Toy simulations¶

So far we have used asymptotic formulae to compute p-values and define confidence regions. As explained in Statistical inference, the asymptotic formulae hold exactly only for linear regression models. Since the model in our examples is non-linear this is at best an approximation. To test the quality of this approximation we have to generate toy observables $\vec y\in\mathcal{X}$ distributed according to the PDF $f(\cdot,\vec\omega_0)$ of our model for some parameter vector $\vec\omega_0\in\Omega_0$ , and determine the distribution of $\Delta\chi^2(\vec y)$ . This is called a toy simulation. If the asymptotic formulae are applicable the $\Delta\chi^2$ should have a chi-square distribution with $\dim\Omega - \dim\Omega_0$ degrees of freedom.

If asymptotic formulae are not applicable you can use toy simulations to compute the (plug-in) p-value numerically. To do this you simply count the fraction of toy observable vectors $\vec y$ for which $\Delta\chi^2(\vec y) > \Delta\chi^2(\vec x_0)$ , where $\vec x_0$ is the observed data. If the p-value is small you need a large number of toy observables to estimate this fraction with a reasonable accuracy. For example, to claim “3 sigma evidence” for something you have to show that the p-value is smaller than $2.7\cdot 10^{-3}$ . To estimate such a small p-value with a (relative) 10% accuracy you need about 40000 toys. For a “5 sigma discovery” you alread need $1.7\cdot 10^8$ toys. Simulating such a large number of toys is often impossible, even with a computing cluster. Remember that computing the $\Delta\chi^2$ for one toy vector $\vec y$ requires two numerical optimisations which, for difficult problems, can take several seconds.

myFitter implements an improved method for numerical p-value computations which is based on importance sampling (and referred to as myFitter method in the following). It exploits “geometric” information about the parameter spaces $\Omega$ and $\Omega_0$ to generate toys which are more likely to give a $\Delta\chi^2$ larger than $\Delta\chi^2(\vec x_0)$ . The details are explained in arXiv:1207.1446, but you don’t need to know them in order to use the method. This section shows you how.

The myFitter class which is responsible for running toy simulations and computing p-values (both with the standard and the myFitter method) is called ToySimulator. Since numerical p-value computations are essentially multi-dimensional numerical integrations we can use adaptive techniques to improve the convergence of these integrations. After a suitable re-parametrisation of the integrand ToySimulator uses the VEGAS algorithm for adaptive Monte Carlo integration by linking to the vegas package.

Generating toy observables¶

In the simplest case, a ToySimulator object can be used as a generator for toy observables. Using our model from previous examples, the following code shows how to compute the p-value for the null hypothesis p1 = 1.25 (with all other parameters treated as nuisance parameters), and how to fill a histogram showing the distribution of the test statistic $\Delta\chi^2$ . (See also toy_example_1.py.)

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

model0 = MyModel()
model0.fix({'p1' : 1.25})
fitter0 = mf.Fitter(model0, tries=1)
fitter0.sample(2000)
fit0 = fitter0.fit()

model1 = MyModel()
fitter1 = mf.Fitter(model1, tries=1)
fitter1.sample(2000)
fit1 = fitter1.fit()

dchisq = fit0.minChiSquare - fit1.minChiSquare
dchisqhist = mf.Histogram1D(xbins=np.linspace(0.0, 3.0, 21))
sim = mf.ToySimulator(model0, fit0.bestFitParameters)

dchisqhist.set(0.0)
pvalue = 0.0
for obs, weight in sim.random(neval=2000):
    fit0 = fitter0.fit(obs=obs)
    fit1 = fitter1.fit(obs=obs)
    dchisq_toy = fit0.minChiSquare - fit1.minChiSquare
    try:
        dchisqhist[dchisq_toy] += weight
    except KeyError:
        pass
    if dchisq_toy > dchisq:
        pvalue += weight

asymptotic_pvalue = mf.pvalueFromChisq(dchisq, 1)
print 'p-value:            {0:f}'.format(pvalue)
print 'asymptotic p-value: {0:f}'.format(asymptotic_pvalue)

dchisqhist = mf.densityHistogram(dchisqhist)
dchisqhist.plot(drawstyle='steps')
plt.plot(dchisqhist.xvals, [stats.chi2.pdf(x, 1) for x in dchisqhist.xvals])
plt.show()

As we will see later myFitter actually provides better mechanisms for computing p-values and filling histograms. Here we choose to do things “by hand” in order to illustrate the use of ToySimulator as a generator for toy observables.

In lines 5 to 9 we construct a model object model0 which represents the null hypothesis, i.e. which has the parameter p1 fixed to 1.25. We then initialise a fitter object, fit the model to the data and store the result of the fit in fit0. In lines 11 to 14 we repeat the procedure for the “full model” (where p1 is not fixed) and store the result of the fit in fit1. In line 16 we then compute the observed $\Delta\chi^2$ value and store it in the variable dchisq.

In line 17 we construct a one-dimensional histogram with 20 evenly-spaced bins between 0 and 3. The ToySimulator object sim is constructed in line 18. Passing model0 and fit0.bestFitParameters to the constructor means that the toy observables are distributed according to the PDF of model0 with the parameters set to the best-fit parameters found in the fit in line 9.

The data values of dchisqhist are initialised to None by default. In line 20 we set them all to 0.0, so that we can increment them later. In line 21 we initialise the variable pvalue to zero. After the loop over the toy observables this will hold (an estimate of) the p-value.

In line 22 we start the loop over the toy observables. ToySimulator.random() returns an iterator over the toy observables. The argument neval=2000 means that at most 2000 toys will be generated. The iterator yields a tuple (obs, weight), where obs is a dict describing a point in observable space and weight is the weight associated with that point. The sum of all weights yielded by the iterator is guaranteed to be 1, but the actual number of points yielded may be smaller than the value specified with the neval keyword. To fill the histogram we have to compute the $\Delta\chi^2$ value for each toy observable obs and then increment the corresponding histogram bin by the associated value of weight. (If your model only contains Gaussian observables the weights will all be the same, but for non-Gaussian observables they will be different.) The $\Delta\chi^2$ value is computed in lines 23 to 25 and assigned to the variable dchisq_toy. Note the use of the obs keyword in the calls to Fitter.fit() to override the observable values stored in the obs dicts of the model objects. In line 27 we increment the data value of the bin containing the value of dchisq_toy by weight. Note that you can use square bracket notation to access the data value of the bin associated with dchisq_toy. If the value of dchisq_toy is outside the range of the histogram (i.e. smaller than 0 or larger than 3 in this case) this throws a KeyError exception, which we catch with a try/except block.

After the loop finishes the variable pvalue holds an estimate of the p-value. In line 33 we compute the asymptotic approximation to the p-value and compare the two results in lines 34 and 35. The output should be something like

p-value:            0.092448
asymptotic p-value: 0.117567

The data values of the histogram dchisqhist are now estimates of the integral of the PDF over the region defined by $a \leq \Delta\chi^2 < b$ , where $a$ and $b$ are the boundaries of the corresponding bin. To estimate the PDF for $\Delta\chi^2$ we have to divide each data value by the bin width. This is done in line 37, using myFitter’s densityHistogram() function. This function works on one and two-dimensional histograms. In line 38 we plot the (density) histogram and in line 39 we superimpose a $\chi^2$ distribution with one degree of freedom (using functions from scipy.stats).

Computing p-values¶

Although ToySimulator can be used as a generator for toy observables, its real purpose is the computation of expectation values of arbitrary functions on observable space. Let

$g:\mathcal{X} \to \R, \vec y \mapsto g(\vec y)$

be such a function and $\vec\omega_0\in\Omega_0$ a point in parameter space. Its expectation value of $g$ at $\vec\omega_0$ is

(1) $E_{\vec\omega_0}[g] = \int d^n\vec y\, g(\vec y)f(\vec y,\vec\omega_0) \eqsep.$

To compute this integral ToySimulator uses the VEGAS algorithm for adaptive Monte Carlo integration implemented in the Python module vegas together with a suitable re-parametrisation of the integrand. Since such integrations can take a long time ToySimulator also supports parallelisation.

As discussed in Statistical inference the (plug-in) p-value of a likelihood ratio test is the probability that the $\Delta\chi^2$ value of a randomly generated set of toy observables is larger than the observed $\Delta\chi^2$ value. It is given by the integral (1) with

(2) $g(\vec y) = \theta(\Delta\chi^2(\vec y) - \Delta\chi^2(\vec x_0))$

where $\theta$ is the Heavyside step function and $\vec x_0$ is the observed data. The LRTIntegrand class implements this function. To instantiate it, you need two Fitter objects which fit the full model and the model representing the null hypothesis, respectively, and the value of $\Delta\chi^2(\vec x_0)$ . Here is an example using the model from previous examples (see also toy_example_2.py):

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

model0 = MyModel()
model0.fix({'p1' : 1.25})
fitter0 = mf.Fitter(model0, tries=1)
fitter0.sample(2000)
fit0 = fitter0.fit()

model1 = MyModel()
fitter1 = mf.Fitter(model1, tries=1)
fitter1.sample(2000)
fit1 = fitter1.fit()

dchisq = fit0.minChiSquare - fit1.minChiSquare
lrt = mf.LRTIntegrand(fitter0, fitter1, dchisq, nested=True)

The object lrt is callable and takes a dict representing the point $\vec y$ in observable space as argument. It returns a numpy array of two elements. The first element is the value of $g(\vec y)$ from (2). The second is the value of $\Delta\chi^2(\vec y)$ and can be used for histogramming (see below). If the third argument in the constructor is omitted the first element of the array returned by lrt is always 1. LRTIntegrand objects count the number of failed evaluations where one of the minimisations did not converge or could not find a feasible solution. This number is stored in LRTIntegrand.nfail. The total number of evaluations is stored in LRTIntegrand.nshots. In a good simulation the fraction of failed evaluations should be small, so this information is important to assess the reliability of your results. The nested=True argument indicates that fitter0.model and fitter1.model have the same parameters, so that the best-fit parameters for fitter0.model can be used as a starting point for the fit of fitter1.model. LRTIntegrand uses this trick to improve the numerical stability.

With the assignments above the p-value can be computed as follows (see also toy_example_2.py):

sim = mf.ToySimulator(model0, fit0.bestFitParameters,
                      parallel={'njobs'  : 2,
                                'runner' : mf.LocalRunner})
result = sim(lrt, nitn=5, neval=2000, drop=1, log=sys.stdout)

print
print 'p-value:            '+str(mf.GVar(result[0]))
print 'asymptotic p-value: {0:f}'.format(mf.pvalueFromChisq(dchisq, 1))

In line 1 we construct the ToySimulator object. The first two arguments mean that $f$ is the PDF of model0 and that $\vec\omega_0$ are the maximum likelihood estimates of the parameters found in our earlier fit of model0. This means we are going to compute the plug-in p-value.

The parallel keyword to the constructor (line 2) enables parallelisation and works the same way as for Profiler1D or Profiler2D (see Parallelisation). In line 4 we perform the integration by calling the object sim. The first argument is the integrand function. Since we want the p-value we use the LRTIntegrand object constructed earlier.The VEGAS integration proceeds in several iterations, and after each iteration the results from all iterations are used to adapt the sampling density to the integrand. (Our integrand function actually returns an array with two values, but the adaptation is always driven by the first element of the array.) The final result is the weighted average of the results from all iterations. The allowed keyword arguments to ToySimulator.__call__() are the same as for the ToySimulator constructor and can be used to override the settings made there. The argument nitn=5 means that we run 5 iterations, and neval=2000 limits the number of integrand evaluations in each iteration to 2000. For difficult integrands the estimates from the first few iterations can be rather poor and their errors can be vastly underestimated. It is therefore usually better to exclude the results of the first few iterations from the weighted average. With drop=1 we instruct sim to exclude the result of the first iteration. The log=sys.stdout argument means that information about the iterations will be written to sys.stdout. Only information about the first integrand (i.e. the first element of the array returned by lrt) is written. The output will be something like

itn                  integral               wgt average  chi2/dof     Q
-----------------------------------------------------------------------
              0.0924(88)                0.0924(88)
              0.0912(60)                0.0912(60)
              0.0854(46)                0.0876(37)       0.6   44%
              0.0965(89)                0.0889(34)       0.7   48%
              0.0910(33)                0.0900(24)       0.6   64%

The second column shows the estimates of the integral from the individual iterations. The number in brackets is the error on the last digit. Thus, $0.0924(88)$ means $0.0924\pm 0.0088$ . The third column shows the weighted average of the results from all iterations (except the ones that were dropped) and the first column shows the number of iterations included in the weighted average. The numbers in the fourth and fifth columns are useful to assess the compatibility of the results included in the weighted average. For a reliable integration result the number in the fourth column should be of order 1 and the number in the fifth column should not be too small—say, above 5%. It is worth mentioning that, at least in this example, the VEGAS adaptation actually doesn’t reduce the integration error in a noticable way: the errors of the results in the second column do not decrease significantly as the integration progresses.

In myFitter the result of a VEGAS integration is represented by instances of the IntegrationResult class. IntegrationResult objects hold the results and errors of the individual iterations as well as their weighted average, chi-square (which has nothing to do with the chi-square function of your model), Q-value, and the error of the weighted average. Since our integrand function returns an array of two values the return value of sim in line 4 is an array of two IntegrationResult objects. Since we only want the p-value we are only interested in the first element of this array. In line 7 we print the p-value and its numerical error by converting result[0] first to a GVar object and then to a string. In line 8 we print the asymptotic approximation of the p-value. The output is

p-value:            0.0900(24)
asymptotic p-value: 0.117566

The difference between the numerical result and the asymptotic approximation is about 11 times larger than the numerical error, which shows that the asymptotic approximation is not exact for this model.

The myFitter method¶

So far we have only computed relatively large p-values with toy simulations. As mentioned at the beginning of this chapter, “naive” toy simulations become very inefficient for small p-values, and the unique feature of myFitter is an advanced method for the numerical computation of very small p-values. This is achived by a clever re-parametrisation of the (p-value) integrand which uses information not only from the model describing the null hypothesis (model0) but also from the full model (model1).

To illustrate this feature let us now choose the null hyptothesis p1 = 1.1 (with all other parameters treated as nuisance parameters). This hypothesis has a $\Delta\chi^2$ value of 11.2 and an asymptotic p-value of $8\cdot 10^{-4}$ . A naive toy simulation would need more than $10^8$ integrand evaluations to estimate such a small p-value with a relative 10% accuracy. The following code estimates the p-value using the myFitter method (see also toy_example_3.py):

model0 = MyModel()
model0.fix({'p1' : 1.1})
fitter0 = mf.Fitter(model0, tries=1)
fitter0.sample(2000)
fit0 = fitter0.fit()

model1 = MyModel()
fitter1 = mf.Fitter(model1, tries=1)
fitter1.sample(2000)
fit1 = fitter1.fit()

dchisq = fit0.minChiSquare - fit1.minChiSquare
lrt = mf.LRTIntegrand(fitter0, fitter1, dchisq, nested=True)

sim = mf.ToySimulator(model0, fit0.bestFitParameters,
                      model1=model1,
                      dchisq=dchisq,
                      parallel={'njobs'  : 2,
                                'runner' : mf.LocalRunner})
result = sim(lrt, nitn=5, neval=10000, drop=1, log=sys.stdout)

print
print 'p-value:            {0:s}'.format(str(mf.GVar(result[0])))
print 'asymptotic p-value: {0:f}'.format(mf.pvalueFromChisq(dchisq, 1))

Apart from changing the value of p1 and increasing the number of evaluations per iteration (neval) a little the only difference to the previous example is the addition of the model1 and dchisq keyword arguments in lines 16 and 17. With this extra information ToySimulator constructs a smart sampling density which is taylored for the p-value integrand constructed in line 13. The output of the program is

itn                  integral               wgt average  chi2/dof     Q
-----------------------------------------------------------------------
  1              0.000468(17)              0.000468(17)
  1              0.000471(14)              0.000471(14)
  2              0.000488(21)              0.000476(11)       0.5   49%
  3              0.000505(11)             0.0004905(81)       1.9   15%
  4              0.000483(10)             0.0004875(63)       1.4   24%

p-value:            0.0004875(63)
asymptotic p-value: 0.000808

Note that the first iteration with $10^4$ evaluations already estimates the p-value with a relative accuracy of less than 10%. With the naive method from Generating toy observables this would require $10^8$ evaluations—a factor $10^4$ improvement in efficiency. The final relative accuracy of 1.3%, achieved with the myFitter method after 50 000 evaluations, would require $2.5\cdot 10^{10}$ evaluations with the naive method. That’s a factor $2\cdot 10^5$ improvement in efficiency! Also note that the numerical result for the p-value differs from the asymptotic approximation by almost a factor of 2.

Filling histograms¶

In our initial example we used a toy simulation to compute the p-value and fill a histogram at the same time. For the advanced integration methods discussed in Computing p-values and The myFitter method this is no longer possible. You can, however, use the ToySimulator.fill() method to fill one or more histograms without programming the loop over toy observables yourself. This has the advantage that you automatically get errors estimates for the bin heights. More importantly, you can let ToySimulator parallelise the task of filling histograms for you (just like it automatically parallelises the VEGAS integration).

To understand how fill() works let us first formalise the problem a little. Assume you want to compute the expectation value (1) for some function $g$ on observable space. At the same time you have functions $h_i$ on observable space which you want to histogram, i.e. you are also interested in the integrals

(3) $H_{ij} = \int d^n\vec y\, g(\vec y)f(\vec y,\vec\omega_0) \chi_{ij}(h_i(\vec y))$

where $\chi_{ij}(h)$ is an indicator function which is 1 if $h$ is in the $j$ -th bin of the $i$ -th histogram and 0 otherwise. The notation also applies to two-dimensional histograms, in which case the functions $h_i$ must return a two-component vector. The example from Toy simulations corresponds to the case

(4) $g(\vec y) = 1 \eqsep,\eqsep h_1(\vec y) = \Delta\chi^2(\vec y) \eqsep.$

To compute $E_{\vec\omega_0}[g]$ and fill all the histograms in one simulation you just have to define a Python function—call it gh—which takes a dict describing the point $\vec y$ in observable space as argument and which returns a list (or numpy array) holding the values of $g(\vec y)$ , $h_1(\vec y)$ etc. (the zeroth element being $g(\vec y)$ ). For the case (4) we can just use an LRTIntegrand object for gh. If we omit the third argument (dchisq) in the constructor the first element of the returned array will always be 1.

The following code uses fill() to compute the same histogram as in Toy simulations (see also toy_example_4.py):

model0 = MyModel()
model0.fix({'p1' : 1.25})
fitter0 = mf.Fitter(model0, tries=1)
fitter0.sample(2000)
fit0 = fitter0.fit()

model1 = MyModel()
fitter1 = mf.Fitter(model1, tries=1)
fitter1.sample(2000)

dchisqhist = mf.Histogram1D(xbins=np.linspace(0.0, 3.0, 21))
sim = mf.ToySimulator(model0, fit0.bestFitParameters,
                      parallel={'njobs'  : 2,
                                'runner' : mf.LocalRunner,
                                'log'    : sys.stdout})
gh = mf.LRTIntegrand(fitter0, fitter1, nested=True)

sim.fill(gh, 10000, (1, dchisqhist))

dchisqhist = mf.densityHistogram(dchisqhist)
dchisqhist.plot(drawstyle='steps', color='b')
dchisqhist.errorbars(color='b')
plt.plot(dchisqhist.xvals, [stats.chi2.pdf(x, 1) for x in dchisqhist.xvals])
plt.show()

Lines 1 to 15 are mostly the same as in previous examples. Note that we have added 'log' : sys.stdout to the dict assigned to the parallel keyword in line 15. This causes ToySimulator to print information about the running jobs to stdout. In line 16 we create a LRTIntegrand object and assign it to gh. Note the omission of the dchisq argument in the constructor. This means the first element of the array returned by gh is always 1.

In line 18 we call fill to carry out the simulation and fill the histogram. The first argument is the integrand function gh. The second argument is the (maximal) number of evaluations. The third argument (1, dchisqhist) instructs fill to use element 1 of the list returned by gh to fill the histogram dchisqhist. This element holds the $\Delta\chi^2$ value. You can fill more than one histogram by adding more tuples to the argument list of fill. The second element of each tuple must be a Histogram1D or Histogram2D object. If it is a Histogram2D object the first element must be a tuple of two integers which specify the positions of the two variables to be histogrammed in the list returned by gh. Alternatively, the first element of the tuple can be a callable object. See the reference of fill for details.

After line 18 the data values of dchisqhist are GVar objects which hold the estimates and errors of the integrals (3). The return value of fill is a GVar object holding the estimate and error of the integral (1). In our example we do not need this estimate (it is 1 with an error of 0 since our function $g$ is identical to 1), so we discard it. In lines 21 and 22 we plot the histogram with errorbars. Note that you don’t need the convert keyword to turn GVar objects into ordinary numbers. Histogram1D and Histogram2D know what to do with data values of type GVar.

Toy simulations¶

Generating toy observables¶

Computing p-values¶

The myFitter method¶

Filling histograms¶

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Toy simulations¶

Generating toy observables¶

Computing p-values¶

The myFitter method¶

Filling histograms¶

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