vstatmp_engl

Weighting the Monte Carlo Observations

When we fit parameters, every parameter change obviously entails a modification of the Monte Carlo prediction. Now we do not want to repeat the full simulation with every fitting step. Apart from the fact that we want to avoid the computational e ort there is another more important reason: With the χ2 fit we find the standard error interval by letting vary χ2 by one unit. On the other hand when we compare experimental data with an optimal simulation, we expect a contribution to χ2 from the simulation of the order of p2BN/M for B histogram bins. Even with a simulation sample which is a hundred times larger than the data sample this value is of the order of one. This means that a repetition of the simulation causes considerable fluctuations of the χ2 value which have nothing to do with parameter changes. These fluctuations can only be reduced if the same Monte Carlo sample is used for all parameter values. We have to adjust the simulation to the modified parameters by weighting its observations.

Also re-weighting produces additional fluctuations. These, however, should be tolerable if the weights do not vary too much. If we are not sure that this assumption is justified, we can verify it: We reduce the number of Monte Carlo observations and check whether the result remains stable. We know that the contribution of the simulation to the parameter errors scales with the inverse square root of the number of simulated events. Alternatively, we can also estimate the Monte Carlo contribution to the error by repeating the simulation and the full estimation process.

The weights are computed in the following way: For each Monte Carlo observation x′ we know the true values x of the variates and the corresponding p.d.f. f(x|θ0) for the parameter θ0, which had been used at the generation. When we modify the parameter we weight each observation by w(θ) = f(x|θ)/f(x|θ0). The weighted distribution of x′ then describes the modified prediction.

The weighting of single observations requires a repetition of the histogramming after each parameter change. This may require too much computing time. A more e cient method is based on a Taylor expansion of the p.d.f. in powers of the parameters. We illustrate it with two examples.

Example 85. Fit of the slope of a linear distribution with Monte Carlo correction

We generate observations x uniformly distributed in the interval 0 ≤ x ≤ 1, simulate the experimental resolution and the acceptance, and histogram the distorted variable x′ into bins i and obtain contents m0i. The same observations are weighted by x and summed up to the histogram m1i. These two distributions are shown in Fig. 6.10 a, b. The dotted histograms correspond to the distributions before the distortion by the measurement process. In Fig. 6.10 c is also depicted the experimental distribution. It should be possible to describe it by a superposition mi of the two Monte Carlo distributions:

We optimize the parameter θ such that the histogram di is described up to a normalization constant as well as possible by a superposition of the two

Fig. 6.10. The superposition of two Monte Carlo distributions, a) flat and b) triangular is adjusted to the experimental data.

Monte Carlo histograms. We have to insert mi from (6.18) into (6.16) and

X set cn = N/ mi.

i

Example 86. Fit of a lifetime with Monte Carlo correction

We expand the p.d.f.

f(t|γ) = γe−γt

into a Taylor expansion at γ0 which is a first guess of the decay rate γ:

The Monte Carlo simulation follows the distribution f0 = γ0e−γ0t. Weighting the events by (1/γ0 − t) and (−t/γ0 + t2/2), we obtain the distributions f1 = (1 − γ0t)e−γ0t, f2 = (−t + γ0t2/2)e−γ0t and

f(t|γ) = f0(t) + Δγf1(t) + (Δγ)2f2(t) + · · · .

If it is justified to neglect the higher powers of Δγ/γ0, we can again describe our experimental distribution this time by a superposition of three distributions f0′ (t′), f1′ (t′), f2′ (t′) which are the distorted versions of f0, f1, f2. The

Fig. 6.11. Lifetime fit. The dotted histogram in b) is the superposition of the three histograms of a) with weights depending on Δγ.

parameter Δγ is determined by a χ2 or likelihood fit. In our special case it is even simpler to weight f0 by t, and t2, respectively, and to superpose the corresponding distributions f0, g1 = tf0, g2 = t2f0 with the factors given in the following expression:

The parameter Δγ is then modified until the correspondingly weighted sum of the distorted histograms agrees optimally with the data. Figure 6.11 shows an example. In case the quadratic term can be neglected, two histograms are su cient.

The general case is treated in an analogous manner. The Taylor expansion is:

we obtain two further histograms m1i, m2i. The parameter inference of Δθ is performed by comparing mi = (m0i+Δθm1i+Δθ2m2i) with the experimental histogram di.

In many cases the quadratic term can be omitted. In other situations it might be necessary to iterate the procedure.

6.5.10 Parameter Estimate of a Signal Contaminated by Background

In this section we will discuss point inference in the presence of background in a specific situation, where we have the chance to record independently from a signal sample also a reference sample containing pure background. The measuring times or fluxes, i.e. the relative normalization, of the two experiments are supposed to be known. The parameters searched for could be the position and width of a Breit-Wigner bump but also the slope of an angular distribution or the lifetime of a particle. The interest in this method rests upon the fact that we do not need to parameterize the background distribution and thus are independent of assumptions about its shape. This feature has to be paid for by a certain loss of precision.

The idea behind the method is simple: The log-likelihood of the wanted signal parameter as derived for the full signal sample is a superposition of the log-likelihood of the genuine signal events and the log-likelihood of the background events. The latter can be estimated from the reference sample and subtracted from the full loglikelihood.

To illustrate the procedure, imagine we want to measure the signal response of a radiation detector by recording a sample of signal heights x1, . . . , xN from a monoenergetic source. For a pure signal, the xi would follow a normal distribution with

resolution σ:

f(x|µ) = √ 1 2πσ

The unknown parameter µ is to be estimated. After removing the source, we can – under identical conditions – take a reference sample x′1, . . . , x′M of background events. They follow a distribution which is of no interest to us.

If we knew, which observations xi in our signal sample were signal (x(iS)), respec-

tively background (x(iB)) events, we could take only the S signal events and calculate the correct log-likelihood function

164 6 Parameter Inference I

1.The constraints are used to reduce the number of parameters.

2.The constraints are approximated by narrow Gaussians.

3.Lagrange multipliers are adjusted to satisfy the constraints.

We will discuss the problem in terms of a χ2 minimization. The transition to a likelihood fit is trivial.

6.6.2 Eliminating Redundant Parameters

Sometimes it is possible to eliminate parameters by expressing them by an unconstrained subset.

Example 88. Fit with constraint: two pieces of a rope

A rope of exactly 1 m length is cut into two pieces. A measurement of both pieces yields l1 = 35.3 cm and l2 = 64.3 cm, both with the same Gaussian

including the constraint λ1 + λ2 = L = 100 cm. We simply replace λ2 by L − λ1 and adjust λ1, minimizing

The minimization relative to λ1 leads to the result:

The constraint has the same e ect as a double measurement, but with the modification that now the results are (maximally) anti-correlated: one finds cov(λ1λ2) = −var(λi).

Example 89. Fit of the particle composition of an event sample

A particle identification variable x has di erent distributions fi(x) for di erent particles. The p.d.f. given the relative particle abundance λm for particle species m out of M di erent particles is

As the constraint relation is linear, we can easily eliminate the parameter λM to get rid of the constraint P λm = 1:

From the MLE we obtain in the usual way the first M −1 parameters and their error matrix E. The remaining parameter λM and the related error matrix elementsPeMj are derived from the constraint (6.23) and the corresponding relation Δλm = 0. The diagonal error is the expected value of (ΔλM )2:

The remaining elements are computed analogously:

MX−1

EMj = EjM = −Ejj − Emj . m6=j

An iterative method, called channel likelihood, to find the particle contributions, is given in [34].

These trivial examples are not really representative for the typical problems we have to solve in particleor astrophysics. Indeed, it is often complicated or even impossible to reduce the parameter set analytically to an unconstrained subset. But we can introduce a new unconstrained parameter set which then predicts the measured quantities. To find such a set is straight forward in the majority of problems: We just have to think how we would simulate the corresponding experimental process. A simulation is always based on a minimum set of parameters. Constraints are obeyed automatically.

Example 90. Kinematical fit with constraints: eliminating parameters

A neutral particle c is decaying into two charged particles a and b, for instance Λ → p + π−. The masses mc, ma, mb are known. Measured are the momenta πa, πb of the decay products and the decay vertex ρ. The measurements of the components of the momentum vectors are correlated. The inverse error matrices be Va and Vb. The origin of the decaying particle be at the origin of the coordinate system. Thus we have 9 measurements (r, pa, pb), 10 parameters, namely the 3 momentum vectors and the distance (πc, πa, πb, ρ), and 4 constraints from momentum and energy conservation:

166 6 Parameter Inference I

X3

+(pbi − πbi)Vbij (pbj − πbj ) .

i,j=1

The vertex parameters ρi are fixed by the vector relation ρ = ρπc/|πc|. Now we would like to remove 4 out of the 10 parameters using the 4 constraints.

A Monte Carlo simulation of the Λ decay would proceed as follows: First we would select the Λ momentum vector (3 parameters). Next the decay length would be generated (1 parameter). The decay of the Λ hyperon into proton and pion is fully determined when we choose the proton direction in the lambda center of mass system (2 parameters). All measured laboratory quantities and thus also χ2 can then be expressed analytically by these 6 unconstrained quantities (we omit here the corresponding relations) which are varied in the fitting procedure until χ2 is minimal. Of course in the fit we would not select random starting values for the parameters but the values which we compute from the experimental decay length and the measured momentum vectors. Once the reduced parameter set has been adjusted, it is easy to compute also the remaining laboratory momenta and their errors which, obviously, are strongly correlated.

Often the reduced parameter set is more relevant than the set corresponding to the measurement, because a simulation usually is based on parameters which are of scientific interest. For example, the investigation of the Λ decay might have the goal to determine the Λ decay parameter which depends on the center of mass direction of the proton relative to the Λ polarization, i.e. on one of the directly fitted quantities.

6.6.3 Gaussian Approximation of Constraints

As suggested by formula (6.22) to fulfil the constraints within the precision of our measurement, we just have to choose the uncertainty δk su ciently small. We will give a precise meaning to the expression su ciently small below.

Example 91. Example 88 continued

Minimizing

produces the same result as the fit presented above. The value δk2 = 10−10δ2 is chosen small compared to δ.