In this paper, we develop a novel variational Bayesian learning method for the Dirichlet process (DP) mixture of the inverted Dirichlet distributions, which has been shown to be very flexible for modeling vectors with positive elements. The recently proposed extended variational inference (EVI) framework is adopted to derive an analytically tractable solution. The convergency of the proposed algorithm is theoretically guaranteed by introducing single lower bound approximation to the original objective function in the EVI framework. In principle, the proposed model can be viewed as an infinite inverted Dirichlet mixture model that allows the automatic determination of the number of mixture components from data. Therefore, the problem of predetermining the optimal number of mixing components has been overcome. Moreover, the problems of overfitting and underfitting are avoided by the Bayesian estimation approach. Compared with several recently proposed DP-related methods and conventional applied methods, the good performance and effectiveness of the proposed method have been demonstrated with both synthesized data and real data evaluations.

For most of non-Gaussian statistical models, the data being modeled represent strongly structured properties, such as scalar data with bounded support (e.g., beta distribution), vector data with unit length (e.g., Dirichlet distribution), and vector data with positive elements (e.g., generalized inverted Dirichlet distribution). In practical implementations of non-Gaussian statistical models, it is infeasible to find an analytically tractable solution to estimating the posterior distributions of the parameters. Variational inference (VI) is a widely used framework in Bayesian estimation. Recently, an improved framework, namely the extended variational inference (EVI), has been introduced and applied successfully to a number of non-Gaussian statistical models. EVI derives analytically tractable solutions, by introducing lower-bound approximations to the variational objective function. In this paper, we compare two approximation strategies, namely the multiple lower-bounds (MLB) approximation and the single lower-bound (SLB) approximation, which can be applied to carry out the EVI. For implementation, two different conditions, the weak and the strong conditions, are discussed. Convergence of the EVI depends on the selection of the lowerbound, regardless of the choice of weak or strong condition. We also discuss the convergence properties to clarify the differences between MLB and SLB. Extensive comparisons are made based on some EVI-based non-Gaussian statistical models. Theoretical analysis is conducted to demonstrate the differences between the weak and strong conditions. Experimental results based on real data show advantages of the SLB approximation over the MLB approximation.

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Image translation tasks based on generative models have become an important research area, such as the general framework for unsupervised image translation-CycleGAN (Cycle-Consistent Generative Adversarial Networks). A typical advantage of CycleGAN is that it can realize the training of two image sets without pairing, but there are still some problems in the preservation of semantic information and the learning of specific features. In this paper, we propose the CycleGAN-AdaIN framework based on the CycleGAN model, which can translate real photos into Chinese ink paintings. In order to retain the content of the image completely, we use one cycle consistency loss to replace two in the structure of the model. To learn the style information of the ink painting, we introduce an AdaIN (Adaptive Instance Normalization) module before the decoding process of the generation network. In addition, to correct the details of the generated image, we add the MS-SSIM (Multi-Scale-Structural Similarity Index) loss in the reconstruction loss to generate a higher quality image. Compared with the existing methods in FID, Kernel MMD, PSNR and SSIM, the experiment results show that our method can accomplish the task of transferring real photos to ink paintings and get better performance than the baseline model.