Abstract:
In mixture model-based clustering applications, it is common to fit several models from a family and report clustering results from only the `best' one. In such circumstances, selection of this best model is achieved using a model selection criterion, most often the Bayesian information criterion. Rather than throw away all but the best model, we average multiple models that are in some sense close to the best one, thereby producing a weighted average of clustering results. Two (weighted) averaging approaches are considered: averaging the component membership probabilities and averaging models. In both cases, Occam's window is used to determine closeness to the best model and weights are computed within a Bayesian model averaging paradigm. In some cases, we need to merge components before averaging; we introduce a method for merging mixture components based on the adjusted Rand index. The effectiveness of our model-based clustering averaging approaches is illustrated using a family of Gaussian mixture models on real and simulated data.

Abstract:
Bayesian Model Averaging is a weighted averaging method based on posterior distribution. It considers comprehensively the prior and sample information of model and parameter, reduces the model uncertainty. Bayesian Model Averaging improves statistical inference accuracy and provides improved out-of-sample predictive performance. In this paper, we outline the details of the Bayesian model averaging principle, introduce the application of Bayesian Model Averaging in macroeconomy and give an example about the application of Bayesian Model Averaging in GDP research. Key words: Bayesian Model Averaging; Model uncertainty; Macroeconomy

Abstract:
An accurate and simple model of stacking fault energy for alloys (solid solutions) has beendeveloped based on the embedded-atom method. The calculated value of stacking fault energy35 mJ/m2 for 304 austenitic stainless steel, is in a good agreement with the experimental one,30 mJ/m2. In the present model we find that the Hirth's empirical relationship is also suitablefor alloy.

Abstract:
This paper considers the volume averaging in the quasispherical Szekeres model. The volume averaging became of considerable interest after it was shown that the volume acceleration calculated within the averaging framework can be positive even though the local expansion rate is always decelerating. This issue was intensively studied within spherically symmetric models. However, since our Universe is not spherically symmetric similar analysis is needed in non symmetrical models. This papers presents the averaging analysis within the quasispherical Szekeres model which is a non-symmetrical generalisation of the spherically symmetric Lema\^itre--Tolman family of models. Density distribution in the quasispherical Szekeres has a structure of a time-dependent mass dipole superposed on a monopole. This paper shows that when calculating the volume acceleration, $\ddot{a}$, within the Szekeres model, the dipole does not contribute to the final result, hence $\ddot{a}$ only depends on a monopole configuration. Thus, the volume averaging within the Szekeres model leads to literally the same solutions as obtained within the Lema\^itre--Tolman model.

Abstract:
This paper presents recent developments in model selection and model averaging for parametric and nonparametric models. While there is extensive literature on model selection under parametric settings, we present recently developed results in the context of nonparametric models. In applications, estimation and inference are often conducted under the selected model without considering the uncertainty from the selection process. This often leads to inefficiency in results and misleading confidence intervals. Thus an alternative to model selection is model averaging where the estimated model is the weighted sum of all the submodels. This reduces model uncertainty. In recent years, there has been significant interest in model averaging and some important developments have taken place in this area. We present results for both the parametric and nonparametric cases. Some possible topics for future research are also indicated.

Abstract:
Autoresonance is a phase locking phenomenon occurring in nonlinear oscislatory system, which is forced by oscillating perturbation. Many physical applicatcons of the autoresonance are known in nonlenear physics. The essence of the phenomecon is that the nonlinear oscillator selfadjusts to the varying external conditions so that it remains in resonance with the orivvr for a long time. This long time resonance leads to a strong increase in the response amplitude under weak drivinc perturbation. An analytic treatment of a simple mathematical model is done here by means of asymptotic analysis using a small driving parameter. The main result is finding threshold for entering the autoresonance.

Abstract:
Although model selection is routinely used in practice nowadays, little is known about its precise effects on any subsequent inference that is carried out. The same goes for the effects induced by the closely related technique of model averaging. This paper is concerned with the use of the same data first to select a model and then to carry out inference, in particular point estimation and point prediction. The properties of the resulting estimator, called a post-model-selection estimator (PMSE), are hard to derive. Using selection criteria such as hypothesis testing, AIC, BIC, HQ and Cp, we illustrate that, in terms of risk function, no single PMSE dominates the others. The same conclusion holds more generally for any penalised likelihood information criterion. We also compare various model averaging schemes and show that no single one dominates the others in terms of risk function. Since PMSEs can be regarded as a special case of model averaging, with 0-1 random-weights, we propose a connection between the two theories, in the frequentist approach, by taking account of the selection procedure when performing model averaging. We illustrate the point by simulating a simple linear regression model.

Abstract:
We review the use of Bayesian Model Averaging in astrophysics. We first introduce the statistical basis of Bayesian Model Selection and Model Averaging. We discuss methods to calculate the model-averaged posteriors, including Markov Chain Monte Carlo (MCMC), nested sampling, Population Monte Carlo, and Reversible Jump MCMC. We then review some applications of Bayesian Model Averaging in astrophysics, including measurements of the dark energy and primordial power spectrum parameters in cosmology, cluster weak lensing and Sunyaev-Zel'dovich effect data, estimating distances to Cepheids, and classifying variable stars.

Abstract:
The model averaging problem is to average multiple models to achieve a prediction accuracy not much worse than that of the best single model in terms of mean squared error. It is known that if the models are misspecified, model averaging is superior to model selection. Specifically, let $n$ be the sample size, then the worst case regret of the former decays at a rate of $O(1/n)$ while the worst case regret of the latter decays at a rate of $O(1/\sqrt{n})$. The recently proposed $Q$-aggregation algorithm \citep{DaiRigZhang12} solves the model averaging problem with the optimal regret of $O(1/n)$ both in expectation and in deviation; however it suffers from two limitations: (1) for continuous dictionary, the proposed greedy algorithm for solving $Q$-aggregation is not applicable; (2) the formulation of $Q$-aggregation appears ad hoc without clear intuition. This paper examines a different approach to model averaging by considering a Bayes estimator for deviation optimal model averaging by using exponentiated least squares loss. We establish a primal-dual relationship of this estimator and that of $Q$-aggregation and propose new greedy procedures that satisfactorily resolve the above mentioned limitations of $Q$-aggregation.

Abstract:
In several instances of statistical
practice, it is not uncommon to use the same data for both model selection and
inference, without taking account of the variability induced by model selection
step. This is usually referred to as post-model selection inference. The
shortcomings of such practice are widely recognized, finding a general solution
is extremely challenging. We propose a model averaging alternative consisting
on taking into account model selection probability and the like-lihood in
assigning the weights. The approach is applied to Bernoulli trials and
outperforms Akaike weights model averaging and post-model selection estimators.