2,226 research outputs found
The MM Alternative to EM
The EM algorithm is a special case of a more general algorithm called the MM
algorithm. Specific MM algorithms often have nothing to do with missing data.
The first M step of an MM algorithm creates a surrogate function that is
optimized in the second M step. In minimization, MM stands for
majorize--minimize; in maximization, it stands for minorize--maximize. This
two-step process always drives the objective function in the right direction.
Construction of MM algorithms relies on recognizing and manipulating
inequalities rather than calculating conditional expectations. This survey
walks the reader through the construction of several specific MM algorithms.
The potential of the MM algorithm in solving high-dimensional optimization and
estimation problems is its most attractive feature. Our applications to random
graph models, discriminant analysis and image restoration showcase this
ability.Comment: Published in at http://dx.doi.org/10.1214/08-STS264 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Data-driven satisficing measure and ranking
We propose an computational framework for real-time risk assessment and
prioritizing for random outcomes without prior information on probability
distributions. The basic model is built based on satisficing measure (SM) which
yields a single index for risk comparison. Since SM is a dual representation
for a family of risk measures, we consider problems constrained by general
convex risk measures and specifically by Conditional value-at-risk. Starting
from offline optimization, we apply sample average approximation technique and
argue the convergence rate and validation of optimal solutions. In online
stochastic optimization case, we develop primal-dual stochastic approximation
algorithms respectively for general risk constrained problems, and derive their
regret bounds. For both offline and online cases, we illustrate the
relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure
ChoiceRank: Identifying Preferences from Node Traffic in Networks
Understanding how users navigate in a network is of high interest in many
applications. We consider a setting where only aggregate node-level traffic is
observed and tackle the task of learning edge transition probabilities. We cast
it as a preference learning problem, and we study a model where choices follow
Luce's axiom. In this case, the marginal counts of node visits are a
sufficient statistic for the transition probabilities. We show how to
make the inference problem well-posed regardless of the network's structure,
and we present ChoiceRank, an iterative algorithm that scales to networks that
contains billions of nodes and edges. We apply the model to two clickstream
datasets and show that it successfully recovers the transition probabilities
using only the network structure and marginal (node-level) traffic data.
Finally, we also consider an application to mobility networks and apply the
model to one year of rides on New York City's bicycle-sharing system.Comment: Accepted at ICML 201
The incomplete Analytic Hierarchy Process and Bradley-Terry model: (in)consistency and information retrieval
Several methods of preference modeling, ranking, voting and multi-criteria
decision making include pairwise comparisons. It is usually simpler to compare
two objects at a time, furthermore, some relations (e.g., the outcome of sports
matches) are naturally known for pairs. This paper investigates and compares
pairwise comparison models and the stochastic Bradley-Terry model. It is proved
that they provide the same priority vectors for consistent (complete or
incomplete) comparisons. For incomplete comparisons, all filling in levels are
considered. Recent results identified the optimal subsets and sequences of
multiplicative/additive/reciprocal pairwise comparisons for small sizes of
items (up to n = 6). Simulations of this paper show that the same subsets and
sequences are optimal in case of the Bradley-Terry and the Thurstone models as
well. This, somehow surprising, coincidence suggests the existence of a more
general result. Further models of information and preference theory are subject
to future investigation in order to identify optimal subsets of input data
Revisiting Random Utility Models
This thesis explores extensions of Random Utility Models (RUMs), providing more flexible models and adopting a computational perspective. This includes building new models and understanding their properties such as identifiability and the log concavity of their likelihood functions as well as the development of estimation algorithms.Engineering and Applied Science
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