Quantum Clustering is a powerful method to detect clusters in data with mixed density. However, it is very sensitive to a length parameter that is inherent to the Schr\"odinger equation. In addition, linking data points into clusters requires local estimates of covariance that are also controlled by length parameters. This raises the question of how to adjust the control parameters of the Schr\"odinger equation for optimal clustering. We propose a probabilistic framework that provides an objective function for the goodness-of-fit to the data, enabling the control parameters to be optimised within a Bayesian framework. This naturally yields probabilities of cluster membership and data partitions with specific numbers of clusters. The proposed framework is tested on real and synthetic data sets, assessing its validity by measuring concordance with known data structure by means of the Jaccard score (JS). This work also proposes an objective way to measure performance in unsupervised learning that correlates very well with JS.

Artificial intelligence, or AI, is largely an experimental science—at least as much progress has been made by building and analyzing programs as by examining theoretical questions. MYCIN is one of several well-known programs that embody some intelligence and provide data on the extent to which intelligent behavior can be programmed. As with other AI programs, its development was slow and not always in a forward direction. But we feel we learned some useful lessons in the course of nearly a decade of work on MYCIN and related programs. In this book we share the results of many experiments performed in that time, and we try to paint a coherent picture of the work. The book is intended to be a critical analysis of several pieces of related research, performed by a large number of scientists. We believe that the whole field of AI will benefit from such attempts to take a detailed retrospective look at experiments, for in this way the scientific foundations of the field will gradually be defined. It is for all these reasons that we have prepared this analysis of the MYCIN experiments.

Variational inference (VI) is a widely used framework in Bayesian estimation. For most of the non-Gaussian statistical models, it is infeasible to find an analytically tractable solution to estimate the posterior distributions of the parameters. Recently, an improved framework, namely the extended variational inference (EVI), has been introduced and applied to derive analytically tractable solution by employing lower-bound approximation to the variational objective function. Two conditions required for EVI implementation, namely the weak condition and the strong condition, are discussed and compared in this paper. In practical implementation, the convergence of the EVI depends on the selection of the lower-bound approximation, no matter with the weak condition or the strong condition. In general, two approximation strategies, the single lower-bound (SLB) approximation and the multiple lower-bounds (MLB) approximation, can be applied to carry out the lower-bound approximation. To clarify the differences between the SLB and the MLB, we will also discuss the convergence properties of the aforementioned two approximations. Extensive comparisons are made based on some existing EVI-based non-Gaussian statistical models. Theoretical analysis are conducted to demonstrate the differences between the weak and the strong conditions. Qualitative and quantitative experimental results are presented to show the advantages of the SLB approximation.

This book has been adapted in large part from the author's doctoral thesis [Shortliffe, l 974b]. Portions of the work appeared previously in Computers And Biomedical Research [Shortliffe, 1973, l 975b], Mathematical Biosciences [Shortliffe, 1975a], and the Proceedings Of The Thirteenth San Diego Biomedical Symposium [Shortliffe, l 974a]. To Stanford's Medical Scientist Training Program, which is supported by the National Institutes of Health Contents

JANICE S. AIKINS Dr. Aikins received her Ph.D. in computer science from Stanford University in 1980. She is currently a research computer scientist at IBM's Palo Alto Scientific Center. She specializes in designing systems with an emphasis on the explicit representation of control knowledge in expert systems. ROBERT L. BLUM Dr. Blum received his M.D. from the University of California Medical School at San Francisco in 1973. From 1973 to 1976 he did an internship and residency in the Department of Internal Medicine at the Kaiser Foundation Hospital in Oakland, California, where he was chief resident in 1976.

Edward H. Shortliffe Chapter 6 Details of the Revised Therapy Algorithm

This paper introduces a new method for model selection and more generally hyperparameter selection in machine learning. The paper first proves a relationship between generalization error and a difference of description lengths of the training data; we call this difference differential description length (DDL). This allows prediction of generalization error from the training data \emph{alone} by performing encoding of the training data. This can now be used for model selection by choosing the model that has the smallest predicted generalization error. We show how this encoding can be done for linear regression and neural networks. We provide experiments showing that this leads to smaller generalization error than cross-validation and traditional MDL and Bayes methods.

We consider a finite time horizon multi-armed bandit (MAB) problem in a Bayesian framework, for which we develop a general set of control policies that leverage ideas from information relaxations of stochastic dynamic optimization problems. In crude terms, an information relaxation allows the decision maker (DM) to have access to the future (unknown) rewards and incorporate them in her optimization problem to pick an action at time $t$, but penalizes the decision maker for using this information. In our setting, the future rewards allow the DM to better estimate the unknown mean reward parameters of the multiple arms, and optimize her sequence of actions. By picking different information penalties, the DM can construct a family of policies of increasing complexity that, for example, include Thompson Sampling and the true optimal (but intractable) policy as special cases. We systematically develop this framework of information relaxation sampling, propose an intuitive family of control policies for our motivating finite time horizon Bayesian MAB problem, and prove associated structural results and performance bounds. Numerical experiments suggest that this new class of policies performs well, in particular in settings where the finite time horizon introduces significant tension in the problem. Finally, inspired by the finite time horizon Gittins index, we propose an index policy that builds on our framework that particularly outperforms to the state-of-the-art algorithms in our numerical experiments.

Consider a setting with $N$ independent individuals, each with an unknown parameter, $p_i \in [0, 1]$ drawn from some unknown distribution $P^\star$. After observing the outcomes of $t$ independent Bernoulli trials, i.e., $X_i \sim \text{Binomial}(t, p_i)$ per individual, our objective is to accurately estimate $P^\star$. This problem arises in numerous domains, including the social sciences, psychology, health-care, and biology, where the size of the population under study is usually large while the number of observations per individual is often limited. Our main result shows that, in the regime where $t \ll N$, the maximum likelihood estimator (MLE) is both statistically minimax optimal and efficiently computable. Precisely, for sufficiently large $N$, the MLE achieves the information theoretic optimal error bound of $\mathcal{O}(\frac{1}{t})$ for $t < c\log{N}$, with regards to the earth mover's distance (between the estimated and true distributions). More generally, in an exponentially large interval of $t$ beyond $c \log{N}$, the MLE achieves the minimax error bound of $\mathcal{O}(\frac{1}{\sqrt{t\log N}})$. In contrast, regardless of how large $N$ is, the naive "plug-in" estimator for this problem only achieves the sub-optimal error of $\Theta(\frac{1}{\sqrt{t}})$.

The posteriors over neural network weights are high dimensional and multimodal. Each mode typically characterizes a meaningfully different representation of the data. We develop Cyclical Stochastic Gradient MCMC (SG-MCMC) to automatically explore such distributions. In particular, we propose a cyclical stepsize schedule, where larger steps discover new modes, and smaller steps characterize each mode. We prove that our proposed learning rate schedule provides faster convergence to samples from a stationary distribution than SG-MCMC with standard decaying schedules. Moreover, we provide extensive experimental results to demonstrate the effectiveness of cyclical SG-MCMC in learning complex multimodal distributions, especially for fully Bayesian inference with modern deep neural networks.