SCQA

Checks performed as per the Statistics Committee (SC) request

Q1)If the statistics had been a bit larger, it would have been possible just to calculate chi-squared at each point in (M_H, tan(beta)) space for agreement between predictions and the 4 observed data numbers. With the current statistics, a likelihood ratio method is probably needed, as suggested by Tom. So you need to check
a) whether the best fit is acceptable (the answer is surely 'YES')
b) to determine the contour that corresponds to a 95% interval. For chi-squared, you probably could just use the appropriate delta(chi-squared) for 95% in 2 variables.

It would have been possible to do it in a frequentist way. However, doing it the suggested way, would answer the question of whether the data can rule out the model at each point in the (mH,tan(b)) plane. Our question is different: we ask, what values of tan(b) are disfavored by data in such a way that they are ruled out at 95% credible level with respect to other values of tan(b) under the same assumptions?. Of course in both cases no point can be absolutely excluded, only at some confidence/credible level.

With that in mind, and having specified that we are dealing with two different questions, we proceed to answer the SC question.

A1)a) it is. See the figure below. I constructed the chi2 distribution using pseudoexperiments (PSE) and then compared to the chi2 obtained in data.

A1)b) Using the likelihood ratio method (LR) (done in a "fast" way) we obtained the 1-sigma and 2-sigma contours in the Benchmark#1 as shown in the picture below.

As expected the LR method results in weaker limits at 95%CL.

Q2) Check coverage for at least one value of M_H, for tan(beta) around the edges of the rejected region. This will help validate what you have done.

A2) We checked the (frequentist) coverage in two points at mH+=120 GeV. The two points are given by tan(b)=0.73 and tan(b) = 39. These points are exactly in the contour of our exclusion regions.
Of course, we do not expect the coverage to be 95% or better since bayesian statistics does not need to cover.
By generating 500 PSE the point at low tan(b) shows a coverage of (93.2+/-1.2)%
By generating 500 PSE the point at high tan(b) shows a coverage of (91.4+/-1.4)%
Both these numbers are close to the expected by a frequentist method.

Q3) Worry about the fact that you exclude the region with BR.gt.0.9 simply because it corresponds to a width of top greater than 15 GeV. This would not be a problem for the likelihood ratio approach.

A3) That is absolutely true. We have just redone the Tauonic Higgs model and the Worst Case Combination plot. The tauonic Higgs model changes are extremely small. The Worst Case BR Combination plot changed a little, specially at a mass of 100 GeV, where the sensitivity is very low. Still working in the sensitivity band.
The first plot is the new limit obtained integrating from 0 to 1, the second one is the old one in which the integration was performed from 0 to 0.9.

WCS.gig

From the above questions it appears you want us to address a different question with our analysis. First let's review what questions our analysis is addressing.:

The motivating questions are three :
A- In the assumption that nature is described by the MSSM, and in the assumption that we know all MSSM parameters, except tan(b) ( MSSM parameters include the value of the charged Higgs mass), what values of tan(b) can be ruled out at 95% credible level ?
B- In the assumption that a charged Higgs boson of a specific mass decays to tau-nu only, what values BR(t->H+b) can be ruled out at 95% Credible level.
C- In the assumption that a charged Higgs boson decays to the four channels (csbar, taunu, t*b, Wh0), can we set a conservative limit that holds for any combination of the charged Higgs branching ratio decays ?

Before proceeding, let's review a frequentist and bayesian analysis. Take for example the case in which we assume that a charged Higgs of mass 120 GeV, decays to tau+nu all the time, and we want to measure BR(t->H+b->tau+nu+b) under those two assumptions.

Frequentist

Bayesian

For each point in BR(t->Hb) the number of expected events in each of the four cross section analyses can be obtained. These are combined with the observed events in a Poisson probability to form the Likelihood. It can be said that each point in BR(t->Hb) is a model. Each point in the BR(t->Hb) is compared to data using a test statistic, Likelihood Ratio for example. The outcome determines whether each point is excluded or not at a certain frequentist Confidence Level. In a sense it can be said that each model point can be excluded or not by itself, in the assumptions that H+ decays to tau-nu and the charged Higgs mass is 120 GeV.

It can be said that each point in BR(t->Hb) is a model. All models are compared in relation to each other, and those disfavor by data are excluded. The integral should add up to one by hypothesis (being that nature is described by one of these models). The comparison includes a prior that puts weight between different models. Notice that SM model is included in the set.
The exclusion region can be determined either by the integration of the Posterior probability over the maximum density region, or by left-to-right integration. Either cases yield the same result in this particular case, since the SM (leftmost model given by BR(t->Hb)=0) is also the maximum density region, and the posterior is monotonically decreasing. The limits will change depending on the prior used.

Both cases are statistically accepted and each person has its own preference.
The Statistics Committee has previously stated that Bayesian is CDF's default method.

Let's look now at our case.
Not only we assume that the mass of the charged Higgs is given, but also that all the other MSSM parameters are known. We plot out exclusion region as a function of tan(b) .

Our case

Again, each point in tan(b), or in log10(tan(b)), is considered a model. Again, the SM is represented in the set by the points closed to tan(b)~7 (log10(7)=0.84). Again we formed the Posterior probability by using a prior, in this case flat in log10(tan(b)). We have decided to integrate over the maximun density region, as opposed to left-to-right or tail integration. The integral over the minimum and maximum allowed tan(b) values should add up to one by hypothesis (the hypothesis being that nature is determined by all the given MSSM parameters, and one unknown tan(b) that should lie in an model-allowed range). Thus, the points outside the 95% are considered excluded values of tan(b) for the given MSSM parameters.
As it is shown here, the posterior (and also the top and Higgs BR's ) are smooth functions of log10(tan(b)). That's the reason we chose the prior in log10(tan(b)); it just seems natural!

There is no difference with respect to the case shown above in the Bayesian column.