# Higgs Boson and statistical issues

## Higgs Boson and statistical issues

Καταγράφω μια ανταλλαγή email που έγινε στη λίστα του ISBA σχετική με το Higgs Boson

[Από Tony O'Hagan - πολύ γνωστό Bayesian]

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of

answers.

We've heard a lot about the Higgs boson. The news reports say that the LHC

needed convincing evidence before they would announce that a particle had

been found that looks like (in the sense of having some of the right

characteristics of) the elusive Higgs boson. Specifically, the news referred

to a confidence interval with 5-sigma limits.

Now this appears to correspond to a frequentist significance test with an

extreme significance level. Five standard deviations, assuming normality,

means a p-value of around 0.0000005. A number of questions spring to mind.

1. Why such an extreme evidence requirement? We know from a Bayesian

perspective that this only makes sense if (a) the existence of the Higgs

boson (or some other particle sharing some of its properties) has extremely

small prior probability and/or (b) the consequences of erroneously announcing

its discovery are dire in the extreme. Neither seems to be the case, so why

5-sigma?

2. Rather than ad hoc justification of a p-value, it is of course better to

do a proper Bayesian analysis. Are the particle physics community completely

wedded to frequentist analysis? If so, has anyone tried to explain what bad

science that is?

3. We know that given enough data it is nearly always possible for a

significance test to reject the null hypothesis at arbitrarily low p-values,

simply because the parameter will never be exactly equal to its null value.

And apparently the LNC has accumulated a very large quantity of data. So

could even this extreme p-value be illusory?

If anyone has any answers to these or related questions, I'd be interested to

know and will be sure to pass them on to Dennis.

Regards,

Tony

----

Professor A O'Hagan Email: a.ohagan@sheffield.ac.uk

Department of Probability and Statistics

University of Sheffield Phone: +44 114 222 3773

Hicks Building

Sheffield S3 7RH, UK Fax: +44 114 222 3759

----------- http://www.tonyohagan.co.uk/ ------------

[Από Tony O'Hagan - πολύ γνωστό Bayesian]

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of

answers.

We've heard a lot about the Higgs boson. The news reports say that the LHC

needed convincing evidence before they would announce that a particle had

been found that looks like (in the sense of having some of the right

characteristics of) the elusive Higgs boson. Specifically, the news referred

to a confidence interval with 5-sigma limits.

Now this appears to correspond to a frequentist significance test with an

extreme significance level. Five standard deviations, assuming normality,

means a p-value of around 0.0000005. A number of questions spring to mind.

1. Why such an extreme evidence requirement? We know from a Bayesian

perspective that this only makes sense if (a) the existence of the Higgs

boson (or some other particle sharing some of its properties) has extremely

small prior probability and/or (b) the consequences of erroneously announcing

its discovery are dire in the extreme. Neither seems to be the case, so why

5-sigma?

2. Rather than ad hoc justification of a p-value, it is of course better to

do a proper Bayesian analysis. Are the particle physics community completely

wedded to frequentist analysis? If so, has anyone tried to explain what bad

science that is?

3. We know that given enough data it is nearly always possible for a

significance test to reject the null hypothesis at arbitrarily low p-values,

simply because the parameter will never be exactly equal to its null value.

And apparently the LNC has accumulated a very large quantity of data. So

could even this extreme p-value be illusory?

If anyone has any answers to these or related questions, I'd be interested to

know and will be sure to pass them on to Dennis.

Regards,

Tony

----

Professor A O'Hagan Email: a.ohagan@sheffield.ac.uk

Department of Probability and Statistics

University of Sheffield Phone: +44 114 222 3773

Hicks Building

Sheffield S3 7RH, UK Fax: +44 114 222 3759

----------- http://www.tonyohagan.co.uk/ ------------

## Απ: Higgs Boson and statistical issues

Dear Tony, dear all,

this paper provides some hints

http://arxiv.org/abs/1112.3620

(see also http://www.roma1.infn.it/~dagos/badmath/index.html#added )

Moreover

- The "higgs p-values" does not seem to be what a professional

(frequentistic) statistician would mean by that term:

-> there is no serious null hypothesis without Higgs, because

a Standard Model without Higgs mechanism loses completely meaning.

-> what has the meaning of a p-value depending on the mass?

(a number, calculated in the hypothesis that the Higgs does not

exist, reported in function of its mass...)

- Also the "95% CL exclusion" regions have dubious meaning

because they are derived by "prescriptions" that do not

provide a quantitative statement of how much we should

be confidence on something.

Regards, and best greetings to Dennis,

Giulio

this paper provides some hints

http://arxiv.org/abs/1112.3620

(see also http://www.roma1.infn.it/~dagos/badmath/index.html#added )

Moreover

- The "higgs p-values" does not seem to be what a professional

(frequentistic) statistician would mean by that term:

-> there is no serious null hypothesis without Higgs, because

a Standard Model without Higgs mechanism loses completely meaning.

-> what has the meaning of a p-value depending on the mass?

(a number, calculated in the hypothesis that the Higgs does not

exist, reported in function of its mass...)

- Also the "95% CL exclusion" regions have dubious meaning

because they are derived by "prescriptions" that do not

provide a quantitative statement of how much we should

be confidence on something.

Regards, and best greetings to Dennis,

Giulio

## Απ: Higgs Boson and statistical issues

Hello Bayesians,

Below are a few answers. Thnaks for the interest

Louis Lyons (organiser of PHYSTAT series of meetings,

and member of CMS Collaboration at CERN.)

________________________________________

From: ISBA Webmaster [hans@stat.duke.edu]

Sent: 11 July 2012 02:46

To: news@bayesian.org

Subject: Higgs boson

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of

answers.

We've heard a lot about the Higgs boson. The news reports say that the LHC

needed convincing evidence before they would announce that a particle had

been found that looks like (in the sense of having some of the right

characteristics of) the elusive Higgs boson.

************ The test statistic we use for looking at p-values is basically

the likelihood

ratio for the two hypotheses (H_0 = Standard Model (S. M.) of Particle

Physics, but no Higgs;

H_1 = S.M with Higgs). A small p_0 (and a reasonable p_1) then implies that

H_1 is a better

description of the data than H_0. This of course does not prove that H_1 is

correct, but

maybe Nature corresponds to some H_2, which is more like H_1 than it is like

H_0. Indeed

in principle data will never prove a theory is true, but the more

experimental tests it survives,

the happier we are to use it - e.g. Newtonian mechanics was fine for

centuries till the arrival

of Relativity.

************* In the case of the Higgs, it can decay to different sets of

particles, and these rates

are defined by the S.M. We measure these ratios, but with large

uncertainties with the present data.

They are consistent with the S.M. predictions, but it could be much more

convincing with more

data. Hence the caution about saying we have discovered the Higgs of the S.

M..

Specifically, the news referred

to a confidence interval with 5-sigma limits.

************** 5-sigma really refers to p_0

Now this appears to correspond to a frequentist significance test with an

extreme significance level. Five standard deviations, assuming normality,

means a p-value of around 0.0000005. A number of questions spring to mind.

1. Why such an extreme evidence requirement? We know from a Bayesian

perspective that this only makes sense if (a) the existence of the Higgs

boson (or some other particle sharing some of its properties) has extremely

small prior probability and/or (b) the consequences of erroneously announcing

its discovery are dire in the extreme. Neither seems to be the case, so why

5-sigma?

********************** This is an unfortunate tradition, that is used more

readily

by journal editors than by Particle Physicists. Reasons are

a) Historically we have had 3 and 4 sigma effects that have gone away

b) The 'Look Elsewhere Effect' (LEE). We are worried about the chance of a

statistical fluctuation mimicking our observation, not only at the given mass

of 125 GeV but anywhere in the spectrum. The quoted p-values are

'local' i.e. the chance of a fluctuation at the observed mass. Unfortunately

the LEE correction factor is not very precisely defined, because of

ambiguities about

what is meant by 'elsewhere'

c) The possibility of some systematic effect (characterised by a nuisance

parameter)

being more important than allowed for in the analysis, or even overlooked -

see the

recent experiment at CERN which claimed that neutrinos travelled faster than

the speed of

light.

d) A subconscious use of Bayes Theorem to turn p-values into probabilities

about the

hypotheses.

All the above vary from experiment to experiment, so we realise that it is a

bit unfair to

use the same standard for discovery for all analyses. We prefer just to quote

the p-values

(or whatever).

2. Rather than ad hoc justification of a p-value, it is of course better to

do a proper Bayesian analysis. Are the particle physics community completely

wedded to frequentist analysis?

************** No we are not anti-Bayesian, and indeed our test statistics is

a likelihood ratio.

If you like, you can regard our p-values as an attempt to calibrate the

meaning of a

particular value of the likelihood ratio.

************** We actually recommend that for parameter determination at the

LHC, it is

useful to compare Bayesian and Frequentist methods. But for comparing

hypotheses

(e.g. an experimental distribution is fitted by H_0 = a smooth distribution;

or by H_1 = a smooth

distribution plus a localised peak), we are worried about what priors to use

for the extra

parameters that occur in the alternative hypothesis. ******* We would welcome

advice.**********

If so, has anyone tried to explain what bad

science that is?

*************** Comment ignored

3. We know that given enough data it is nearly always possible for a

significance test to reject the null hypothesis at arbitrarily low p-values,

simply because the parameter will never be exactly equal to its null value.

And apparently the LHC has accumulated a very large quantity of data. So

could even this extreme p-value be illusory?

************** We are aware of this. But in fact, although the LHC has

accumulated enormous

amounts of data, the Higgs search is like looking for a needle in a

haystack. The final samples

of events that are used to look for the Higgs contain only tens to thousands

of events.

If anyone has any answers to these or related questions,

***************** These and related issues are discussed to some extent in my

article "Open statistical issues in Particle Physics", Ann. Appl. Stat.

Volume 2, Number 3 (2008),

887-915. It is supposed to be statistician-friendly

Below are a few answers. Thnaks for the interest

Louis Lyons (organiser of PHYSTAT series of meetings,

and member of CMS Collaboration at CERN.)

________________________________________

From: ISBA Webmaster [hans@stat.duke.edu]

Sent: 11 July 2012 02:46

To: news@bayesian.org

Subject: Higgs boson

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of

answers.

We've heard a lot about the Higgs boson. The news reports say that the LHC

needed convincing evidence before they would announce that a particle had

been found that looks like (in the sense of having some of the right

characteristics of) the elusive Higgs boson.

************ The test statistic we use for looking at p-values is basically

the likelihood

ratio for the two hypotheses (H_0 = Standard Model (S. M.) of Particle

Physics, but no Higgs;

H_1 = S.M with Higgs). A small p_0 (and a reasonable p_1) then implies that

H_1 is a better

description of the data than H_0. This of course does not prove that H_1 is

correct, but

maybe Nature corresponds to some H_2, which is more like H_1 than it is like

H_0. Indeed

in principle data will never prove a theory is true, but the more

experimental tests it survives,

the happier we are to use it - e.g. Newtonian mechanics was fine for

centuries till the arrival

of Relativity.

************* In the case of the Higgs, it can decay to different sets of

particles, and these rates

are defined by the S.M. We measure these ratios, but with large

uncertainties with the present data.

They are consistent with the S.M. predictions, but it could be much more

convincing with more

data. Hence the caution about saying we have discovered the Higgs of the S.

M..

Specifically, the news referred

to a confidence interval with 5-sigma limits.

************** 5-sigma really refers to p_0

Now this appears to correspond to a frequentist significance test with an

extreme significance level. Five standard deviations, assuming normality,

means a p-value of around 0.0000005. A number of questions spring to mind.

1. Why such an extreme evidence requirement? We know from a Bayesian

perspective that this only makes sense if (a) the existence of the Higgs

boson (or some other particle sharing some of its properties) has extremely

small prior probability and/or (b) the consequences of erroneously announcing

its discovery are dire in the extreme. Neither seems to be the case, so why

5-sigma?

********************** This is an unfortunate tradition, that is used more

readily

by journal editors than by Particle Physicists. Reasons are

a) Historically we have had 3 and 4 sigma effects that have gone away

b) The 'Look Elsewhere Effect' (LEE). We are worried about the chance of a

statistical fluctuation mimicking our observation, not only at the given mass

of 125 GeV but anywhere in the spectrum. The quoted p-values are

'local' i.e. the chance of a fluctuation at the observed mass. Unfortunately

the LEE correction factor is not very precisely defined, because of

ambiguities about

what is meant by 'elsewhere'

c) The possibility of some systematic effect (characterised by a nuisance

parameter)

being more important than allowed for in the analysis, or even overlooked -

see the

recent experiment at CERN which claimed that neutrinos travelled faster than

the speed of

light.

d) A subconscious use of Bayes Theorem to turn p-values into probabilities

about the

hypotheses.

All the above vary from experiment to experiment, so we realise that it is a

bit unfair to

use the same standard for discovery for all analyses. We prefer just to quote

the p-values

(or whatever).

2. Rather than ad hoc justification of a p-value, it is of course better to

do a proper Bayesian analysis. Are the particle physics community completely

wedded to frequentist analysis?

************** No we are not anti-Bayesian, and indeed our test statistics is

a likelihood ratio.

If you like, you can regard our p-values as an attempt to calibrate the

meaning of a

particular value of the likelihood ratio.

************** We actually recommend that for parameter determination at the

LHC, it is

useful to compare Bayesian and Frequentist methods. But for comparing

hypotheses

(e.g. an experimental distribution is fitted by H_0 = a smooth distribution;

or by H_1 = a smooth

distribution plus a localised peak), we are worried about what priors to use

for the extra

parameters that occur in the alternative hypothesis. ******* We would welcome

advice.**********

If so, has anyone tried to explain what bad

science that is?

*************** Comment ignored

3. We know that given enough data it is nearly always possible for a

significance test to reject the null hypothesis at arbitrarily low p-values,

simply because the parameter will never be exactly equal to its null value.

And apparently the LHC has accumulated a very large quantity of data. So

could even this extreme p-value be illusory?

************** We are aware of this. But in fact, although the LHC has

accumulated enormous

amounts of data, the Higgs search is like looking for a needle in a

haystack. The final samples

of events that are used to look for the Higgs contain only tens to thousands

of events.

If anyone has any answers to these or related questions,

***************** These and related issues are discussed to some extent in my

article "Open statistical issues in Particle Physics", Ann. Appl. Stat.

Volume 2, Number 3 (2008),

887-915. It is supposed to be statistician-friendly

## Απ: Higgs Boson and statistical issues

Dear Tony

I have written a bit about the explanation of the P-value here

http://understandinguncertainty.org/explaining-5-sigma-higgs-how-well-did-they-do#comment-1449

The CERN teams' reports also discuss what they would expect were the

Higgs there, so there seems a real possibility of a likelihood ratio

being computed, which would be a start. Not sure why they don't do this.

d(avid Spigelhalter)

I have written a bit about the explanation of the P-value here

http://understandinguncertainty.org/explaining-5-sigma-higgs-how-well-did-they-do#comment-1449

The CERN teams' reports also discuss what they would expect were the

Higgs there, so there seems a real possibility of a likelihood ratio

being computed, which would be a start. Not sure why they don't do this.

d(avid Spigelhalter)

## Απ: Higgs Boson and statistical issues

[Moderator's note: this message is from

Harrison B. Prosper

Kirby W. Kemper Professor of Physics

Distinguished Research Professor

Florida State University

harry@hep.fsu.edu

]

Dear Tony,

First some general remarks, then I'll try to answer your questions.

I am in an interesting position regarding the "Higgs" boson discovery: I am

thrilled to be an insider with respect to the discovery and I happen also to

be one of the relatively few particle physicists who actually regard Bayesian

reasoning as "exactly what is needed" to make sense of what we do. The vast

majority of my colleagues believe that p-values are objective and therefore

"scientific". Therefore, many of my colleagues move mountains, or at any rate

consume prodigious amounts of computing power, to check that some (typically

ad hoc) procedure covers.

For your edification I've attached a (PUBLIC!) plot [Moderator's note:

available at http://bayesian.org/webfm_send/274 ] of (slightly massaged)

binned data from my collaboration (CMS) that shows a spectrum arising from

proton-proton collisions that resulted in the creation of a pair of photons

(gammas in high energy argot). The Standard Model predicts that the Higgs

boson should decay (that is break up) into a pair of photons. (The Higgs is

predicted to decay in other ways too, such as a pair of Z bosons.) The bump

in the plot at around 125 GeV is evidence for the existence of some particle

of a definite mass that decays into a pair of photons. That something, as far

as we've been able to ascertain, is likely to be the Higgs boson. These data,

along with data in which proton-proton collisions yield two Z bosons are the

basis of our 5-sigma claim.

These data can be modeled with the function

f(x) = exp(a0 + a1*x + a2*x^2) + s * Gaussian(x|m, w)

where "x" is the mass the di-photon (pair of photons), and the first term

describes the smoothly falling (background) spectrum, while the second term

models the bump. "s" is the total expected signal, "m" is the mass of the new

particle and "w" is the width of the bump. The total background, that is, the

"noise" is just the integral of the first term. "a0", "a1", "a2" are nuisance

parameters. This is therefore a 6-parameter problem for which we have no

prior information (or choose to act as if this is so) for the six parameters.

The analysis of this spectrum has caused a lot of angst about the

"look-elsewhere-effect" (multiple hypothesis testing), which I think is a red

herring in this context.

Now for your questions. See below.

Harrison

Quoting ISBA Webmaster :

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of

answers.

We've heard a lot about the Higgs boson. The news reports say that the LHC

needed convincing evidence before they would announce that a particle had

been found that looks like (in the sense of having some of the right

characteristics of) the elusive Higgs boson. Specifically, the news referred

to a confidence interval with 5-sigma limits.

Now this appears to correspond to a frequentist significance test with an

extreme significance level. Five standard deviations, assuming normality,

means a p-value of around 0.0000005. A number of questions spring to mind.

1. Why such an extreme evidence requirement? We know from a Bayesian

perspective that this only makes sense if (a) the existence of the Higgs

boson (or some other particle sharing some of its properties) has extremely

small prior probability and/or (b) the consequences of erroneously announcing

its discovery are dire in the extreme. Neither seems to be the case, so why

5-sigma?

The "5-sigma" (p-value = 3.0e-7) is an historical artifact. Over the past

several decades, we have made many a "discovery" that turned out not to be

so. As a consequence, we gradually settled on a p-value thought to be small

enough to reduce the chance that we are fooling ourselves. In fact, we do

have high standards because in our view we are trying to arrive at "true"

statements about the world in the pragmatic sense that these statements yield

predictions that turn out to be correct. Given that the search for the Higgs

took some 45 years, tens of thousands of scientists and engineers, billions

of dollars, not to mention numerous divorces, huge amounts of sleep

deprivation, tens of thousands of bad airline meals, etc., etc., we want to

be sure as is humanly possible that this is real.

2. Rather than ad hoc justification of a p-value, it is of course better to

do a proper Bayesian analysis. Are the particle physics community completely

wedded to frequentist analysis? If so, has anyone tried to explain what bad

science that is?

I for one would be delighted to see a Bayesian analysis of these data from

you guys! Unfortunately, however, I am forbidden from e-mailing you the

50,000 "x"s I have right here on my laptop...very frustrating...

3. We know that given enough data it is nearly always possible for a

significance test to reject the null hypothesis at arbitrarily low p-values,

simply because the parameter will never be exactly equal to its null value.

And apparently the LNC has accumulated a very large quantity of data. So

could even this extreme p-value be illusory?

As noted above, small p-value-based "discoveries" have come and gone.

However, the reason I am convinced this is real is not because of the

p-value, nor frankly because of the (pseudo)-frequentist method of analysis

showcased on July 4th during the announcement at CERN, a method that I have

repeatedly criticized within my collaboration. Rather it is because when I

study the profile likelihood in the variables "s", "m", and "w" for the 2011

dataset and for the 2012 dataset, I find visually convincing structures in

the profile likelihood at m ~ 125 GeV in both independent datasets, obtained

at different proton beam energies (7 TeV and 8 TeV). Of course, I would

rather have preferred to have done a Bayesian analysis, marginalizing over

"a0", "a1", "a2", and "w", and to study the posterior density in the

variables "s" and "m", but constructing a non-evidence-based prior in

4-dimensions that would pass muster seems quite a chore. Any advice from You

would be welcome. (I favor the recursive reference prior algorithm of

Bernardo, but this would have to be done numerically and I have not yet

figured out how to do so efficiently, while taking into account the "nested

compact sets". The whole thing seems rather daunting.)

Harrison B. Prosper

Kirby W. Kemper Professor of Physics

Distinguished Research Professor

Florida State University

harry@hep.fsu.edu

]

Dear Tony,

First some general remarks, then I'll try to answer your questions.

I am in an interesting position regarding the "Higgs" boson discovery: I am

thrilled to be an insider with respect to the discovery and I happen also to

be one of the relatively few particle physicists who actually regard Bayesian

reasoning as "exactly what is needed" to make sense of what we do. The vast

majority of my colleagues believe that p-values are objective and therefore

"scientific". Therefore, many of my colleagues move mountains, or at any rate

consume prodigious amounts of computing power, to check that some (typically

ad hoc) procedure covers.

For your edification I've attached a (PUBLIC!) plot [Moderator's note:

available at http://bayesian.org/webfm_send/274 ] of (slightly massaged)

binned data from my collaboration (CMS) that shows a spectrum arising from

proton-proton collisions that resulted in the creation of a pair of photons

(gammas in high energy argot). The Standard Model predicts that the Higgs

boson should decay (that is break up) into a pair of photons. (The Higgs is

predicted to decay in other ways too, such as a pair of Z bosons.) The bump

in the plot at around 125 GeV is evidence for the existence of some particle

of a definite mass that decays into a pair of photons. That something, as far

as we've been able to ascertain, is likely to be the Higgs boson. These data,

along with data in which proton-proton collisions yield two Z bosons are the

basis of our 5-sigma claim.

These data can be modeled with the function

f(x) = exp(a0 + a1*x + a2*x^2) + s * Gaussian(x|m, w)

where "x" is the mass the di-photon (pair of photons), and the first term

describes the smoothly falling (background) spectrum, while the second term

models the bump. "s" is the total expected signal, "m" is the mass of the new

particle and "w" is the width of the bump. The total background, that is, the

"noise" is just the integral of the first term. "a0", "a1", "a2" are nuisance

parameters. This is therefore a 6-parameter problem for which we have no

prior information (or choose to act as if this is so) for the six parameters.

The analysis of this spectrum has caused a lot of angst about the

"look-elsewhere-effect" (multiple hypothesis testing), which I think is a red

herring in this context.

Now for your questions. See below.

Harrison

Quoting ISBA Webmaster :

Dear Bayesians,

A question from Dennis Lindley prompts me to consult this list in search of

answers.

We've heard a lot about the Higgs boson. The news reports say that the LHC

needed convincing evidence before they would announce that a particle had

been found that looks like (in the sense of having some of the right

characteristics of) the elusive Higgs boson. Specifically, the news referred

to a confidence interval with 5-sigma limits.

Now this appears to correspond to a frequentist significance test with an

extreme significance level. Five standard deviations, assuming normality,

means a p-value of around 0.0000005. A number of questions spring to mind.

1. Why such an extreme evidence requirement? We know from a Bayesian

perspective that this only makes sense if (a) the existence of the Higgs

boson (or some other particle sharing some of its properties) has extremely

small prior probability and/or (b) the consequences of erroneously announcing

its discovery are dire in the extreme. Neither seems to be the case, so why

5-sigma?

The "5-sigma" (p-value = 3.0e-7) is an historical artifact. Over the past

several decades, we have made many a "discovery" that turned out not to be

so. As a consequence, we gradually settled on a p-value thought to be small

enough to reduce the chance that we are fooling ourselves. In fact, we do

have high standards because in our view we are trying to arrive at "true"

statements about the world in the pragmatic sense that these statements yield

predictions that turn out to be correct. Given that the search for the Higgs

took some 45 years, tens of thousands of scientists and engineers, billions

of dollars, not to mention numerous divorces, huge amounts of sleep

deprivation, tens of thousands of bad airline meals, etc., etc., we want to

be sure as is humanly possible that this is real.

2. Rather than ad hoc justification of a p-value, it is of course better to

do a proper Bayesian analysis. Are the particle physics community completely

wedded to frequentist analysis? If so, has anyone tried to explain what bad

science that is?

I for one would be delighted to see a Bayesian analysis of these data from

you guys! Unfortunately, however, I am forbidden from e-mailing you the

50,000 "x"s I have right here on my laptop...very frustrating...

3. We know that given enough data it is nearly always possible for a

significance test to reject the null hypothesis at arbitrarily low p-values,

simply because the parameter will never be exactly equal to its null value.

And apparently the LNC has accumulated a very large quantity of data. So

could even this extreme p-value be illusory?

As noted above, small p-value-based "discoveries" have come and gone.

However, the reason I am convinced this is real is not because of the

p-value, nor frankly because of the (pseudo)-frequentist method of analysis

showcased on July 4th during the announcement at CERN, a method that I have

repeatedly criticized within my collaboration. Rather it is because when I

study the profile likelihood in the variables "s", "m", and "w" for the 2011

dataset and for the 2012 dataset, I find visually convincing structures in

the profile likelihood at m ~ 125 GeV in both independent datasets, obtained

at different proton beam energies (7 TeV and 8 TeV). Of course, I would

rather have preferred to have done a Bayesian analysis, marginalizing over

"a0", "a1", "a2", and "w", and to study the posterior density in the

variables "s" and "m", but constructing a non-evidence-based prior in

4-dimensions that would pass muster seems quite a chore. Any advice from You

would be welcome. (I favor the recursive reference prior algorithm of

Bernardo, but this would have to be done numerically and I have not yet

figured out how to do so efficiently, while taking into account the "nested

compact sets". The whole thing seems rather daunting.)

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