Posterior Covariance Information Criterion for Weighted Inference.

For predictive evaluation based on quasi-posterior distributions, we develop a new information criterion, the posterior covariance information criterion (PCIC). PCIC generalizes the widely applicable information criterion (WAIC) so as to effectively handle predictive scenarios where likelihoods for the estimation and the evaluation of the model may be different. A typical example of such scenarios is the weighted likelihood inference, including prediction under covariate shift and counterfactual prediction. The proposed criterion uses a posterior covariance form and is computed by using only one Markov chain Monte Carlo run. Through numerical examples, we demonstrate how PCIC can apply in practice. Further, we show that PCIC is asymptotically unbiased to the quasi-Bayesian generalization error under mild conditions in weighted inference with both regular and singular statistical models.

Bayesian estimation of phase response curves.

Phase response curve (PRC) of an oscillatory neuron describes the response of the neuron to external perturbation. The PRC is useful to predict synchronized dynamics of neurons; hence, its measurement from experimental data attracts increasing interest in neural science. This paper introduces a Bayesian method for estimating PRCs from data, which allows for the correlation of errors in explanatory and response variables of the PRC. The method is implemented with a replica exchange Monte Carlo technique; this avoids local minima and enables efficient calculation of posterior averages. A test with artificial data generated by the noisy Morris-Lecar equation shows that the proposed method outperforms conventional regression that ignores errors in the explanatory variable. Experimental data from the pyramidal cells in the rat motor cortex is also analyzed with the method; a case is found where the result with the proposed method is considerably different from that obtained by conventional regression.

Multicanonical sampling of rare events in random matrices.

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble, sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of ∼10(-200), a region where simple random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.