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]

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

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

$$\tilde{\lambda }\left(\mu \right)=\{\begin{array}{ll}\frac{L(\mu ,\widehat{\hat{\mathit{\theta }}}(\mu ))}{L(\hat{\mu },\widehat{\hat{\mathit{\theta }}}(0))},& \hat{\mu }<0,\\ \frac{L(\mu ,\widehat{\hat{\mathit{\theta }}}(\mu ))}{L(\hat{\mu },\hat{\mathit{\theta }})},& \hat{\mu }\ge 0.\end{array}$$

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