spey.backends.default_pdf.CorrelatedBackground

class spey.backends.default_pdf.CorrelatedBackground(signal_yields: ndarray, background_yields: ndarray, data: ndarray, covariance_matrix: ndarray, signal_uncertainty_configuration: Dict[str, Any] | None = None)

Correlated multi-region statistical model. The correlation between each nuisance parameter has been captured via Multivariate Normal distribution and the log-probability distribution is combination of Multivariate Normal along with Poisson distribution. The Multivariate Normal distribution is constructed by the help of a covariance matrix provided by the user which captures the uncertainties and background correlations between each histogram bin. The probability distribution of a simplified likelihood can be formed as follows;

\[\mathcal{L}_{SL}(\mu,\theta) = \underbrace{\left[\prod_i^N {\rm Poiss}\left(n^i_{obs} | \lambda_i(\mu, \theta)\right) \right]}_{\rm main\ model} \cdot \underbrace{\mathcal{N}(\theta | 0, \rho)}_{\rm constraint\ model}\]

Here the first term is the so-called main model based on Poisson distribution centred around \(\lambda_i(\mu, \theta) = \mu n^i_{sig} + \theta + n^i_{bkg}\) and the second term is the multivariate normal distribution centred around zero with the correlation matrix \(\rho\).

Parameters:

• signal_yields (np.ndarray) – signal yields

• background_yields (np.ndarray) – background yields

• data (np.ndarray) – observations

• covariance_matrix (np.ndarray) – covariance matrix (square matrix)

• signal_uncertainty_configuration (Dict[Text, Any]], default None) –

  Configuration input for signal uncertainties

  • absolute_uncertainties (List[float]): Absolute uncertainties for the signal

  • absolute_uncertainty_envelops (List[Tuple[float, float]]): upper and lower

    uncertainty envelops

  • correlation_matrix (List[List[float]]): Correlation matrix

  • third_moments (List[float]): diagonal elemetns of the third moment

Note

Each input should have the same dimensionality, i.e. if data has three regions, signal_yields and background_yields inputs should have three regions as well. Additionally covariance_matrix is expected to be square matrix, thus for a three region statistical model it is expected to be 3x3 matrix.

Example:

>>> import spey

>>> stat_wrapper = spey.get_backend('default_pdf.correlated_background')

>>> signal_yields = [12.0, 11.0]
>>> background_yields = [50.0, 52.0]
>>> data = [51, 48]
>>> covariance_matrix = [[3.,0.5], [0.6,7.]]

>>> statistical_model = stat_wrapper(signal_yields,background_yields,data,covariance_matrix)
>>> print(statistical_model.exclusion_confidence_level())

__init__(signal_yields: ndarray, background_yields: ndarray, data: ndarray, covariance_matrix: ndarray, signal_uncertainty_configuration: Dict[str, Any] | None = None)

Methods

__init__(signal_yields, background_yields, ...)
asimov_negative_loglikelihood([poi_test, ...])	Compute negative log-likelihood at fixed \(\mu\) for Asimov data.
combine(other, **kwargs)	A routine to combine to statistical models.
config([allow_negative_signal, poi_upper_bound])	Model configuration.
expected_data(pars[, include_auxiliary])	Compute the expected value of the statistical model
get_hessian_logpdf_func([expected, data])	Currently Hessian of \(\log\mathcal{L}(\mu, \theta)\) is only used to compute variance on \(\mu\).
get_logpdf_func([expected, data])	Generate function to compute \(\log\mathcal{L}(\mu, \theta)\) where \(\mu\) is the parameter of interest and \(\theta\) are nuisance parameters.
get_objective_function([expected, data, do_grad])	Objective function i.e. twice negative log-likelihood, \(-2\log\mathcal{L}(\mu, \theta)\).
get_sampler(pars)	Retreives the function to sample from.
minimize_asimov_negative_loglikelihood([...])	A backend specific method to minimize negative log-likelihood for Asimov data.
minimize_negative_loglikelihood([expected, ...])	A backend specific method to minimize negative log-likelihood.
negative_loglikelihood([poi_test, expected])	Backend specific method to compute negative log-likelihood for a parameter of interest \(\mu\).

Attributes

constraints	Constraints to be used during optimisation process
signal_uncertainty_configuration
author	Author of the backend
constraint_model	retreive constraint model distribution
is_alive	Returns True if at least one bin has non-zero signal yield.
main_model	retreive the main model distribution
name	Name of the backend
spey_requires	Spey version required for the backend
version	Version of the backend