On asymptotic validity of naive inference with an approximate likelihood

Many statistical models have likelihoods which are intractable: it is impossible or too expensive to compute the likelihood exactly. In such settings, a common approach is to replace the likelihood with an approximation, and proceed with inference as if the approximate likelihood were the exact likelihood. In this paper, we describe conditions on the approximate likelihood which guarantee that this naive inference with an approximate likelihood has the same first-order asymptotic properties as inference with the exact likelihood. We investigate the implications of these results for inference using a Laplace approximation to the likelihood in a simple two-level latent variable model, and using reduced dependence approximations to the likelihood in an Ising model on a lattice.Comment: Updated to add an additional example (inference for an Ising model on a lattice using reduced dependence approximations to the likelihood

Bounds for Hamiltonians with arbitrary kinetic parts

A method is presented to compute approximate solutions for eigenequations in quantum mechanics with an arbitrary kinetic part. In some cases, the approximate eigenvalues can be analytically determined and they can be lower or upper bounds. A semiclassical interpretation of the generic formula obtained for the eigenvalues supports a new definition of the effective particle mass used in solid state physics. An analytical toy model with a Gaussian dependence in the momentum is studied in order to check the validity of the method.Comment: Improved version with new refernce

Nonequilibrium Fields: Exact and Truncated Dynamics

The nonperturbative real-time evolution of quantum fields out of equilibrium is often solved using a mean-field or Hartree approximation or by applying effective action methods. In order to investigate the validity of these truncations, we implement similar methods in classical scalar field theory and compare the approximate dynamics with the full nonlinear evolution. Numerical results are shown for the early-time behaviour, the role of approximate fixed points, and thermalization.Comment: 5 pages, 6 eps figures, talk presented at Strong and Electroweak Matter (SEWM2000), Marseille, France, 14-17 June, 200

Statistical Mechanics of Learning: A Variational Approach for Real Data

Using a variational technique, we generalize the statistical physics approach of learning from random examples to make it applicable to real data. We demonstrate the validity and relevance of our method by computing approximate estimators for generalization errors that are based on training data alone.Comment: 4 pages, 2 figure

Effective Potential and Quantum Dynamical Correlators

The approach to the calculation of quantum dynamical correlation functions is presented in the framework of the Mori theory. An unified treatment of classic and quantum dynamics is given in terms of Weyl representation of operators and holomorphic variables. The range of validity of an approximate molucular dynamics is discussedComment: 8 pages, Latex fil

Non-Markovian dissipative dynamics of two coupled qubits in independent reservoirs: a comparison between exact solutions and master equation approaches

The reduced dynamics of two interacting qubits coupled to two independent bosonic baths is investigated. The one-excitation dynamics is derived and compared with that based on the resolution of appropriate non-Markovian master equations. The Nakajima-Zwanzig and the time-convolutionless projection operator techniques are exploited to provide a description of the non-Markovian features of the dynamics of the two-qubits system. The validity of such approximate methods and their range of validity in correspondence to different choices of the parameters describing the system are brought to light.Comment: 6 pages, 3 figures. Submitted to PR

Assessing the Approximate Validity of Moment Restrictions

We propose a new theoretical framework to assess the approximate validity of overidentifying moment restrictions. Their validity is evaluated by the divergence between the true probability measure and the closest measure that imposes the moment restrictions of interest. The divergence can be chosen as any of the Cressie-Read family. The considered alternative hypothesis states that the divergence is smaller than some user-chosen tolerance. Tests are constructed based on the minimum empirical divergence that attain the local semiparametric power envelope of invariant tests. We show how the tolerance can be chosen by reformulating the hypothesis under test as a set of admissible misspecifications. Two empirical applications illustrate the practical usefulness of the new tests for providing evidence on the potential extent of misspecification