3 ms·

So you're basically just Monday morning quarterbacking here. It's kind of telling you didn't understand Wasserman's reply to you, it was perfectly clear to any

So you're basically just Monday morning quarterbacking here. It's kind of telling you didn't understand Wasserman's reply to you, it was perfectly clear to any practitioner. Here's an example of a real-world problem I had to solve in my last job. We wanted to compare two treatments on an object, OLD and NEW, and evaluate which was better. We got a bunch of volunteers and showed them each a random sample of objects, the OLD treatment, and the NEW treatment (suitably blinded of course) and asked them to rate each on a scale of 1-5. The goal was to determine if the OLD treatment should be replaced by the NEW treatment. What's your Bayesian approach to solving this? I was the closest thing to a domain expert, and I certainly didn't have any prior beliefs about what the ratings would be (except for a weak expectation that OLD>NEW). Tell me the tests you would have run, and then I'll tell you the non-Bayesian thing I did that gave a very solid answer and took well under a second of computation.

loup-vaillant 13y ago
> It's kind of telling you didn't understand Wasserman's reply to you, it was perfectly clear to any practitioner. Condescension doesn't impress me. Explanations do. Could you please explain his "perfectly clear" reply to me? > What's your Bayesian approach to solving this? Applying probability theory as best I could. If it turns out to be too complicated, or computationally intractable, I'll resort to approximations. For this particular case… Well, first, we don't have 2 treatments, we have as many treatments as we have objects. It's like flipping a coin. When you flip a coin in the obvious way, you have no idea which side will eventually come up, because even if you know it came up heads the previous time, and you tried to perform the same move again: in fact, you don't perform the same move, and you don't know how exactly your second move differed from the first. So first, there is an unpredictable variability in the way the treatment will be performed. This is the first source of uncertainty. Second, the objects aren't all the same. I expect them to be more or less clean to begin with, and other characteristics such as sharp angles may more or less hinder the treatment. This is the second source of uncertainty. Once you perform a treatment on an object, it had a definite effect. Unfortunately, it is hard to define how well it actually went. Maybe you can come up with a definite criterion, but apparently, since you needed to ask a sample of human volunteers, that criterion is either not well understood, or hard to measure properly. Anyway, you have access to an uncertain measure, which in this case is a human giving you a scale from 1 to 5. This is the third and fourth sources of uncertainty (akin to the first and second one respectively: variability in how a given human will assess an object; and variability across humans). From what I have understood of what you told me, there were 2 sets of objects, and 2 groups of people. One group looked at every object before any treatment. Then you applied OLD to half objects, and NEW to half the other. You then showed the newly cleaned up objects to the second group. Of course, you don't tell the group which objects they get to examine. An alternative would be to photograph every objects before and after the treatment, then show both sets of photos to everyone, and ask them to rank the objects (both before and after), and the perceived efficiency of the treatment. I don't like it much however, because photos add an additional source of uncertainty. This looks like a complicated version of a compound estimation problem, described in chapter 6 of Jayne's PT:TLOS (elementary parameter estimation). Pretty basic stuff. We're not finished yet. You wanted to know which treatment should be chosen: OLD or NEW. You basically have 3 alternatives: always apply OLD, always apply NEW, or look at the object then decide which treatment you apply. For each alternative, you should have a probability distribution over the "cleanness" distribution of the objects. Decide a utility function for the cleanness distribution, and chose the method that maximizes expected utility. And now I'm stuck because I don't know probability theory deeply enough to actually give you a non-ambiguous procedure in less than a couple days. Not to mention that I don't have your prior information about OLD and NEW (why you expect OLD to be better etc.) Anyway, I believe a specialist would only need a couple minutes, or a couple hours tops. (To find the procedure, that is. Actually running the numbers in a computer may or may not be expensive. I personally have no idea.)
- Jach 13y ago
  I've only read the first two chapters of Jaynes' book, and bits and pieces of various other books. (One of the late chapters in http://uncertainty.stat.cmu.edu/ http://uncertainty.stat.cmu.edu/ is a great reference for outlining the problems with significance and hypothesis testing.) Anyway, to me the problem described above sounds like it can generally be phrased as the common problem of ranking user-rated products on an e-commerce page. (A good Bayesian write-up is here, with simple code at the end: http://masanjin.net/blog/how-to-rank-products-based-on-user-input http://masanjin.net/blog/how-to-rank-products-based-on-user-...) At least if one assumes that treatments are applied more-or-less the same, and sets of objects amongst a treatment are more-or-less the same. There's also the question of how rankings change when both treatments are presented to a volunteer instead of just one. I interpreted the problem as just showing the OLD sets or just showing the NEW sets, never showing the original sets pre-treatment. How does sanskritabelt solve the problem, I wonder? Set an arbitrary number of volunteers for each group to say "this is the point we can stop collecting data", assume normalcy of ratings, take the means and use some test to see if the difference is significant or not? Really the problem of the problem is defining what is meant by 'better'. Is 'better' a higher score? (Which scoring function do you use -- mean, median, most 5-stars, least 1-stars, something else that takes into account the number of ratings per rating bucket?) Does it have to be higher by a certain amount (how meaningful are deltas in your scoring function)? Once it's defined what is meant by "A is better than B", then you can go about the business of computing the likelihood of your data given "A is better than B" to fulfill the RHS of Bayes' theorem.
  - sanskritabelt 13y ago
    In my career, and maybe this is just the kinds of problems I've been presented with, I've found that assuming normality is, by and large, like drinking tequila. It's fun at the time, but I end up regretting it in the morning. EDIT: its this kind of experience that make me reach for things like nonparametric tests and the bootstrap.
- sanskritabelt 13y ago
  Maybe I was unclear. To be explicit, the experimental setup was: * Draw a sample of objects (in this case, an 'object' was a set of text documents) * Apply the OLD and NEW treatments to each sampled object * Show the results of OLD and NEW to volunteers (double blind, of course), get an evaluation. And so the data I get is: (object 1, SCORE OLD 1, SCORE NEW 1) (object 2, SCORE OLD 2, SCORE NEW 2) etc So again: * I don't have any prior knowledge of how the scores are going to be distributed * The primary reason I expected OLD to be 'better' than NEW is because OLD had theory behind it, and NEW was an ad hoc tweak of a previously rejected treatment * The very initial text corpus was fixed * The objects were selected from the initial corpus through a multistep process, at least one of those steps was a stochastic optimization of an objective function * One of the treatments was also obtained through a constrained optimization * The initial text corpus wasn't all that large, and the samples were pretty small So what I did was look at all observations for which OLD SCORE was different than NEW SCORE. Say there were n of them, and say that for p of those OLD<NEW. If NEW wasn't producing results that scored better than OLD, we would expect p/n to be less than 0.5, up to some uncertainty from the random sample. This is just a straight up test of a binomial parameter, so I computed confidence intervals and found that 0.5 was several standard deviations below the observed p/n, from which I concluded that NEW was better than OLD and we should switch. (as described here: http://en.wikipedia.org/wiki/Sign_test http://en.wikipedia.org/wiki/Sign_test) In a more responsible analysis I could have done something more sophisticated and modeled the scores as a sum of object effect, tester effect, treatment effect, fixed term, noise, etc. Given the magnitude of the sign test score, I didn't feel that was necessary.

loup-vaillant 13y ago

> It's kind of telling you didn't understand Wasserman's reply to you, it was perfectly clear to any practitioner. Condescension doesn't impress me. Explanations do. Could you please explain his "perfectly clear" reply to me? > What's your Bayesian approach to solving this? Applying probability theory as best I could. If it turns out to be too complicated, or computationally intractable, I'll resort to approximations. For this particular case… Well, first, we don't have 2 treatments, we have as many treatments as we have objects. It's like flipping a coin. When you flip a coin in the obvious way, you have no idea which side will eventually come up, because even if you know it came up heads the previous time, and you tried to perform the same move again: in fact, you don't perform the same move, and you don't know how exactly your second move differed from the first. So first, there is an unpredictable variability in the way the treatment will be performed. This is the first source of uncertainty. Second, the objects aren't all the same. I expect them to be more or less clean to begin with, and other characteristics such as sharp angles may more or less hinder the treatment. This is the second source of uncertainty. Once you perform a treatment on an object, it had a definite effect. Unfortunately, it is hard to define how well it actually went. Maybe you can come up with a definite criterion, but apparently, since you needed to ask a sample of human volunteers, that criterion is either not well understood, or hard to measure properly. Anyway, you have access to an uncertain measure, which in this case is a human giving you a scale from 1 to 5. This is the third and fourth sources of uncertainty (akin to the first and second one respectively: variability in how a given human will assess an object; and variability across humans). From what I have understood of what you told me, there were 2 sets of objects, and 2 groups of people. One group looked at every object before any treatment. Then you applied OLD to half objects, and NEW to half the other. You then showed the newly cleaned up objects to the second group. Of course, you don't tell the group which objects they get to examine. An alternative would be to photograph every objects before and after the treatment, then show both sets of photos to everyone, and ask them to rank the objects (both before and after), and the perceived efficiency of the treatment. I don't like it much however, because photos add an additional source of uncertainty. This looks like a complicated version of a compound estimation problem, described in chapter 6 of Jayne's PT:TLOS (elementary parameter estimation). Pretty basic stuff. We're not finished yet. You wanted to know which treatment should be chosen: OLD or NEW. You basically have 3 alternatives: always apply OLD, always apply NEW, or look at the object then decide which treatment you apply. For each alternative, you should have a probability distribution over the "cleanness" distribution of the objects. Decide a utility function for the cleanness distribution, and chose the method that maximizes expected utility. And now I'm stuck because I don't know probability theory deeply enough to actually give you a non-ambiguous procedure in less than a couple days. Not to mention that I don't have your prior information about OLD and NEW (why you expect OLD to be better etc.) Anyway, I believe a specialist would only need a couple minutes, or a couple hours tops. (To find the procedure, that is. Actually running the numbers in a computer may or may not be expensive. I personally have no idea.)

Jach 13y ago
I've only read the first two chapters of Jaynes' book, and bits and pieces of various other books. (One of the late chapters in http://uncertainty.stat.cmu.edu/ http://uncertainty.stat.cmu.edu/ is a great reference for outlining the problems with significance and hypothesis testing.) Anyway, to me the problem described above sounds like it can generally be phrased as the common problem of ranking user-rated products on an e-commerce page. (A good Bayesian write-up is here, with simple code at the end: http://masanjin.net/blog/how-to-rank-products-based-on-user-input http://masanjin.net/blog/how-to-rank-products-based-on-user-...) At least if one assumes that treatments are applied more-or-less the same, and sets of objects amongst a treatment are more-or-less the same. There's also the question of how rankings change when both treatments are presented to a volunteer instead of just one. I interpreted the problem as just showing the OLD sets or just showing the NEW sets, never showing the original sets pre-treatment. How does sanskritabelt solve the problem, I wonder? Set an arbitrary number of volunteers for each group to say "this is the point we can stop collecting data", assume normalcy of ratings, take the means and use some test to see if the difference is significant or not? Really the problem of the problem is defining what is meant by 'better'. Is 'better' a higher score? (Which scoring function do you use -- mean, median, most 5-stars, least 1-stars, something else that takes into account the number of ratings per rating bucket?) Does it have to be higher by a certain amount (how meaningful are deltas in your scoring function)? Once it's defined what is meant by "A is better than B", then you can go about the business of computing the likelihood of your data given "A is better than B" to fulfill the RHS of Bayes' theorem.
- sanskritabelt 13y ago
  In my career, and maybe this is just the kinds of problems I've been presented with, I've found that assuming normality is, by and large, like drinking tequila. It's fun at the time, but I end up regretting it in the morning. EDIT: its this kind of experience that make me reach for things like nonparametric tests and the bootstrap.
sanskritabelt 13y ago
Maybe I was unclear. To be explicit, the experimental setup was: * Draw a sample of objects (in this case, an 'object' was a set of text documents) * Apply the OLD and NEW treatments to each sampled object * Show the results of OLD and NEW to volunteers (double blind, of course), get an evaluation. And so the data I get is: (object 1, SCORE OLD 1, SCORE NEW 1) (object 2, SCORE OLD 2, SCORE NEW 2) etc So again: * I don't have any prior knowledge of how the scores are going to be distributed * The primary reason I expected OLD to be 'better' than NEW is because OLD had theory behind it, and NEW was an ad hoc tweak of a previously rejected treatment * The very initial text corpus was fixed * The objects were selected from the initial corpus through a multistep process, at least one of those steps was a stochastic optimization of an objective function * One of the treatments was also obtained through a constrained optimization * The initial text corpus wasn't all that large, and the samples were pretty small So what I did was look at all observations for which OLD SCORE was different than NEW SCORE. Say there were n of them, and say that for p of those OLD<NEW. If NEW wasn't producing results that scored better than OLD, we would expect p/n to be less than 0.5, up to some uncertainty from the random sample. This is just a straight up test of a binomial parameter, so I computed confidence intervals and found that 0.5 was several standard deviations below the observed p/n, from which I concluded that NEW was better than OLD and we should switch. (as described here: http://en.wikipedia.org/wiki/Sign_test http://en.wikipedia.org/wiki/Sign_test) In a more responsible analysis I could have done something more sophisticated and modeled the scores as a sum of object effect, tester effect, treatment effect, fixed term, noise, etc. Given the magnitude of the sign test score, I didn't feel that was necessary.