spey.backends.default_pdf.UncorrelatedBackground

class spey.backends.default_pdf.UncorrelatedBackground(signal_yields: List[float] | Callable[[ndarray], ndarray], background_yields: List[float], data: List[int], absolute_uncertainties: List[float], modifiers: List[Dict[str, Any]] | None = None, n_signal_parameters: int = 0, signal_parameter_bounds: List[Tuple[float | None, float | None]] | None = None)

Single- or multi-bin simplified likelihood with uncorrelated background uncertainties (default.uncorrelated_background).

Each bin is assigned its own independent Gaussian nuisance parameter \(\theta_i\), scaled by the absolute background uncertainty \(\sigma_i\). The full likelihood is

\[\mathcal{L}(\mu, \boldsymbol{\theta}) = \prod_{i=1}^{N} \mathrm{Poiss}\!\left(n^{\rm obs}_i \,\Big|\, \mu\, n^s_i \cdot f(\boldsymbol{\theta}_{\rm sig}) + n^b_i + \theta_i \sigma_i\right) \cdot \prod_{i=1}^{N} \mathcal{N}(\theta_i \mid 0, 1),\]

where

• \(n^s_i\) — signal yield in bin \(i\),

• \(n^b_i\) — expected background yield in bin \(i\),

• \(\sigma_i\) — absolute background uncertainty in bin \(i\) (e.g. 0.5 for a yield reported as \(3.1 \pm 0.5\)),

• \(f(\boldsymbol{\theta}_{\rm sig})\) — optional signal-uncertainty modifier (1 by default).

The positivity constraint \(n^b_i + \theta_i \sigma_i \geq 0\) is enforced during optimisation via a NonlinearConstraint.

Because the bins are independent the log-likelihood factorises into a sum over bins, making this the fastest backend for quick estimates.

Parameters:

• signal_yields (np.ndarray | Callable[[np.ndarray], np.ndarray]) – Per-bin signal yields \(\{n^s_i\}\), or a callable that accepts the extra signal parameters pars[1 : 1 + n_signal_parameters] and returns the per-bin yields as a np.ndarray.

• background_yields (List[float]) – Per-bin expected background yields \(\{n^b_i\}\).

• data (List[int]) – Per-bin observed counts \(\{n^{\rm obs}_i\}\).

• absolute_uncertainties (List[float]) – Absolute (not relative) background uncertainties \(\{\sigma_i\}\). Must have the same length as the other array inputs.

• modifiers (List[Dict[str, Any]], default None) –

  Optional list of signal-uncertainty modifier configuration dictionaries. Each dictionary must contain:

  • "type" (str, required): morphing mode, either

    • "normalization" — one shared nuisance parameter for all bins (e.g. PDF uncertainties, luminosity);

    • "shape" — one independent nuisance parameter per bin (e.g. scale uncertainties, theory prediction statistics).

  • "uncertainties" (List[float] | List[Tuple[float, float]], required): absolute uncertainty values per bin. Use a flat list[float] for symmetric uncertainties, or a list[(up, down)] for asymmetric ones.

  • "name" (str, optional): label used for parameter naming; defaults to "mod0", "mod1", …

  Not supported when signal_yields is a callable.

  Example:

  modifiers = [
      {"type": "normalization", "name": "pdf",
       "uncertainties": [0.6, 1.0]},
      {"type": "shape", "name": "scale",
       "uncertainties": [(0.3, 0.4), (0.5, 0.6)]},
  ]
  

• n_signal_parameters (int, default 0) – Number of additional free parameters accepted by a callable signal_yields. Has no effect when signal_yields is a plain array. See DefaultPDFBase for the parameter-vector layout.

• signal_parameter_bounds (List[Tuple[Optional[float], Optional[float]]] | None) – Optimiser bounds for each extra signal parameter. Each entry is a (lower, upper) pair; use None for an unbounded side. When None, every extra signal parameter receives (None, None). Must have exactly n_signal_parameters entries when provided.

Note

All input lists must have the same length \(N\) (number of bins/regions).

Example:

>>> import spey
>>> stat_wrapper = spey.get_backend('default.uncorrelated_background')
>>> data = [1, 3]
>>> signal = [0.5, 2.0]
>>> background = [2.0, 2.8]
>>> background_unc = [1.1, 0.8]
>>> stat_model = stat_wrapper(
...     signal, background, data, background_unc, analysis="multi-bin", xsection=0.123
... )
>>> print("CLs : %.3f" % tuple(stat_model.exclusion_confidence_level()))

__init__(signal_yields: List[float] | Callable[[ndarray], ndarray], background_yields: List[float], data: List[int], absolute_uncertainties: List[float], modifiers: List[Dict[str, Any]] | None = None, n_signal_parameters: int = 0, signal_parameter_bounds: List[Tuple[float | None, float | None]] | None = None)

Methods

__init__(signal_yields, background_yields, ...)
asimov_negative_loglikelihood([poi_test, ...])	Compute the profiled negative log-likelihood at fixed \(\mu\) on Asimov data.
combine(other, **kwargs)	Combine this statistical model with another backend instance.
config([allow_negative_signal, poi_upper_bound])	Model configuration.
expected_data(pars[, include_auxiliary])	Compute the expected data vector at the given parameter point.
get_hessian_logpdf_func([expected, data])	Return a callable that evaluates the Hessian of \(\ln\mathcal{L}(\mu, \boldsymbol{\theta})\).
get_logpdf_func([expected, data])	Return a callable that evaluates \(\ln\mathcal{L}(\mu, \boldsymbol{\theta})\).
get_objective_function([expected, data, do_grad])	Return the objective function \(-\ln\mathcal{L}(\mu, \boldsymbol{\theta})\) used by the optimiser.
get_sampler(pars)	Return a callable that draws pseudo-data from the statistical model.
minimize_asimov_negative_loglikelihood([...])	Find the global minimum of the negative log-likelihood on Asimov data (free fit).
minimize_negative_loglikelihood([expected, ...])	Find the global minimum of the negative log-likelihood (free fit).
negative_loglikelihood([poi_test, expected])	Compute the profiled negative log-likelihood at a fixed \(\mu\).

Attributes

constraints	Constraints to be used during optimisation process
signal_uncertainty_configuration
author	Author of the backend
constraint_model	Constraint model distribution \(\mathcal{C}(\boldsymbol{\theta})\).
is_alive	Returns True if at least one bin has non-zero signal yield.
main_model	Main model distribution — the Poisson term of the likelihood.
name	Name of the backend
spey_requires	Spey version required for the backend
version	Version of the backend