from .. import get_backend
from .mle import fixed_poi_fit, fit
[docs]def qmu(mu, data, pdf, init_pars, par_bounds):
r"""
The test statistic, :math:`q_{\mu}`, for establishing an upper
limit on the strength parameter, :math:`\mu`, as defiend in
Equation (14) in :xref:`arXiv:1007.1727`.
.. math::
:nowrap:
\begin{equation}
q_{\mu} = \left\{\begin{array}{ll}
-2\ln\lambda\left(\mu\right), &\hat{\mu} < \mu,\\
0, & \hat{\mu} > \mu
\end{array}\right.
\end{equation}
Example:
>>> import pyhf
>>> pyhf.set_backend("numpy")
>>> model = pyhf.simplemodels.hepdata_like(
... signal_data=[12.0, 11.0], bkg_data=[50.0, 52.0], bkg_uncerts=[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()
>>> pyhf.infer.test_statistics.qmu(test_mu, data, model, init_pars, par_bounds)
3.938244920380498
Args:
mu (Number or Tensor): The signal strength parameter
data (Tensor): The data to be considered
pdf (~pyhf.pdf.Model): The HistFactory statistical model used in the likelihood ratio calculation
init_pars (`list`): Values to initialize the model parameters at for the fit
par_bounds (`list` of `list`\s or `tuple`\s): The extrema of values the model parameters are allowed to reach in the fit
Returns:
Float: The calculated test statistic, :math:`q_{\mu}`
"""
tensorlib, optimizer = get_backend()
mubhathat, fixed_poi_fit_lhood_val = fixed_poi_fit(
mu, data, pdf, init_pars, par_bounds, return_fitted_val=True
)
muhatbhat, unconstrained_fit_lhood_val = fit(
data, pdf, init_pars, par_bounds, return_fitted_val=True
)
qmu = fixed_poi_fit_lhood_val - unconstrained_fit_lhood_val
qmu = tensorlib.where(
muhatbhat[pdf.config.poi_index] > mu, tensorlib.astensor(0.0), qmu
)[0]
return tensorlib.clip(qmu, 0, max_value=None)