Second-order asymptotic theory of predictive distributions is investigated. It is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model. The model is assumed to be a multidimensional curved exponential family. Predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model. The average Kullback-Leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components. One of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model. The optimal shift orthogonal to the model is obtained. A quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced. The difference in average Kullback-Leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model. An asymptotic expression for Bayesian predictive distributions is also obtained.

We investigate shrinkage methods for constructing predictive distributions. We consider the multivariate Normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the Bayesian predictive distribution based on the vague prior when the dimension is not less than three. Kullback–Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function.

We investigate shrinkage priors for constructing Bayesian predictive distributions. It is shown that there exist shrinkage predictive distributions asymptotically dominating Bayesian predictive distributions based on the Jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions. Kullback–Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function. Conformal transformations of model manifolds corresponding to vague priors are introduced. We show several examples where shrinkage predictive distributions dominate Bayesian predictive distributions based on vague priors.

Simultaneous predictive distributions for independent Poisson observables are investigated. A class of improper prior distributions for Poisson means is introduced. The Bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the Kullback–Leibler loss. A Bayesian predictive distribution based on a prior in this class dominates the Bayesian predictive distribution based on the Jeffreys prior.

Many time series are naturally considered as a superposition of several oscillation components. For example, electroencephalogram (EEG) time series include oscillation components such as alpha, beta, and gamma. We propose a method for decomposing time series into such oscillation components using state-space models. Based on the concept of random frequency modulation, gaussian linear state-space models for oscillation components are developed. In this model, the frequency of an oscillator fluctuates by noise. Time series decomposition is accomplished by this model like the Bayesian seasonal adjustment method. Since the model parameters are estimated from data by the empirical Bayes’ method, the amplitudes and the frequencies of oscillation components are determined in a data-driven manner. Also, the appropriate number of oscillation components is determined with the Akaike information criterion (AIC). In this way, the proposed method provides a natural decomposition of the given time series into oscillation components. In neuroscience, the phase of neural time series plays an important role in neural information processing. The proposed method can be used to estimate the phase of each oscillation component and has several advantages over a conventional method based on the Hilbert transform. Thus, the proposed method enables an investigation of the phase dynamics of time series. Numerical results show that the proposed method succeeds in extracting intermittent oscillations like ripples and detecting the phase reset phenomena. We apply the proposed method to real data from various fields such as astronomy, ecology, tidology, and neuroscience.