@article{SchuettHarmelingMackeetal.2016,
  author    = {Sch{\"u}tt, Heiko Herbert and Harmeling, Stefan and Macke, Jakob H. and Wichmann, Felix A.},
  title     = {Painfree and accurate Bayesian estimation of psychometric functions for (potentially) overdispersed data},
  series = {Vision research : an international journal for functional aspects of vision.},
  volume    = {122},
  journal   = {Vision research : an international journal for functional aspects of vision.},
  publisher = {Elsevier},
  address   = {Oxford},
  issn      = {0042-6989},
  doi       = {10.1016/j.visres.2016.02.002},
  pages     = {105 -- 123},
  year      = {2016},
  abstract  = {The psychometric function describes how an experimental variable, such as stimulus strength, influences the behaviour of an observer. Estimation of psychometric functions from experimental data plays a central role in fields such as psychophysics, experimental psychology and in the behavioural neurosciences. Experimental data may exhibit substantial overdispersion, which may result from non-stationarity in the behaviour of observers. Here we extend the standard binomial model which is typically used for psychometric function estimation to a beta-binomial model. We show that the use of the beta-binomial model makes it possible to determine accurate credible intervals even in data which exhibit substantial overdispersion. This goes beyond classical measures for overdispersion goodness-of-fit which can detect overdispersion but provide no method to do correct inference for overdispersed data. We use Bayesian inference methods for estimating the posterior distribution of the parameters of the psychometric function. Unlike previous Bayesian psychometric inference methods our software implementation-psignifit 4 performs numerical integration of the posterior within automatically determined bounds. This avoids the use of Markov chain Monte Carlo (MCMC) methods typically requiring expert knowledge. Extensive numerical tests show the validity of the approach and we discuss implications of overdispersion for experimental design. A comprehensive MATLAB toolbox implementing the method is freely available; a python implementation providing the basic capabilities is also available. (C) 2016 The Authors. Published by Elsevier Ltd.},
  language  = {en}
}
@article{GianniotisSchnoerrMolkenthinetal.2016,
  author    = {Gianniotis, Nikolaos and Schnoerr, Christoph and Molkenthin, Christian and Bora, Sanjay Singh},
  title     = {Approximate variational inference based on a finite sample of Gaussian latent variables},
  series = {Pattern Analysis \& Applications},
  volume    = {19},
  journal   = {Pattern Analysis \& Applications},
  publisher = {Springer},
  address   = {New York},
  issn      = {1433-7541},
  doi       = {10.1007/s10044-015-0496-9},
  pages     = {475 -- 485},
  year      = {2016},
  abstract  = {Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower bound on the desired integral to be approximated, e.g. marginal likelihood. The lower bound is then optimised with respect to its free parameters, the so-called variational parameters. However, this is not always possible as for certain integrals it is very challenging (or tedious) to come up with a suitable lower bound. Here, we propose a simple scheme that overcomes some of the awkward cases where the usual variational treatment becomes difficult. The scheme relies on a rewriting of the lower bound on the model log-likelihood. We demonstrate the proposed scheme on a number of synthetic and real examples, as well as on a real geophysical model for which the standard variational approaches are inapplicable.},
  language  = {en}
}