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)[source]#

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

$$\begin{array}{r}\begin{array}{rl}{C}_i& =-sign(m_i^{(3)})\sqrt{2m_{ii}^{(2)}}\cos (\frac{4\pi }{3}+\frac{1}{3}\arctan \left(\sqrt{\frac{8(m_{ii}^{(2)}{)}^3}{(m_i^{(3)}{)}^2}-1}\right))\\ B_i& =\sqrt{m_{ii}^{(2)}-2C_i^2}\\ A_i& =m_i^{(1)}-C_i\\ {\rho }_{ij}& =\frac{1}{4C_i{C}_j}(\sqrt{(B_i{B}_j{)}^2+8C_i{C}_j{m}_{ij}^{(2)}}-B_i{B}_j)\end{array}\end{array}$$

which further modifies ${\lambda }_i(\mu ,\theta )=\mu n_{sig}^i+A_i+B_i{\theta }_i+C_i{\theta }_i^2$ 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)[source]#

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