\small
\begin{center}
{\it John Conway, 
     Howard Haber,
     John Hobbs,
     Harrison Prosper}
\end{center}
\normalsize\nopagebreak

The statistical method employed here for combining channels follows an
essentially Bayesian approach by forming a joint likelihood for a
given set of experimental outcomes, as a function of a dummy
multiplier $f$ on the theoretical cross section.

For each search channel $i$ assume that we expect $s_i$ events from
the Higgs signal and $b_i$ events from all backgrounds, in 1 fb$^{-1}$
integrated luminosity.  These values, then, can equivalently be thought
of as the accepted signal and background cross sections in units of
femtobarns.

In the actual experiments, after some integrated luminosity L has been
recorded, one expects L$s_i$ signal and L$b_i$ background in a given
channel $i$, and observes some number $n_i$.  The Poisson probability
${\cal P}(n_i|\mu_i)$ for observing $n_i$ events, where
$\mu_i\equiv{\rm L}(s_i+b_i)$ events are expected in this channel is
well known:
\begin{eqnarray}
     {\cal P}(n_i|\mu_i) = \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!} \ \ .
\end{eqnarray}
The joint likelihood ${\cal L}(n_1,n_2,...|s_1,s_2,...,b_1,b_2,...)$ 
for all channels is simply the product of these probabilities:
\begin{eqnarray}
     {\cal L}(n_1,n_2,...|s_1,s_2,...,b_1,b_2,...) = 
              \prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!} \ \ .
\end{eqnarray}
This likelihood tells us the absolute probability of our particular
experimental outcome, but gives no information about whether or not
a signal is actually present.  To accomplish this one can introduce 
a factor $f$ which multiplies the expected signal, and determine 
confidence intervals on its true value from a probability density
function (p.d.f.) based on the joint likelihood above.  Thus, we
now have $\mu_i(f)\equiv{\rm L}(fs_i+b_i)$, and from Bayes'
Theorem we find that this ``posterior'' p.d.f. can be written as
\begin{eqnarray}
     {\cal P}(f|\bar{n},\bar{s},\bar{b}) = 
          \frac{\displaystyle \prod_i \mu_i(f)^{n_i} e^{-\mu_i(f)}/n_i!}
               {\displaystyle \prod_i \int_0^\infty \mu_i(f^\prime)^{n_i} e^{-\mu_i(f^\prime)}/n_i! df^\prime}
\end{eqnarray}
where we have expressed the sets $\{n_1,n_2,...\}$, $\{s_1,s_2,...\}$,
and $\{b_1,b_2,...\}$ as vectors for clarity.  If there is no Higgs
then the function ${\cal P}(f|\bar{n},\bar{s},\bar{b})$ will be
maximum at $f=0$, and fall more and more sharply as the integrated
luminosity increases.  If, on the other hand, there is a Higgs (at 
the mass were are considering), then the posterior p.d.f. will be
more and more sharply peaked at $f=1$ as L increases.  

For a given integrated luminosity L and a given experimental outcome,
then, how can one use the posterior p.d.f. in $f$ to either claim a
discovery at some level of significance or exclude the predicted Higgs
cross section with come confidence?  

For discoveries, one usually quotes significance in units of $\sigma$,
(say, $3\sigma$ or $5\sigma$) where it is understood that this refers
to equivalent Gaussian standard deviations.  In the method used here
one finds that value of $f$ at which ${\cal P}(f|\bar{n},\bar{s},\bar{b})$ 
is a maximum, ${\cal P}_{max}$, and
compares with the value ${\cal P}_0$ evaluated for $f=0$.  That is,
one compares the maximum likelihood to that of the null
hypothesis. One can then find the number of ``equivalent Gaussian
$\sigma$'' from the ratio of these values:
\begin{eqnarray}
  \frac{{\cal P}_{max}}{{\cal P}_0} = \frac{e^0}{e^{-x^2/2\sigma^2}}
\end{eqnarray}
and so the significance $x$ can be written as
\begin{eqnarray}
  x = \left( 2 \ln \frac{{\cal P}_{max}}{{\cal P}_0} \right)^{\frac{1}{2}}\sigma \ \ .
\end{eqnarray}

For exclusion limits, or confidence intervals on $f$ in general, one 
simply performs integrals of the posterior p.d.f., which by definition
is normalized to unity.  Thus if we desire a 95\% confidence level
exclusion, we need to find that value of $f$ for which
\begin{eqnarray}
   \epsilon = \int_f^\infty {\cal P}(f^\prime |\bar{n},\bar{s},\bar{b}) df^\prime
\end{eqnarray}
where $\epsilon=0.05$.  If, for example, there is no Higgs and we find
that we can exclude at 95\% CL any value of $f>1$, then we conclude that
the Higgs is excluded at that confidence level, since $f=1$ corresponds
to the Standard Model prediction for the Higgs cross section.

One can also take into account systematic errors using this method, by
convoluting Gaussians representing the relative uncertainties in the
$s_i$ and $b_i$ with the expression for the joint likelihood.  In the
method used here this is approximated by a coarse numerical integration
in the likelihood calculation.

The main goal of the Higgs studies presented here, however, is to
estimate the integrated luminosity needed to reach certain thresholds
of significance.  But clearly for a given set of expected number of
signal and background events, at a certain integrate luminosity, there
are many possible future experimental outcomes, some of which will 
and some of which will not meet the desired level of significance.
Following the precedent of the LEP-II Working Group, we simply seek
that integrated luminosity at which 50\% of the possible future 
outcomes would meet the desired level.

To actually calculate these integrated luminosity thresholds requires
simulating a large ensemble of pseudoexperiments, each representing a
certain outcome, distributed according to their actual probability of
occurrence.  Then, one varies the integrated luminosity until the 50\%
criterion is met.   Note that in the case of determining 95\% CL
limits, the program generates no signal events in each
pseudoexperiment, whereas in the case of determining discovery
thresholds, the program does generate signal events in each
pseudoexperiment.

A few notes about the statistical method are in order.  Firstly, in
the case of a single channel with no systematic errors, for setting
95\% CL limits, the Bayesian method outlined above is in exact
agreement with the commonly used frequentist method described in the
1996 PDG Review of Particle Properties.   In that approach one
excludes at some confidence level $1-\epsilon$ any value of $f$
greater than that for which
\begin{eqnarray}
   \epsilon = \frac {\displaystyle \sum_{j=0}^n (fs+b)^n e^{-(fs+b)}/n! }
                    {\displaystyle\sum_{j=0}^n b^n e^{-b}/n! } \ \ .
\end{eqnarray}
This formula simply asks for that level of expected signal such that
the probability of seeing $n$ or fewer events, and having $n$ or fewer
background, is less than some value $\epsilon$.

Also, in the limit of high statistics, the results of the Bayesian
method outlined above agree closely with those from what one might
call the ``naive Gaussian approximation.''  This approximation uses
the values of the $s_i$ and $b_i$ to form ``significances''
$S_i(L)\equiv \sqrt{L}s_i/\sqrt{b_i}$, and then finds those values of L
for which $S(L)=\sqrt{\sum_i S_i^2(L)}$ exceeds some value, namely 3 or
5 for $3\sigma$ and $5\sigma$ discoveries, respectively, and 1.96 for
a 95\% CL limit.  This commonly used approximation breaks down, of
course, in the low-statistics regime where the distributions are
Poisson and not well approximated by Gaussians.

