### Merge branch 'docstrings_torsten' into 'NIFTy_5'

Docstrings torsten

See merge request ift/nifty-dev!167
parents 4f692156 661d1746
 ... ... @@ -19,7 +19,7 @@ There is a full toolbox of methods that can be used, like the classical approxim ..  T.A. Enßlin (2014), "Astrophysical data analysis with information field theory", AIP Conference Proceedings, Volume 1636, Issue 1, p.49; arXiv:1405.7701 _ ..  Wikipedia contributors (2018), "Information field theory" _, Wikipedia, The Free Encyclopedia. ..  Wikipedia contributors (2018), "Information field theory" _, Wikipedia, The Free Encyclopedia. ..  T.A. Enßlin (2019), "Information theory for fields", accepted by Annalen der Physik; arXiv:1804.03350 _ ... ... @@ -85,7 +85,7 @@ The above line of argumentation analogously applies to the discretization of ope The proper discretization of spaces, fields, and operators, as well as the normalization of position integrals, is essential for the conservation of the continuum limit. Their consistent implementation in NIFTY allows a pixelization independent coding of algorithms. Free Theory & Implicit Operators Free Theory & Implicit Operators -------------------------------- A free IFT appears when the signal field :math:{s} and the noise :math:{n} of the data :math:{d} are independent, zero-centered Gaussian processes of kown covariances :math:{S} and :math:{N}, respectively, ... ... @@ -94,7 +94,7 @@ A free IFT appears when the signal field :math:{s} and the noise :math:{n} o \mathcal{P}(s,n) = \mathcal{G}(s,S)\,\mathcal{G}(n,N), and the measurement equation is linear in both, signal and noise, and the measurement equation is linear in both signal and noise, .. math:: ... ... @@ -109,15 +109,15 @@ associate professor \mathcal{H}(d,s)= -\log \mathcal{P}(d,s)= \frac{1}{2} s^\dagger S^{-1} s + \frac{1}{2} (d-R\,s)^\dagger N^{-1} (d-R\,s) + \mathrm{const} is only of quadratic order in :math:{s}, which leads to a linear relation between the data and the posterior mean field. is only of quadratic order in :math:{s}, which leads to a linear relation between the data and the posterior mean field. In this case, the posterior is In this case, the posterior is .. math:: \mathcal{P}(s|d) = \mathcal{G}(s-m,D) with with .. math:: ... ... @@ -129,7 +129,7 @@ the posterior mean field, D = \left( S^{-1} + R^\dagger N^{-1} R\right)^{-1} the posterior covariance operator, and the posterior covariance operator, and .. math:: ... ... @@ -137,7 +137,7 @@ the posterior covariance operator, and the information source. The operation in :math:{d= D\,R^\dagger N^{-1} d} is also called the generalized Wiener filter. NIFTy permits to define the involved operators :math:{R}, :math:{R^\dagger}, :math:{S}, and :math:{N} implicitely, as routines that can be applied to vectors, but which do not require the explicit storage of the matrix elements of the operators. NIFTy permits to define the involved operators :math:{R}, :math:{R^\dagger}, :math:{S}, and :math:{N} implicitely, as routines that can be applied to vectors, but which do not require the explicit storage of the matrix elements of the operators. Some of these operators are diagonal in harmonic (Fourier) basis, and therefore only require the specification of a (power) spectrum and :math:{S= F\,\widehat{P_s} F^\dagger}. Here :math:{F = \mathrm{HarmonicTransformOperator}}, :math:{\widehat{P_s} = \mathrm{DiagonalOperator}(P_s)}, and :math:{P_s(k)} is the power spectrum of the process that generated :math:{s} as a function of the (absolute value of the) harmonic (Fourier) space koordinate :math:{k}. For those, NIFTy can easily also provide inverse operators, as :math:{S^{-1}= F\,\widehat{\frac{1}{P_s}} F^\dagger} in case :math:{F} is unitary, :math:{F^\dagger=F^{-1}}. ... ... @@ -170,7 +170,7 @@ The joint information Hamiltonian for the whitened signal field :math:{\xi} re \mathcal{H}(d,\xi)= -\log \mathcal{P}(d,s)= \frac{1}{2} \xi^\dagger \xi + \frac{1}{2} (d-R\,A\,\xi)^\dagger N^{-1} (d-R\,A\,\xi) + \mathrm{const}. NIFTy takes advantage of this formulation in several ways: NIFTy takes advantage of this formulation in several ways: 1) All prior degrees of freedom have unit covariance which improves the condition number of operators which need to be inverted. 2) The amplitude operator can be regarded as part of the response, :math:{R'=R\,A}. In general, more sophisticated responses can be constructed out of the composition of simpler operators. ... ...
 ... ... @@ -48,7 +48,7 @@ from .operators.simple_linear_operators import ( FieldAdapter, ducktape, GeometryRemover, NullOperator) from .operators.energy_operators import ( EnergyOperator, GaussianEnergy, PoissonianEnergy, InverseGammaLikelihood, BernoulliEnergy, Hamiltonian, SampledKullbachLeiblerDivergence) BernoulliEnergy, Hamiltonian, AveragedEnergy) from .probing import probe_with_posterior_samples, probe_diagonal, \ StatCalculator ... ...
 ... ... @@ -442,7 +442,7 @@ class Field(object): ---------- spaces : None, int or tuple of int The summation is only carried out over the sub-domains in this tuple. If None, it is carried out over all sub-domains. Default: None. tuple. If None, it is carried out over all sub-domains. Returns ------- ... ... @@ -463,7 +463,6 @@ class Field(object): spaces : None, int or tuple of int The summation is only carried out over the sub-domains in this tuple. If None, it is carried out over all sub-domains. Default: None. Returns ------- ... ... @@ -547,7 +546,7 @@ class Field(object): ---------- spaces : None, int or tuple of int The operation is only carried out over the sub-domains in this tuple. If None, it is carried out over all sub-domains. Default: None. tuple. If None, it is carried out over all sub-domains. Returns ------- ... ...
 ... ... @@ -15,11 +15,14 @@ # # NIFTy is being developed at the Max-Planck-Institut fuer Astrophysik. import numpy as np from .. import utilities from ..domain_tuple import DomainTuple from ..field import Field from ..linearization import Linearization from ..sugar import makeOp, makeDomain from ..sugar import makeDomain, makeOp from .linear_operator import LinearOperator from .operator import Operator from .sampling_enabler import SamplingEnabler from .sandwich_operator import SandwichOperator ... ... @@ -27,11 +30,28 @@ from .simple_linear_operators import VdotOperator class EnergyOperator(Operator): """Operator which has a scalar domain as target domain.""" """Operator which has a scalar domain as target domain. It is intended as an objective function for field inference. Examples -------- - Information Hamiltonian, i.e. negative-log-probabilities. - Gibbs free energy, i.e. an averaged Hamiltonian, aka Kullbach-Leibler divergence. """ _target = DomainTuple.scalar_domain() class SquaredNormOperator(EnergyOperator): """Computes the L2-norm of the output of an operator. Parameters ---------- domain : Domain, DomainTuple or tuple of Domain Target domain of the operator in which the L2-norm shall be computed. """ def __init__(self, domain): self._domain = domain ... ... @@ -45,12 +65,24 @@ class SquaredNormOperator(EnergyOperator): class QuadraticFormOperator(EnergyOperator): def __init__(self, op): """Computes the L2-norm of a Field or MultiField with respect to a specific metric endo. .. math :: E(f) = \\frac12 f^\\dagger \\text{endo}(f) Parameters ---------- endo : EndomorphicOperator Kernel of quadratic form. """ def __init__(self, endo): from .endomorphic_operator import EndomorphicOperator if not isinstance(op, EndomorphicOperator): if not isinstance(endo, EndomorphicOperator): raise TypeError("op must be an EndomorphicOperator") self._op = op self._domain = op.domain self._op = endo self._domain = endo.domain def apply(self, x): self._check_input(x) ... ... @@ -63,7 +95,38 @@ class QuadraticFormOperator(EnergyOperator): class GaussianEnergy(EnergyOperator): """Class for energies of fields with Gaussian probability distribution. Represents up to constants in :math:m: .. math :: E(f) = - \\log G(f-m, D) = 0.5 (f-m)^\\dagger D^{-1} (f-m), an information energy for a Gaussian distribution with mean m and covariance D. Parameters ---------- mean : Field Mean of the Gaussian. Default is 0. covariance : LinearOperator Covariance of the Gaussian. Default is the identity operator. domain : Domain, DomainTuple, tuple of Domain or MultiDomain Operator domain. By default it is inferred from mean or covariance if specified Note ---- At least one of the arguments has to be provided. """ def __init__(self, mean=None, covariance=None, domain=None): if mean is not None and not isinstance(mean, Field): raise TypeError if covariance is not None and not isinstance(covariance, LinearOperator): raise TypeError self._domain = None if mean is not None: self._checkEquivalence(mean.domain) ... ... @@ -90,7 +153,7 @@ class GaussianEnergy(EnergyOperator): def apply(self, x): self._check_input(x) residual = x if self._mean is None else x-self._mean residual = x if self._mean is None else x - self._mean res = self._op(residual).real if not isinstance(x, Linearization) or not x.want_metric: return res ... ... @@ -99,7 +162,29 @@ class GaussianEnergy(EnergyOperator): class PoissonianEnergy(EnergyOperator): """Class for likelihood Hamiltonians of expected count field constrained by Poissonian count data. Represents up to an f-independent term :math:log(d!): .. math :: E(f) = -\\log \\text{Poisson}(d|f) = \\sum f - d^\\dagger \\log(f), where f is a :class:Field in data space with the expectation values for the counts. Parameters ---------- d : Field Data field with counts. Needs to have integer dtype and all field values need to be non-negative. """ def __init__(self, d): if not isinstance(d, Field) or not np.issubdtype(d.dtype, np.integer): raise TypeError if np.any(d.local_data < 0): raise ValueError self._d = d self._domain = DomainTuple.make(d.domain) ... ... @@ -115,7 +200,13 @@ class PoissonianEnergy(EnergyOperator): class InverseGammaLikelihood(EnergyOperator): """ FIXME """ def __init__(self, d): if not isinstance(d, Field): raise TypeError self._d = d self._domain = DomainTuple.make(d.domain) ... ... @@ -131,23 +222,78 @@ class InverseGammaLikelihood(EnergyOperator): class BernoulliEnergy(EnergyOperator): """Computes likelihood energy of expected event frequency constrained by event data. .. math :: E(f) = -\\log \\text{Bernoulli}(d|f) = -d^\\dagger \\log f - (1-d)^\\dagger \\log(1-f), where f is a field defined on d.domain with the expected frequencies of events. Parameters ---------- d : Field Data field with events (1) or non-events (0). """ def __init__(self, d): print(d.dtype) if not isinstance(d, Field) or not np.issubdtype(d.dtype, np.integer): raise TypeError if not np.all(np.logical_or(d.local_data == 0, d.local_data == 1)): raise ValueError self._d = d self._domain = DomainTuple.make(d.domain) def apply(self, x): self._check_input(x) v = x.log().vdot(-self._d) - (1.-x).log().vdot(1.-self._d) v = -(x.log().vdot(self._d) + (1. - x).log().vdot(1. - self._d)) if not isinstance(x, Linearization): return Field.scalar(v) if not x.want_metric: return v met = makeOp(1./(x.val*(1.-x.val))) met = makeOp(1./(x.val*(1. - x.val))) met = SandwichOperator.make(x.jac, met) return v.add_metric(met) class Hamiltonian(EnergyOperator): """Computes an information Hamiltonian in its standard form, i.e. with the prior being a Gaussian with unit covariance. Let the likelihood energy be :math:E_{lh}. Then this operator computes: .. math :: H(f) = 0.5 f^\\dagger f + E_{lh}(f): Other field priors can be represented via transformations of a white Gaussian field into a field with the desired prior probability structure. By implementing prior information this way, the field prior is represented by a generative model, from which NIFTy can draw samples and infer a field using the Maximum a Posteriori (MAP) or the Variational Bayes (VB) method. The metric of this operator can be used as covariance for drawing Gaussian samples. Parameters ---------- lh : EnergyOperator The likelihood energy. ic_samp : IterationController Tells an internal :class:SamplingEnabler which convergence criterium to use to draw Gaussian samples. See also -------- Encoding prior knowledge in the structure of the likelihood, Jakob Knollmüller, Torsten A. Ensslin, _ """ def __init__(self, lh, ic_samp=None): self._lh = lh self._prior = GaussianEnergy(domain=lh.domain) ... ... @@ -156,14 +302,14 @@ class Hamiltonian(EnergyOperator): def apply(self, x): self._check_input(x) if (self._ic_samp is None or not isinstance(x, Linearization) or not x.want_metric): return self._lh(x)+self._prior(x) if (self._ic_samp is None or not isinstance(x, Linearization) or not x.want_metric): return self._lh(x) + self._prior(x) else: lhx, prx = self._lh(x), self._prior(x) mtr = SamplingEnabler(lhx.metric, prx.metric.inverse, self._ic_samp, prx.metric.inverse) return (lhx+prx).add_metric(mtr) return (lhx + prx).add_metric(mtr) def __repr__(self): subs = 'Likelihood:\n{}'.format(utilities.indent(self._lh.__repr__())) ... ... @@ -171,18 +317,62 @@ class Hamiltonian(EnergyOperator): return 'Hamiltonian:\n' + utilities.indent(subs) class SampledKullbachLeiblerDivergence(EnergyOperator): class AveragedEnergy(EnergyOperator): """Computes Kullbach-Leibler (KL) divergence or Gibbs free energies. A sample-averaged energy, e.g. an Hamiltonian, approximates the relevant part of a KL to be used in Variational Bayes inference if the samples are drawn from the approximating Gaussian: .. math :: \\text{KL}(m) = \\frac1{\\#\{v_i\}} \\sum_{v_i} H(m+v_i), where :math:v_i are the residual samples and :math:m is the mean field around which the samples are drawn. Parameters ---------- h: Hamiltonian The energy to be averaged. res_samples : iterable of Fields Set of residual sample points to be added to mean field for approximate estimation of the KL. Note ---- Having symmetrized residual samples, with both v_i and -v_i being present ensures that the distribution mean is exactly represented. This reduces sampling noise and helps the numerics of the KL minimization process in the variational Bayes inference. See also -------- Let :math:Q(f) = G(f-m,D) be the Gaussian distribution which is used to approximate the accurate posterior :math:P(f|d) with information Hamiltonian :math:H(d,f) = -\\log P(d,f) = -\\log P(f|d) + \\text{const}. In Variational Bayes one needs to optimize the KL divergence between those two distributions for m. It is: :math:KL(Q,P) = \\int Df Q(f) \\log Q(f)/P(f)\\\\ = \\left< \\log Q(f) \\right>_Q(f) - \\left< \\log P(f) \\right>_Q(f)\\\\ = \\text{const} + \\left< H(f) \\right>_G(f-m,D) in essence the information Hamiltonian averaged over a Gaussian distribution centered on the mean m. :class:AveragedEnergy(h) approximates :math:\\left< H(f) \\right>_{G(f-m,D)} if the residuals :math:f-m are drawn from a Gaussian distribution with covariance :math:D. """ def __init__(self, h, res_samples): """ # MR FIXME: does h have to be a Hamiltonian? Couldn't it be any energy? h: Hamiltonian N: Number of samples to be used """ self._h = h self._domain = h.domain self._res_samples = tuple(res_samples) def apply(self, x): self._check_input(x) mymap = map(lambda v: self._h(x+v), self._res_samples) return utilities.my_sum(mymap) * (1./len(self._res_samples)) mymap = map(lambda v: self._h(x + v), self._res_samples) return utilities.my_sum(mymap)*(1./len(self._res_samples))
 ... ... @@ -73,7 +73,7 @@ def test_hamiltonian_and_KL(field): ift.extra.check_value_gradient_consistency(hamiltonian, field) S = ift.ScalingOperator(1., space) samps = [S.draw_sample() for i in range(3)] kl = ift.SampledKullbachLeiblerDivergence(hamiltonian, samps) kl = ift.AveragedEnergy(hamiltonian, samps) ift.extra.check_value_gradient_consistency(kl, field) ... ...
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