pygsti.tools.likelihoodfns.logl_approximate_hessian

pygsti.tools.likelihoodfns.logl_approximate_hessian#

logl_approximate_hessian(model, dataset, circuits=None, min_prob_clip=1e-06, prob_clip_interval=(-1000000.0, 1000000.0), radius=0.0001, poisson_picture=True, op_label_aliases=None, mdc_store=None, comm=None, mem_limit=None, verbosity=0)#

An approximate Hessian of the log-likelihood function.

An approximation to the true Hessian is computed using just the Jacobian (and not the Hessian) of the probabilities w.r.t. the model parameters. Let J = d(probs)/d(params) and denote the Hessian of the log-likelihood w.r.t. the probabilities as d2(logl)/dprobs2 (a diagonal matrix indexed by the term, i.e. probability, of the log-likelihood). Then this function computes:

H = J * d2(logl)/dprobs2 * J.T

Which simply neglects the d2(probs)/d(params)2 terms of the true Hessian. Since this curvature is expected to be small at the MLE point, this approximation can be useful for computing approximate error bars.

Parameters:

  • model (Model) – Model of parameterized gates (including SPAM)

  • dataset (DataSet) – Probability data

  • circuits (list of (tuples or Circuits), optional) – Each element specifies a circuit to include in the log-likelihood sum. Default value of None implies all the circuits in dataset should be used.

  • min_prob_clip (float, optional) – The minimum probability treated normally in the evaluation of the log-likelihood. A penalty function replaces the true log-likelihood for probabilities that lie below this threshold so that the log-likelihood never becomes undefined (which improves optimizer performance).

  • prob_clip_interval (2-tuple or None, optional) – (min,max) values used to clip the probabilities predicted by models during MLEGST’s search for an optimal model (if not None). if None, no clipping is performed.

  • radius (float, optional) – Specifies the severity of rounding used to “patch” the zero-frequency terms of the log-likelihood.

  • poisson_picture (boolean, optional) – Whether the Poisson-picutre log-likelihood should be differentiated.

  • op_label_aliases (dictionary, optional) – Dictionary whose keys are operation label “aliases” and whose values are tuples corresponding to what that operation label should be expanded into before querying the dataset. Defaults to the empty dictionary (no aliases defined) e.g. op_label_aliases[‘Gx^3’] = (‘Gx’,’Gx’,’Gx’)

  • mdc_store (ModelDatasetCircuitsStore, optional) – An object that bundles cached quantities along with a given model, dataset, and circuit list. If given, model and dataset and circuits should be set to None.

  • comm (mpi4py.MPI.Comm, optional) – When not None, an MPI communicator for distributing the computation across multiple processors.

  • mem_limit (int, optional) – A rough memory limit in bytes which restricts the amount of intermediate values that are computed and stored.

  • verbosity (int, optional) – How much detail to print to stdout.

Returns:

On the root processor, the approximate Hessian matrix of shape (nModelParams, nModelParams), where nModelParams = model.num_params. None on non-root processors.

Return type:

numpy array or None