pyhf.infer.test_statistics.qmu_tilde¶
-
pyhf.infer.test_statistics.
qmu_tilde
(mu, data, pdf, init_pars, par_bounds, fixed_params)[source]¶ The test statistic, \(\tilde{q}_{\mu}\), for establishing an upper limit on the strength parameter, \(\mu\), for models with bounded POI, as defiend in Equation (16) in [1007.1727]
\begin{equation} \tilde{q}_{\mu} = \left\{\begin{array}{ll} -2\ln\tilde{\lambda}\left(\mu\right), &\hat{\mu} < \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.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() >>> 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)
- 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) – Values to initialize the model parameters at for the fit
par_bounds (list of lists or tuples) – The extrema of values the model parameters are allowed to reach in the fit
fixed_params (list) – Parameters held constant in the fit
- Returns
The calculated test statistic, \(\tilde{q}_{\mu}\)
- Return type
Float