Normative expert systems have not become commonplace because they have been difficult to build and use. Over the past decade, however, researchers have developed the influence diagram, a graphical representation of a decision maker's beliefs, alternatives, and preferences that serves as the knowledge base of a normative expert system. Most people who have seen the representation find it intuitive and easy to use. Consequently, the influence diagram has overcome significantly the barriers to constructing normative expert systems. Nevertheless, building influence diagrams is not practical for extremely large and complex domains. In this book, I address the difficulties associated with the construction of the probabilistic portion of an influence diagram, called a knowledge map, belief network, or Bayesian network. I introduce two representations that facilitate the generation of large knowledge maps. In particular, I introduce the similarity network, a tool for building the network structure of a knowledge map, and the partition, a tool for assessing the probabilities associated with a knowledge map. I then use these representations to build Pathfinder, a large normative expert system for the diagnosis of lymph-node diseases (the domain contains over 60 diseases and over 100 disease findings). In an early version of the system, I encoded the knowledge of the expert using an erroneous assumption that all disease findings were independent, given each disease. When the expert and I attempted to build a more accurate knowledge map for the domain that would capture the dependencies among the disease findings, we failed. Using a similarity network, however, we built the knowledge-map structure for the entire domain in approximately 40 hours. Furthermore, the partition representation reduced the number of probability assessments required by the expert from 75,000 to 14,000.

Probabilistic Similarity Networks

Published nearly seventy years ago, Jeffreys' Theory of Probability (1939) has had a unique impact on the Bayesian community and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the objective Bayes school. In particular, its advances on the derivation of noninformative priors as well as on the scaling of Bayes factors have had a lasting impact on the field. However, the book reflects the characteristics of the time, especially in terms of mathematical rigorousness. In this paper, we point out the fundamental aspects of this reference work, especially the thorough coverage of testing problems and the construction of both estimation and testing noninformative priors based on functional divergences. Our major aim here is to help modern readers in navigating in this difficult text and in concentrating on passages that are still relevant today.

Harold Jeffreys' Theory of Probability revisited

In this paper Portfolio Optimization techniques were used to determine the most favorable investment portfolio In particular, stock indices of three companies, namely Microsoft Corporation, Christian Dior Fashion House and Shevron Corporation were evaluated Using this data the amounts invested in each asset when a portfolio is chosen on the efficient frontier were calculated In addition, the Portfolio with minimum variance, tangency portfolio and optimal Markowitz portfolio are presented

Portfolio Optimization

Given a long list of anomaly detection algorithms developed in the last few decades, how do they perform with regard to (i) varying levels of supervision, (ii) different types of anomalies, and (iii) noisy and corrupted data? In this work, we answer these key questions by conducting (to our best knowledge) the most comprehensive anomaly detection benchmark with 30 algorithms on 57 benchmark datasets, named ADBench. Our extensive experiments (98,436 in total) identify meaningful insights into the role of supervision and anomaly types, and unlock future directions for researchers in algorithm selection and design. With ADBench, researchers can easily conduct comprehensive and fair evaluations for newly proposed methods on the datasets (including our contributed ones from natural language and computer vision domains) against the existing baselines. To foster accessibility and reproducibility, we fully open-source ADBench and the corresponding results.

ADBench: Anomaly Detection Benchmark

The first observational run of the Advanced LIGO detectors, from September 12, 2015 to January 19, 2016, saw the first detections of gravitational waves from binary black hole mergers. In this paper we present full results from a search for binary black hole merger signals with total masses up to 100M⊙ and detailed implications from our observations of these systems. Our search, based on general-relativistic models of gravitational wave signals from binary black hole systems, unambiguously identified two signals, GW150914 and GW151226, with a significance of greater than 5σ over the observing period. It also identified a third possible signal, LVT151012, with substantially lower significance, which has a 87% probability of being of astrophysical origin. We provide detailed estimates of the parameters of the observed systems. Both GW150914 and GW151226 provide an unprecedented opportunity to study the two-body motion of a compact-object binary in the large velocity, highly nonlinear regime. We do not observe any deviations from general relativity, and place improved empirical bounds on several high-order post-Newtonian coefficients. From our observations we infer stellar-mass binary black hole merger rates lying in the range 9−240Gpc−3yr−1. These observations are beginning to inform astrophysical predictions of binary black hole formation rates, and indicate that future observing runs of the Advanced detector network will yield many more gravitational wave detections.

Binary Black Hole Mergers in the First Advanced LIGO Observing Run

Problem Given the number of times in which an unknown event has happened and failed: Required the chance that the probability of its happening in a single trial lies somewhere between any two degrees of probability that can be named. SECTION 1 Definition 1. Several events are inconsistent, when if one of them happens, none of the rest can. 2. Two events are contrary when one, or other of them must; and both together cannot happen. 3. An event is said to fail, when it cannot happen; or, which comes to the same thing, when its contrary has happened. 4. An event is said to be determined when it has either happened or failed. 5. The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon it’s 2 happening.

Thomas Bayes

Papers

An essay towards solving a problem in the doctrine of chances. [Facsimil]