from .test_statistics import qmu
from .utils import (
generate_asimov_data,
pvals_from_teststat,
pvals_from_teststat_expected,
)
from .. import get_backend
[docs]def hypotest(
poi_test, data, pdf, init_pars=None, par_bounds=None, qtilde=False, **kwargs
):
r"""
Computes :math:`p`-values and test statistics for a single value of the parameter of interest
Args:
poi_test (Number or Tensor): The value of the parameter of interest (POI)
data (Number or Tensor): The root of the calculated test statistic given the Asimov data, :math:`\sqrt{q_{\mu,A}}`
pdf (~pyhf.pdf.Model): The HistFactory statistical model
init_pars (Array or Tensor): The initial parameter values to be used for minimization
par_bounds (Array or Tensor): The parameter value bounds to be used for minimization
qtilde (Bool): When ``True`` perform the calculation using the alternative test statistic, :math:`\tilde{q}`, as defined in Equation (62) of :xref:`arXiv:1007.1727`
Keyword Args:
return_tail_probs (bool): Bool for returning :math:`\textrm{CL}_{s+b}` and :math:`\textrm{CL}_{b}`
return_expected (bool): Bool for returning :math:`\textrm{CL}_{\textrm{exp}}`
return_expected_set (bool): Bool for returning the :math:`(-2,-1,0,1,2)\sigma` :math:`\textrm{CL}_{\textrm{exp}}` --- the "Brazil band"
return_test_statistics (bool): Bool for returning :math:`q_{\mu}` and :math:`q_{\mu,A}`
Returns:
Tuple of Floats and lists of Floats:
- :math:`\textrm{CL}_{s}`: The :math:`p`-value compared to the given threshold :math:`\alpha`, typically taken to be :math:`0.05`, defined in :xref:`arXiv:1007.1727` as
.. math::
\textrm{CL}_{s} = \frac{\textrm{CL}_{s+b}}{\textrm{CL}_{b}} = \frac{p_{s+b}}{1-p_{b}}
to protect against excluding signal models in which there is little sensitivity. In the case that :math:`\textrm{CL}_{s} \leq \alpha` the given signal model is excluded.
- :math:`\left[\textrm{CL}_{s+b}, \textrm{CL}_{b}\right]`: The signal + background :math:`p`-value and 1 minus the background only :math:`p`-value as defined in Equations (75) and (76) of :xref:`arXiv:1007.1727`
.. math::
\textrm{CL}_{s+b} = p_{s+b} = \int\limits_{q_{\textrm{obs}}}^{\infty} f\left(q\,\middle|s+b\right)\,dq = 1 - \Phi\left(\frac{q_{\textrm{obs}} + 1/\sigma_{s+b}^{2}}{2/\sigma_{s+b}}\right)
.. math::
\textrm{CL}_{b} = 1- p_{b} = 1 - \int\limits_{-\infty}^{q_{\textrm{obs}}} f\left(q\,\middle|b\right)\,dq = 1 - \Phi\left(\frac{q_{\textrm{obs}} - 1/\sigma_{b}^{2}}{2/\sigma_{b}}\right)
with Equations (73) and (74) for the mean
.. math::
E\left[q\right] = \frac{1 - 2\mu}{\sigma^{2}}
and variance
.. math::
V\left[q\right] = \frac{4}{\sigma^{2}}
of the test statistic :math:`q` under the background only and and signal + background hypotheses. Only returned when ``return_tail_probs`` is ``True``.
- :math:`\textrm{CL}_{s,\textrm{exp}}`: The expected :math:`\textrm{CL}_{s}` value corresponding to the test statistic under the background only hypothesis :math:`\left(\mu=0\right)`. Only returned when ``return_expected`` is ``True``.
- :math:`\textrm{CL}_{s,\textrm{exp}}` band: The set of expected :math:`\textrm{CL}_{s}` values corresponding to the median significance of variations of the signal strength from the background only hypothesis :math:`\left(\mu=0\right)` at :math:`(-2,-1,0,1,2)\sigma`. That is, the :math:`p`-values that satisfy Equation (89) of :xref:`arXiv:1007.1727`
.. math::
\textrm{band}_{N\sigma} = \mu' + \sigma\,\Phi^{-1}\left(1-\alpha\right) \pm N\sigma
for :math:`\mu'=0` and :math:`N \in \left\{-2, -1, 0, 1, 2\right\}`. These values define the boundaries of an uncertainty band sometimes referred to as the "Brazil band". Only returned when ``return_expected_set`` is ``True``.
- :math:`\left[q_{\mu}, q_{\mu,A}\right]`: The test statistics for the observed and Asimov datasets respectively. Only returned when ``return_test_statistics`` is ``True``.
"""
init_pars = init_pars or pdf.config.suggested_init()
par_bounds = par_bounds or pdf.config.suggested_bounds()
tensorlib, _ = get_backend()
asimov_mu = 0.0
asimov_data = generate_asimov_data(asimov_mu, data, pdf, init_pars, par_bounds)
qmu_v = tensorlib.clip(
qmu(poi_test, data, pdf, init_pars, par_bounds), 0, max_value=None
)
sqrtqmu_v = tensorlib.sqrt(qmu_v)
qmuA_v = tensorlib.clip(
qmu(poi_test, asimov_data, pdf, init_pars, par_bounds), 0, max_value=None
)
sqrtqmuA_v = tensorlib.sqrt(qmuA_v)
CLsb, CLb, CLs = pvals_from_teststat(sqrtqmu_v, sqrtqmuA_v, qtilde=qtilde)
_returns = [CLs]
if kwargs.get('return_tail_probs'):
_returns.append([CLsb, CLb])
if kwargs.get('return_expected_set'):
CLs_exp = []
for n_sigma in [-2, -1, 0, 1, 2]:
CLs_exp.append(pvals_from_teststat_expected(sqrtqmuA_v, nsigma=n_sigma)[-1])
CLs_exp = tensorlib.astensor(CLs_exp)
if kwargs.get('return_expected'):
_returns.append(CLs_exp[2])
_returns.append(CLs_exp)
elif kwargs.get('return_expected'):
_returns.append(pvals_from_teststat_expected(sqrtqmuA_v)[-1])
if kwargs.get('return_test_statistics'):
_returns.append([qmu_v, qmuA_v])
# Enforce a consistent return type of the observed CLs
return tuple(_returns) if len(_returns) > 1 else _returns[0]
__all__ = ['qmu', 'hypotest']