Concept Bottleneck Model (CBM) is a methods for explaining neural networks. In CBM, concepts which correspond to reasons of outputs are inserted in the last intermediate layer as observed values. It is expected that we can interpret the relationship between the output and concept similar to linear regression. However, this interpretation requires observing all concepts and decreases the generalization performance of neural networks. Partial CBM (PCBM), which uses partially observed concepts, has been devised to resolve these difficulties. Although some numerical experiments suggest that the generalization performance of PCBMs is almost as high as that of the original neural networks, the theoretical behavior of its generalization error has not been yet clarified since PCBM is singular statistical model. In this paper, we reveal the Bayesian generalization error in PCBM with a three-layered and linear architecture. The result indcates that the structure of partially observed concepts decreases the Bayesian generalization error compared with that of CBM (full-observed concepts).

Reviewer's response to the rebuttal "line 308: [10, 400] range for alpha We chose 400 because higher values of alpha caused computational difficulties in the Gibbs sampling. This upper bound is a bit arbitrary; however, we found the exact upper limit above about 100 had little effect on the estimates of transition probabilities. Similarly, adjusting the lower bound below 60 or so had little effect. While we do hope for a more well justified hyperprior for alpha in future work, we believe our choice did not overly influence the results." Please add this information to the paper.

For spiking neurons, the entropy rate places an upper bound on the rate at which the spike train can convey stimulus information, and a large literature has focused on the problem of estimating entropy rate from spike train data.

The paper introduces probabilistic principal component analysis on Riemannian manifolds, extending earlier non-probabilistic versions to a probabilistic latent variable model, and derives maximum likelihood estimation procedures for a broad class of manifolds. The methods are demonstrated on toy data (maniold is a sphere) and shape analysis on images. This is a very interesting advancement, and the paper is well written, making it reasonably accessible in spite of the difficult topic. I have a set of interrelated questions; explicating and clarifying them would clarify the potential impact of the paper to the reader: - Is essential generality lost by assuming an Euclidean latent space? Locally on a tangent space it makes sense, and may be practically necessary, of course.

Principal geodesic analysis (PGA) is a generalization of principal component analysis (PCA) for dimensionality reduction of data on a Riemannian manifold. Currently PGA is defined as a geometric fit to the data, rather than as a probabilistic model. Inspired by probabilistic PCA, we present a latent variable model for PGA that provides a probabilistic framework for factor analysis on manifolds. To compute maximum likelihood estimates of the parameters in our model, we develop a Monte Carlo Expectation Maximization algorithm, where the expectation is approximated by Hamiltonian Monte Carlo sampling of the latent variables. We demonstrate the ability of our method to recover the ground truth parameters in simulated sphere data, as well as its effectiveness in analyzing shape variability of a corpus callosum data set from human brain images.

Movement Primitives (MP) are a well-established approach for representing modular and re-usable robot movement generators. Many state-of-the-art robot learning successes are based MPs, due to their compact representation of the inherently continuous and high dimensional robot movements. A major goal in robot learning is to combine multiple MPs as building blocks in a modular control architecture to solve complex tasks. To this effect, a MP representation has to allow for blending between motions, adapting to altered task variables, and co-activating multiple MPs in parallel. We present a probabilistic formulation of the MP concept that maintains a distribution over trajectories. Our probabilistic approach allows for the derivation of new operations which are essential for implementing all aforementioned properties in one framework. In order to use such a trajectory distribution for robot movement control, we analytically derive a stochastic feedback controller which reproduces the given trajectory distribution. We evaluate and compare our approach to existing methods on several simulated as well as real robot scenarios.

A long term goal of Interactive Reinforcement Learning is to incorporate nonexpert human feedback to solve complex tasks. Some state-of-the-art methods have approached this problem by mapping human information to rewards and values and iterating over them to compute better control policies. In this paper we argue for an alternate, more effective characterization of human feedback: Policy Shaping. We introduce Advise, a Bayesian approach that attempts to maximize the information gained from human feedback by utilizing it as direct policy labels. We compare Advise to state-of-the-art approaches and show that it can outperform them and is robust to infrequent and inconsistent human feedback.

There are two major routes to address linear inverse problems. Whereas regularization-based approaches build estimators as solutions of penalized regression optimization problems, Bayesian estimators rely on the posterior distribution of the unknown, given some assumed family of priors. While these may seem radically different approaches, recent results have shown that, in the context of additive white Gaussian denoising, the Bayesian conditional mean estimator is always the solution of a penalized regression problem. The contribution of this paper is twofold. First, we extend the additive white Gaussian denoising results to general linear inverse problems with colored Gaussian noise. Second, we characterize conditions under which the penalty function associated to the conditional mean estimator can satisfy certain popular properties such as convexity, separability, and smoothness. This sheds light on some tradeoff between computational efficiency and estimation accuracy in sparse regularization, and draws some connections between Bayesian estimation and proximal optimization.

"Reciprocally Coupled Local Estimators Implement Bayesian Information Integration Distributively" puts forward a new take on Bayesian integration of multimodal cues. Instead of assuming a special area in the brain, where evidence from various sensory cues is combined (as in Ma and all, 2006), the authors consider a scenario, whereby each area receiving direct afferent input from a single modality (i.e. In the example analysed by the authors, and under a number of suitable assumptions, the cue integration they observe in their networks is close to Bayes-optimal. Building up on work of Fung and all (2010), the authors derive theoretical predictions for the integration of information in reciprocally coupled ring attractors (CANNs), which they also confirm by simulations. The reader is led through the general steps of the analysis, while details are provided in the supplementary material.

In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process. In particular, there may be certain time intervals exhibiting rapid changes and others in which changes are slow. If such locally adaptive smoothness is not accounted for, one can obtain misleading inferences and predictions, with over-smoothing across erratic time intervals and under-smoothing across times exhibiting slow variation. This can lead to miscalibration of predictive intervals, which can be substantially too narrow or wide depending on the time. We propose a continuous multivariate stochastic process for time series having locally varying smoothness in both the mean and covariance matrix. This process is constructed utilizing latent dictionary functions in time, which are given nested Gaussian process priors and linearly related to the observed data through a sparse mapping. Using a differential equation representation, we bypass usual computational bottlenecks in obtaining MCMC and online algorithms for approximate Bayesian inference. The performance is assessed in simulations and illustrated in a financial application.