5 ms·

Here is one paper: http://arxiv.org/pdf/1207.7214v2.pdf http://arxiv.org/pdf/1207.7214v2.pdf Look at Figures 8 and 9. They show the p-value (at whatever level

Here is one paper: http://arxiv.org/pdf/1207.7214v2.pdf http://arxiv.org/pdf/1207.7214v2.pdf Look at Figures 8 and 9. They show the p-value (at whatever level of data was collected when the paper was written) over the parameter space that is being searched. You can see that the values observed have a clear separation -- most are close to 1 (null-hypothesis holds) with just one significant dip towards 0 (null-hypothesis doesn't hold). If you were to animate this graph with the p-values over time (as more observations are made), you would see the trend towards 0 or 1 much clearer. The Boson experimenters would expect a (1 - p) reproduction rate for the next observation made (if the null-hypothesis holds). That is, the next observation has a p probability of fitting within the parameters of the null-hypothesis and (1-p) probability that it is inconsistent with the null-hypothesis. Why would they expect that? Because the math involved in telling you whether or not that is what you should expect is exactly what p-value calculates (again, assuming a well-formed experiement -- which the Higgs experiments probably are). But again, when the null-hypothesis doesn't hold, p-value tells you very little (it's actually undefined in the math).

kgwgk 11y ago
> But again, when the null-hypothesis doesn't hold, p-value tells you very little (it's actually undefined in the math). The p-value is well defined whether the null hypothesis holds or not. You calculate it assuming it does. There you go, you have a properly calculated p-value. That's what physicists do: "Taking into account the entire mass range of the search, 110– 600 GeV, the global significance of the excess is 5.1 σ, which corresponds to p0 = 1.7 × 10−7." You see, they have calculated a p-value. Does the null hypothesis hold? I don't think they had any expectations consistent with the null hypothesis being true before the experiment. After the experiment they clearly think that the null hypothesis is false: "These results provide conclusive evidence for the discovery of a new particle with mass 126.0 ± 0.4 (stat) ± 0.4 (sys) GeV." They don't see any problem in stating a p-value and rejecting the null hypothesis at the same time (in fact, it's because the p-value that they calculated is very small that they conclude that the null hypothesis doesn't hold). Apparently you see a problem, because if the Higgs boson exists and produces the signal in the experiment then the null hypothesis is false and all the p-value calculations they did to get to that conclusion are "wrong". Anyway, I have no need to convince you of anything. I can live with people being wrong on the internet.
- jsprogrammer 11y ago
  Well, before you go, I implore you to look into the actual computation and theory of 'p-value'. A p-value is simply P(X|H). P(X|H) only means something when H is true. If H is false, P(X|H) tells you nothing. Since H is your null-hypothesis, if it does not actually hold in the real-world, P(X|H) is meaningless. If you read the paper I linked, they never explicitly call out the null hypothesis (nor do, I believe, they show the work for their calculations). There should be another paper somewhere that describes exactly what it is, in the terms I am using. So, phrases like, "[t]hey don't see any problem in stating a p-value and rejecting the null hypothesis" make me think you have no idea what you're talking about. The null hypothesis can never be 'rejected' (ie. p-value can never reach 0). I don't think you will find anyone working on the Higgs boson that will claim otherwise.
  - kgwgk 11y ago
    I think we agree that their null hypothesis is "there is a background, with events coming from all the known particles". I think we agree that their conclusion is "these results provide conclusive evidence for the discovery of a new particle". I don't see how can they say that there is a new particle without rejecting the hypothesis that there is no such new particle. Of course you can say that the null hypothesis can never be rejected (relevant Dilbert strip: http://dilbert.com/strip/2001-10-25 http://dilbert.com/strip/2001-10-25) but then they can never discover a new particle either. Regarding p-values in general, your definition is the same I've been using all along. But I don't think it is meaningless when the null hypothesis does not hold. The meaning is clear: "the probability of getting a value for the statistic as high as the observed one if the null hypothesis was true". For example, there would be one chance in several millions of observing the kind of data they found at the LHC if the Higgs boson didn't exist. You might want to look into the theory yourself, because the notion of p-values trending towards 1 if the null hypothesis is true is nonsense. By definition, if the null hypothesis is true the p-value is uniformly distributed between 0 and 1. If you have at some point a p-value close to one (or to any other number for that matter) and keep adding data, in the long run it will still be uniformly distributed between 0 and 1.
    - 11y ago

kgwgk 11y ago

> But again, when the null-hypothesis doesn't hold, p-value tells you very little (it's actually undefined in the math). The p-value is well defined whether the null hypothesis holds or not. You calculate it assuming it does. There you go, you have a properly calculated p-value. That's what physicists do: "Taking into account the entire mass range of the search, 110– 600 GeV, the global significance of the excess is 5.1 σ, which corresponds to p0 = 1.7 × 10−7." You see, they have calculated a p-value. Does the null hypothesis hold? I don't think they had any expectations consistent with the null hypothesis being true before the experiment. After the experiment they clearly think that the null hypothesis is false: "These results provide conclusive evidence for the discovery of a new particle with mass 126.0 ± 0.4 (stat) ± 0.4 (sys) GeV." They don't see any problem in stating a p-value and rejecting the null hypothesis at the same time (in fact, it's because the p-value that they calculated is very small that they conclude that the null hypothesis doesn't hold). Apparently you see a problem, because if the Higgs boson exists and produces the signal in the experiment then the null hypothesis is false and all the p-value calculations they did to get to that conclusion are "wrong". Anyway, I have no need to convince you of anything. I can live with people being wrong on the internet.

jsprogrammer 11y ago
Well, before you go, I implore you to look into the actual computation and theory of 'p-value'. A p-value is simply P(X|H). P(X|H) only means something when H is true. If H is false, P(X|H) tells you nothing. Since H is your null-hypothesis, if it does not actually hold in the real-world, P(X|H) is meaningless. If you read the paper I linked, they never explicitly call out the null hypothesis (nor do, I believe, they show the work for their calculations). There should be another paper somewhere that describes exactly what it is, in the terms I am using. So, phrases like, "[t]hey don't see any problem in stating a p-value and rejecting the null hypothesis" make me think you have no idea what you're talking about. The null hypothesis can never be 'rejected' (ie. p-value can never reach 0). I don't think you will find anyone working on the Higgs boson that will claim otherwise.
- kgwgk 11y ago
  I think we agree that their null hypothesis is "there is a background, with events coming from all the known particles". I think we agree that their conclusion is "these results provide conclusive evidence for the discovery of a new particle". I don't see how can they say that there is a new particle without rejecting the hypothesis that there is no such new particle. Of course you can say that the null hypothesis can never be rejected (relevant Dilbert strip: http://dilbert.com/strip/2001-10-25 http://dilbert.com/strip/2001-10-25) but then they can never discover a new particle either. Regarding p-values in general, your definition is the same I've been using all along. But I don't think it is meaningless when the null hypothesis does not hold. The meaning is clear: "the probability of getting a value for the statistic as high as the observed one if the null hypothesis was true". For example, there would be one chance in several millions of observing the kind of data they found at the LHC if the Higgs boson didn't exist. You might want to look into the theory yourself, because the notion of p-values trending towards 1 if the null hypothesis is true is nonsense. By definition, if the null hypothesis is true the p-value is uniformly distributed between 0 and 1. If you have at some point a p-value close to one (or to any other number for that matter) and keep adding data, in the long run it will still be uniformly distributed between 0 and 1.
  - 11y ago