spey.backends.default_pdf.ThirdMomentExpansion

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

Simplified likelihood interface with third moment expansion. Third moment expansion follows simplified likelihood construction and modifies the \(\lambda\) and \(\Sigma\). Using the expected background yields, \(m^{(1)}_i\), diagonal elements of the third moments, \(m^{(3)}_i\) and the covariance matrix, \(m^{(2)}_{ij}\), one can write a modified correlation matrix and \(\lambda\) function as follows

\[ \begin{align}\begin{aligned}C_i &= -sign(m^{(3)}_i) \sqrt{2 m^{(2)}_{ii}} \cos\left( \frac{4\pi}{3} + \frac{1}{3}\arctan\left(\sqrt{ \frac{8(m^{(2)}_{ii})^3}{(m^{(3)}_i)^2} - 1}\right) \right)\\B_i &= \sqrt{m^{(2)}_{ii} - 2 C_i^2}\\A_i &= m^{(1)}_i - C_i\\\rho_{ij} &= \frac{1}{4C_iC_j} \left( \sqrt{(B_iB_j)^2 + 8C_iC_jm^{(2)}_{ij}} - B_iB_j \right)\end{aligned}\end{align} \]

which further modifies \(\lambda_i(\mu, \theta) = \mu n^i_{sig} + A_i + B_i \theta_i + C_i \theta_i^2\) and the multivariate normal has been modified via the inverse of the correlation matrix, \(\mathcal{N}(\theta | 0, \rho^{-1})\). See [arXiv:1809.05548] Sec. 2 for details.

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)

• third_moment (np.ndarray) – third moment for each region.

• 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. Following these, third_moment should also have three inputs.

__init__(signal_yields: ndarray, background_yields: ndarray, data: ndarray, covariance_matrix: ndarray, third_moment: 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
arXiv	arXiv reference for the backend
author	Author of the backend
constraint_model	retreive constraint model distribution
doi	Citable DOI for the backend
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