The technique of principal component analysis (PCA) has recently been expressed as the maximum likelihood solution for a generative latent variable model. In this paper we use this probabilistic reformulation as the basis for a Bayesian treatment of PCA. Our key result is that ef(cid:173) fective dimensionality of the latent space (equivalent to the number of retained principal components) can be determined automatically as part of the Bayesian inference procedure. An important application of this framework is to mixtures of probabilistic PCA models, in which each component can determine its own effective complexity.

We consider probabilistic PCA and related factor models from a Bayesian perspective. These models are in general not identifiable as the likelihood has a rotational symmetry. This gives rise to complicated posterior distributions with continuous subspaces of equal density and thus hinders efficiency of inference as well as interpretation of obtained parameters. In particular, posterior averages over factor loadings become meaningless and only model predictions are unambiguous. Here, we propose a parameterization based on Householder transformations, which remove the rotational symmetry of the posterior. Furthermore, by relying on results from random matrix theory, we establish the parameter distribution which leaves the model unchanged compared to the original rotationally symmetric formulation. In particular, we avoid the need to compute the Jacobian determinant of the parameter transformation. This allows us to efficiently implement probabilistic PCA in a rotation invariant fashion in any state of the art toolbox. Here, we implemented our model in the probabilistic programming language Stan and illustrate it on several examples.

While the Matrix Generalized Inverse Gaussian ($\mathcal{MGIG}$) distribution arises naturally in some settings as a distribution over symmetric positive semi-definite matrices, certain key properties of the distribution and effective ways of sampling from the distribution have not been carefully studied. In this paper, we show that the $\mathcal{MGIG}$ is unimodal, and the mode can be obtained by solving an Algebraic Riccati Equation (ARE) equation [7]. Based on the property, we propose an importance sampling method for the $\mathcal{MGIG}$ where the mode of the proposal distribution matches that of the target. The proposed sampling method is more efficient than existing approaches [32, 33], which use proposal distributions that may have the mode far from the $\mathcal{MGIG}$'s mode. Further, we illustrate that the the posterior distribution in latent factor models, such as probabilistic matrix factorization (PMF) [25], when marginalized over one latent factor has the $\mathcal{MGIG}$ distribution. The characterization leads to a novel Collapsed Monte Carlo (CMC) inference algorithm for such latent factor models. We illustrate that CMC has a lower log loss or perplexity than MCMC, and needs fewer samples.

Extreme Components Analysis (XCA) is a statistical method based on a single eigenvalue decomposition to recover the optimal combination of principal and minor components in the data. Unfortunately, minor components are notoriously sensitive to overfitting when the number of data items is small relative to the number of attributes. We present a Bayesian extension of XCA by introducing a conjugate prior for the parameters of the XCA model. This Bayesian-XCA is shown to outperform plain vanilla XCA as well as Bayesian-PCA and XCA based on a frequentist correction to the sample spectrum. Moreover, we show that minor components are only picked when they represent genuine constraints in the data, even for very small sample sizes. An extension to mixtures of Bayesian XCA models is also explored.

The technique of principal component analysis (PCA) has recently been expressed as the maximum likelihood solution for a generative latent variable model. In this paper we use this probabilistic reformulation as the basis for a Bayesian treatment of PCA. Our key result is that effective dimensionality of the latent space (equivalent to the number of retained principal components) can be determined automatically as part of the Bayesian inference procedure. An important application of this framework is to mixtures of probabilistic PCA models, in which each component can determine its own effective complexity.

The technique of principal component analysis (PCA) has recently been expressed as the maximum likelihood solution for a generative latent variable model. In this paper we use this probabilistic reformulation as the basis for a Bayesian treatment of PCA. Our key result is that effective dimensionality of the latent space (equivalent to the number of retained principal components) can be determined automatically as part of the Bayesian inference procedure. An important application of this framework is to mixtures of probabilistic PCA models, in which each component can determine its own effective complexity.

The technique of principal component analysis (PCA) has recently been expressed as the maximum likelihood solution for a generative latent variable model. In this paper we use this probabilistic reformulation as the basis for a Bayesian treatment of PCA. Our key result is that effective dimensionalityof the latent space (equivalent to the number of retained principal components) can be determined automatically as part of the Bayesian inference procedure. An important application of this framework is to mixtures of probabilistic PCA models, in which each component can determine its own effective complexity. 1 Introduction Principal component analysis (PCA) is a widely used technique for data analysis. Recently Tipping and Bishop (1997b) showed that a specific form of generative latent variable model has the property that its maximum likelihood solution extracts the principal subspace of the observed data set.