave a money pump. There are few cases in which I would expect my calibration to be superior to what you could predict with complete knowledge of the distribution.

*1 point [-]Phooey. There goes plan A.

*3 points [-]Of course not. If you choose to care only about the things Bayes can give you, it's a mathematical fact that you can't do better.

I didn't like the "by Bayesian lights" phrase either. What I take as the relevant part of the question is this:

Can you provide an example of a frequentist concept that can be used to make predictions in the real world for which a bayesian prediction will fail?

"Bayesian answers don't give coverage guarantees" doesn't demonstrate anything by itself. The question is

could the application of Bayes give a prediction equal to or superior to the prediction about the real world implicit in a coverage guarantee?If you can provide such an example then you will have proved many people to be wrong in a significant, fundamental way. But I haven't seen anything in this thread or in either of Cyan's which fits that category.

*2 points [-]Once again: the real-world performance (as opposed to internal coherence) of the Bayesian method on any given problem depends on the prior you choose for that problem. If you have a well-calibrated prior, Bayes gives well-calibrated results equal or superior to any frequentist methods. If you don't, science knows no general way to invent a prior that will reliably yield results superior to

anything at all, not just frequentist methods. For example, Jaynes spent a large part of his life searching for a method to create uninformative priors with maxent, but maxent still doesn't guarantee you anything beyond "cross your fingers".If your prior is screwed up enough, you'll also misunderstand the experimental setup and the likelihood ratios. Frequentism depends on prior knowledge just as much as Bayesianism, it just doesn't have a good formal way of treating it.

*3 points [-]I give you some numbers taken from a normal distribution with unknown mean and variance. If you're a frequentist, your honest estimate of the mean will be the sample mean. If you're a Bayesian, it will be some number off to the side, depending on whatever bullshit prior you managed to glean from my words above - and you don't have the option of skipping that step, and don't have the option of devising a prior that will always exactly match the frequentist conclusion because math doesn't allow it in the general case . (I kinda equivocate on "honest estimate", but refusing to ever give point estimates doesn't speak well of a mathematician anyway.) So nah, Bayesianism depends on priors

more, not "just as much".If tomorrow Bayesians find a good formalization of "uninformative prior" and a general formula to devise them, you'll happily discard your old bullshit prior and go with the flow, thus admitting that your careful analysis of my words about "unknown normal distribution" today wasn't relevant at all. This is the most fishy part IMO.

(Disclaimer: I am not a crazy-convinced frequentist. I'm a newbie trying to get good answers out of Bayesians, and some of the answers already given in these threads satisfy me perfectly well.)The normal distribution with unknown mean and variance was a bad choice for this example. It's the one case where everyone agrees what the uninformative prior is. (It's flat with respect to the mean and the log-variance.) This uninformative prior is also a matching prior -- posterior intervals are confidence intervals.

*2 points [-]I didn't know that was possible, thanks. (Wow, a prior with integral=infinity! One that can't be reached as a posterior after

anyobservation! How'd a Bayesian come bythat? But seems to work regardless.) What would be a better example?ETA: I believe the point raised in that comment still deserves an answer from Bayesians.

*1 point [-]A Bayesian does not have the option of 'just skipping that step' and choosing to accept whichever prior was mandated by Fisher (or whichever other statistitian created or insisted upon the use of the particular tool in question). It does not follow that the Bayesian is relying on 'Bullshit' more than the frequentist. In fact, when I use the label 'bullshit' I usually mean 'the use of authority or social power mechanisms in lieu of or in direct defiance of reason'. I obviously apply 'bullshit prior' to the frequentist option in this case.

*2 points [-]Why in the world doesn't a Bayesian have that option? I thought you were a free people. :-) How'd you decide to reject those priors in favor of other ones, anyway? As far as I currently understand, there's no universally accepted mathematical way to pick the best prior for every given problem and no psychologically coherent way to pick it of your head either, because it ain't there. In addition to that, here's some anecdotal evidence: I never ever heard of a Bayesian agent accepting or rejecting a prior.

Vocabulary nitpick: I believe you wrote "in luew of" in lieu of "in lieu of".

Sorry, couldn't help it. IAWYC, anyhow.