instruction:
In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
question:
The EM-algorithm is a general procedure to get maximum likelihood estimates if part of the observations on the variables of a network are missing. In this paper a stochastic version of the algorithm is adapted to probabilistic neural networks describing the associative dependency of variables. These networks have a proba­ bility distribution, which is a special case of the distribution generated by proba­ bilistic inference networks. Hence both types of networks can be combined al­ lowing to integrate probabilistic rules as well as unspecified associations in a sound way. The resulting network may have a number of interesting features including cycles of probabilistic rules, hidden 'un­ observable' variables, and uncertain and contradictory evidence.
answer:
Integrating Probabilistic Rules into Neural Networks: A Stochastic EM Learning Algorithm


question:
Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an approximate Expectation Propagation procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning. We evaluate the new method for non-linear regression on eleven real-world datasets, showing that it always outperforms GP regression and is almost always better than state-of-the-art deterministic and sampling-based approximate inference methods for Bayesian neural networks. As a by-product, this work provides a comprehensive analysis of six approximate Bayesian methods for training neural networks.
answer:
Deep Gaussian Processes for Regression using Approximate Expectation Propagation


question:
Markov chain model is widely applied in many fields, especially the field of prediction. The classical Discrete-time Markov chain(DTMC) is a widely used method for prediction. However, the classical DTMC model has some limitation when the system is complex with uncertain information or state space is not discrete. To address it, a new belief Markov chain model is proposed by combining Dempster-Shafer evidence theory with Markov chain. In our model, the uncertain data is allowed to be handle in the form of interval number and the basic probability assignment(BPA) is generated based on the distance between interval numbers. The new belief Markov chain model overcomes the shortcomings of classical Markov chain and has an efficient ability in dealing with uncertain information. Moreover, an example of inventory prediction and the comparison between our model and classi∗Corresponding author at: School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China. Tel: (86-29)88431267. E-mail address: jiangwen@nwpu.edu.cn, jiangwenpaper@hotmail.com Preprint submitted to Elsevier March 7, 2017 cal DTMC model can show the effectiveness and rationality of our proposed model.
answer:

A new belief Markov chain model and its application in inventory prediction