pyhf.infer.test_statistics.qmu_tilde

pyhf.infer.test_statistics.qmu_tilde(mu, data, pdf, init_pars, par_bounds, fixed_params, return_fitted_pars=False)[source]

The “alternative” test statistic, \(\tilde{q}_{\mu}\), for establishing an upper limit on the strength parameter, \(\mu\), for models with bounded POI, as defined in Equation (16) in [1007.1727]

\begin{equation} \tilde{q}_{\mu} = \left\{\begin{array}{ll} -2\ln\tilde{\lambda}\left(\mu\right), &\hat{\mu} \leq \mu,\\ 0, & \hat{\mu} > \mu \end{array}\right. \end{equation}

where \(\tilde{\lambda}\left(\mu\right)\) is the constrained profile likelihood ratio as defined in Equation (10)

\begin{equation} \tilde{\lambda}\left(\mu\right) = \left\{\begin{array}{ll} \frac{L\left(\mu, \hat{\hat{\boldsymbol{\theta}}}(\mu)\right)}{L\left(\hat{\mu}, \hat{\hat{\boldsymbol{\theta}}}(0)\right)}, &\hat{\mu} < 0,\\ \frac{L\left(\mu, \hat{\hat{\boldsymbol{\theta}}}(\mu)\right)}{L\left(\hat{\mu}, \hat{\boldsymbol{\theta}}\right)}, &\hat{\mu} \geq 0. \end{array}\right. \end{equation}

Example

>>> import pyhf
>>> pyhf.set_backend("numpy")
>>> model = pyhf.simplemodels.uncorrelated_background(
...     signal=[12.0, 11.0], bkg=[50.0, 52.0], bkg_uncertainty=[3.0, 7.0]
... )
>>> observations = [51, 48]
>>> data = pyhf.tensorlib.astensor(observations + model.config.auxdata)
>>> test_mu = 1.0
>>> init_pars = model.config.suggested_init()
>>> par_bounds = model.config.suggested_bounds()
>>> fixed_params = model.config.suggested_fixed()
>>> pyhf.infer.test_statistics.qmu_tilde(
...     test_mu, data, model, init_pars, par_bounds, fixed_params
... )
array(3.93824492)
>>> pyhf.infer.test_statistics.qmu_tilde(
...     test_mu, data, model, init_pars, par_bounds, fixed_params, return_fitted_pars=True
... )
(array(3.93824492), (array([1.        , 0.97224597, 0.87553894]), array([0.        , 1.0030512 , 0.96266961])))
Parameters:

  • mu (Number or Tensor) – The signal strength parameter

  • data (tensor) – The data to be considered

  • pdf (Model) – The statistical model adhering to the schema model.json

  • init_pars (list of float) – The starting values of the model parameters for minimization.

  • par_bounds (list of list/tuple) – The extrema of values the model parameters are allowed to reach in the fit. The shape should be (n, 2) for n model parameters.

  • fixed_params (tuple or list of bool) – The flag to set a parameter constant to its starting value during minimization.

  • return_fitted_pars (bool) – Return the best-fit parameter tensors the fixed-POI and unconstrained fits have converged on (i.e. \(\mu, \hat{\hat{\theta}}\) and \(\hat{\mu}, \hat{\theta}\))

Returns:

  • The calculated test statistic, \(\tilde{q}_{\mu}\)

  • The parameter tensors corresponding to the constrained best fit, \(\mu, \hat{\hat{\theta}}\), and the unconstrained best fit, \(\hat{\mu}, \hat{\theta}\). Only returned if return_fitted_pars is True.

Return type:

Tuple of a Float and a Tuple of Tensors