Quick Start

Quick Start#

Installation#

spey is available at pypi , so it can be installed by running:

>>> pip install spey

Python >=3.9 is required. spey heavily relies on numpy, scipy and autograd which are all packaged during the installation with the necessary versions. Note that some versions may be restricted due to numeric stability and validation.

To install spey with iminuit, use

>>> pip install spey[iminuit]

and iminuit functionality can be activated by set_optimiser() function.

What is Spey?#

Spey is a plug-in-based statistics tool designed to consolidate a wide range of likelihood prescriptions in a comprehensive platform. Spey empowers users to integrate different statistical models seamlessly and explore their properties through a unified interface by offering a flexible workspace. To ensure compatibility with existing and future statistical model prescriptions, Spey adopts a versatile plug-in system. This approach enables developers to propose and integrate their statistical model prescriptions, thereby expanding the capabilities and applicability of Spey.

First Steps#

First, one needs to choose which backend to work with. By default, Spey is shipped with various types of likelihood prescriptions which can be checked via AvailableBackends() function

>>> import spey
>>> print(spey.AvailableBackends())
>>> # ['default.correlated_background',
>>> #  'default.effective_sigma',
>>> #  'default.third_moment_expansion',
>>> #  'default.uncorrelated_background']

For details on all the backends, see Plug-ins section.

Using 'default.uncorrelated_background' one can simply create single or multi-bin statistical models. This backend implements the following likelihood prescription:

\[\mathcal{L}(\mu, \theta) = \prod_{i\in{\rm bins}}{\rm Poiss}(n^i|\mu n_s^i + n_b^i + \theta^i\sigma_b^i) \cdot \prod_{j\in{\rm nui}}\mathcal{N}(\theta^j|0, 1)\ ,\]

where \(\mu\) is the signal strength (the parameter of interest or POI), \(\theta^i\) are nuisance parameters that represent deviations from the nominal background estimate, \(n^i\) is the observed event count, \(n_s^i\) and \(n_b^i\) are the signal and background expected yields, and \(\sigma_b^i\) are the absolute uncertainties. The Gaussian constraint ensures the nuisance parameters remain bounded near zero (nominal values).

>>> pdf_wrapper = spey.get_backend('default.uncorrelated_background')

>>> data = [1]
>>> signal_yields = [0.5]
>>> background_yields = [2.0]
>>> background_unc = [1.1]

>>> stat_model = pdf_wrapper(
...     signal_yields=signal_yields,
...     background_yields=background_yields,
...     data=data,
...     absolute_uncertainties=background_unc,
...     analysis="single_bin",
...     xsection=0.123,
... )

where data indicates the observed events (\(n^i\)), signal_yields and background_yields represent the expected signal and background samples (\(n_s^i\) and \(n_b^i\)), and background_unc denotes the absolute uncertainties on the background estimates (\(\sigma_b^i\)). In this case, the background is \(2.0\pm1.1\). The analysis and xsection arguments are optional: analysis provides a unique identifier for the statistical model, and xsection specifies the cross-section value of the signal (used only for computing excluded cross-section values).

During the computation of any probability distribution, Spey relies on the so-called “expectation type”. This can be set via ExpectationType, which includes three different expectation modes that specify which dataset to use in the likelihood evaluation.

  • observed: Uses the actual experimental observation (\(n^i\)) in the likelihood computation. For exclusion limit computation, this yields the observed \(1-CL_s\) value, which reflects what the experiment has actually measured. This is the default mode and is set as default throughout the package.

  • aposteriori: Performs post-fit expected limit computation. Rather than using the observed data, the likelihood uses an “Asimov dataset” (synthetic data at the expected value) generated under the background-only hypothesis (\(\mu=0\)). The expected limits are computed at five discrete points: the central value and \(\pm1\sigma\) and \(\pm2\sigma\) fluctuations around the background expectation. This approach assesses the experiment’s sensitivity assuming no real signal is present.

  • apriori: Performs pre-fit expected limit computation, assuming the Standard Model predictions are the absolute truth and no experimental observation has occurred. The likelihood replaces the observed data with the background-only expected yields (\(n^i = n_b^i\)) everywhere. This ignores experimental data and computes expected limits based purely on the theoretical predictions. This mode is primarily used by theory collaborations to estimate expected sensitivity independent of actual observations.

To compute the observed exclusion limit for the above example, one can type

>>> for expectation in spey.ExpectationType:
>>>     print(f"1-CLs ({expectation}): {stat_model.exclusion_confidence_level(expected=expectation)}")
>>> # 1-CLs (apriori): [0.49026742260475775, 0.3571003642744075, 0.21302512037071475, 0.1756147641077802, 0.1756147641077802]
>>> # 1-CLs (aposteriori): [0.6959976874809755, 0.5466491036450178, 0.3556261845401908, 0.2623335168616665, 0.2623335168616665]
>>> # 1-CLs (observed): [0.40145846656558726]

Note that apriori and aposteriori expectation types resulted in a list of 5 elements which indicates \(-2\sigma,\ -1\sigma,\ 0,\ +1\sigma,\ +2\sigma\) standard deviations from the background hypothesis. observed, on the other hand, resulted in a single value, which is the observed exclusion limit. Notice that the bounds on aposteriori are slightly more potent than apriori; this is due to the data value has been replaced with background yields, which are larger than the observations. apriori is mainly used in theory collaborations to estimate the difference from the Standard Model rather than the experimental observations.

Note

For details on exclusion limit and upper limit computations, see ref. [1].

One can play the same game using the same backend for multi-bin histograms as follows;

>>> pdf_wrapper = spey.get_backend('default.uncorrelated_background')

>>> data = [36, 33]
>>> signal_yields = [12.0, 15.0]
>>> background_yields = [50.0,48.0]
>>> background_unc = [12.0,16.0]

>>> stat_model = pdf_wrapper(
...     signal_yields=signal_yields,
...     background_yields=background_yields,
...     data=data,
...     absolute_uncertainties=background_unc,
...     analysis="multi_bin",
...     xsection=0.123,
... )

Note that our statistical model still represents individual bins of the histograms independently however, it sums up the log-likelihood of each bin. Hence, all bins are completely uncorrelated from each other. Computing the exclusion limits for each ExpectationType will yield

>>> for expectation in spey.ExpectationType:
>>>     print(f"1-CLs ({expectation}): {stat_model.exclusion_confidence_level(expected=expectation)}")
>>> # 1-CLs (apriori): [0.971099302028661, 0.9151646569018123, 0.7747509673901924, 0.5058089246145081, 0.4365406649302913]
>>> # 1-CLs (aposteriori): [0.9989818194986659, 0.9933308419577298, 0.9618669253593897, 0.8317680908087413, 0.5183060229282643]
>>> # 1-CLs (observed): [0.9701795436411219]

It is also possible to compute \(1-CL_s\) value with respect to the parameter of interest, \(\mu\). This can be achieved by including a value for poi_test argument

 1>>> import matplotlib.pyplot as plt
 2>>> import numpy as np
 3
 4>>> poi = np.linspace(0,10,20)
 5>>> poiUL = np.array([stat_model.exclusion_confidence_level(poi_test=p, expected=spey.ExpectationType.aposteriori) for p in poi])
 6>>> plt.plot(poi, poiUL[:,2], color="tab:red")
 7>>> plt.fill_between(poi, poiUL[:,1], poiUL[:,3], alpha=0.8, color="green", lw=0)
 8>>> plt.fill_between(poi, poiUL[:,0], poiUL[:,4], alpha=0.5, color="yellow", lw=0)
 9>>> plt.plot([0,10], [.95,.95], color="k", ls="dashed")
10>>> plt.xlabel(r"${\rm signal\ strength}\ (\mu)$")
11>>> plt.ylabel("$1-CL_s$")
12>>> plt.xlim([0,10])
13>>> plt.ylim([0.6,1.01])
14>>> plt.text(0.5,0.96, r"$95\%\ {\rm CL}$")
15>>> plt.show()

Here in the first line, we extract \(1-CL_s\) values per POI for aposteriori expectation type, and we plot specific standard deviations, which provides the following plot:

Exclusion confidence level with respect to the parameter of interest, :math:`\mu`.

The excluded value of POI can also be retrieved by poi_upper_limit() function. The upper limit \(\mu_{\rm UL}\) is defined as the smallest signal strength for which the experiment excludes the hypothesis at the specified confidence level (typically 95%). Mathematically, it satisfies:

\[{\rm CL}_s(\mu_{\rm UL}) = 1 - {\rm CL} = 0.05 \quad \text{(for 95\% CL)}\]

where \({\rm CL}_s = p_{s+b} / p_b\) is the CLs ratio (signal+background p-value divided by background-only p-value). This ratio protects against excluding signals to which the experiment has no sensitivity.

>>> print("POI UL: %.3f" % stat_model.poi_upper_limit(expected=spey.ExpectationType.aposteriori))
>>> # POI UL:  0.920

The upper limit is the exact point where the red curve (observed \(1-{\rm CL}_s\)) and black dashed line (0.05 threshold) meet in the plot above. The expected upper limits for the \(\pm1\sigma\) or \(\pm2\sigma\) bands can be extracted by setting expected_pvalue to "1sigma" or "2sigma" respectively:

>>> stat_model.poi_upper_limit(expected=spey.ExpectationType.aposteriori, expected_pvalue="1sigma")
>>> # [0.5507713378348318, 0.9195052042538805, 1.4812721449679866]

The three values correspond to the lower \(-1\sigma\), central, and upper \(+1\sigma\) bounds on the expected limit.

At a lower level, one can extract the likelihood information for the statistical model by calling likelihood() and maximize_likelihood() functions. By default, these return negative log-likelihood (NLL) values, but this can be changed via return_nll=False to return the actual likelihood \(\mathcal{L}(\mu, \theta)\).

maximize_likelihood() performs the global optimization:

\[(\hat{\mu}, \hat{\theta}) = \arg\max_{\mu, \theta} \log \mathcal{L}(\mu, \theta)\]

and returns both the optimal POI value \(\hat{\mu}\) and the corresponding likelihood value.

1>>> muhat_obs, maxllhd_obs = stat_model.maximize_likelihood(return_nll=False)
2>>> muhat_apri, maxllhd_apri = stat_model.maximize_likelihood(return_nll=False, expected=spey.ExpectationType.apriori)
3
4>>> poi = np.linspace(-3,4,60)
5
6>>> llhd_obs = np.array([stat_model.likelihood(p, return_nll=False) for p in poi])
7>>> llhd_apri = np.array([stat_model.likelihood(p, expected=spey.ExpectationType.apriori, return_nll=False) for p in poi])

In the first two lines, we extract the maximum likelihood point \((\hat{\mu}, \mathcal{L}_{\rm max})\) for two different expectation types. Subsequently, we compute the likelihood function \(\mathcal{L}(\mu)\) at multiple POI values, which can then be plotted and visualized:

>>> plt.plot(poi, llhd_obs/maxllhd_obs, label=r"${\rm observed\ or\ aposteriori}$")
>>> plt.plot(poi, llhd_apri/maxllhd_apri, label=r"${\rm apriori}$")
>>> plt.scatter(muhat_obs, 1)
>>> plt.scatter(muhat_apri, 1)
>>> plt.legend(loc="upper right")
>>> plt.ylabel(r"$\mathcal{L}(\mu,\theta_\mu)/\mathcal{L}(\hat\mu,\hat\theta)$")
>>> plt.xlabel(r"${\rm signal\ strength}\ (\mu)$")
>>> plt.ylim([0,1.3])
>>> plt.xlim([-3,4])
>>> plt.show()
Likelihood distribution for a multi-bin statistical model.

Notice the slight difference between likelihood distributions because of the use of different expectation types. The dots on the likelihood distribution represent the point where the likelihood is maximised. Since for an apriori likelihood distribution observed and background values are the same, the likelihood should peak at \(\mu=0\).