# pyhf.infer.test_statistics.tmu_tilde

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

The test statistic, $$\tilde{t}_{\mu}$$, for establishing a two-sided interval on the strength parameter, $$\mu$$, for models with bounded POI, as defined in Equation (11) in [1007.1727]

$\tilde{t}_{\mu} = -2\ln\tilde{\lambda}\left(\mu\right)$

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

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

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.tmu_tilde(
...     test_mu, data, model, init_pars, par_bounds, fixed_params
... )
array(3.93824492)
>>> pyhf.infer.test_statistics.tmu_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 (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, $$\tilde{t}_{\mu}$$

• 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