334 research outputs found
Optimality of Thompson Sampling for Gaussian Bandits Depends on Priors
In stochastic bandit problems, a Bayesian policy called Thompson sampling
(TS) has recently attracted much attention for its excellent empirical
performance. However, the theoretical analysis of this policy is difficult and
its asymptotic optimality is only proved for one-parameter models. In this
paper we discuss the optimality of TS for the model of normal distributions
with unknown means and variances as one of the most fundamental example of
multiparameter models. First we prove that the expected regret of TS with the
uniform prior achieves the theoretical bound, which is the first result to show
that the asymptotic bound is achievable for the normal distribution model. Next
we prove that TS with Jeffreys prior and reference prior cannot achieve the
theoretical bound. Therefore the choice of priors is important for TS and
non-informative priors are sometimes risky in cases of multiparameter models
Asymptotic Distribution of Wishart Matrix for Block-wise Dispersion of Population Eigenvalues
This paper deals with the asymptotic distribution of Wishart matrix and its
application to the estimation of the population matrix parameter when the
population eigenvalues are block-wise infinitely dispersed. We show that the
appropriately normalized eigenvectors and eigenvalues asymptotically generate
two Wishart matrices and one normally distributed random matrix, which are
mutually independent. For a family of orthogonally equivariant estimators, we
calculate the asymptotic risks with respect to the entropy or the quadratic
loss function and derive the asymptotically best estimator among the family. We
numerically show 1) the convergence in both the distributions and the risks are
quick enough for a practical use, 2) the asymptotically best estimator is
robust against the deviation of the population eigenvalues from the block-wise
infinite dispersion
A generalization of the integer linear infeasibility problem
Does a given system of linear equations with nonnegative constraints have an
integer solution? This is a fundamental question in many areas. In statistics
this problem arises in data security problems for contingency table data and
also is closely related to non-squarefree elements of Markov bases for sampling
contingency tables with given marginals. To study a family of systems with no
integer solution, we focus on a commutative semigroup generated by a finite
subset of and its saturation. An element in the difference of the
semigroup and its saturation is called a ``hole''. We show the necessary and
sufficient conditions for the finiteness of the set of holes. Also we define
fundamental holes and saturation points of a commutative semigroup. Then, we
show the simultaneous finiteness of the set of holes, the set of non-saturation
points, and the set of generators for saturation points. We apply our results
to some three- and four-way contingency tables. Then we will discuss the time
complexities of our algorithms.Comment: This paper has been published in Discrete Optimization, Volume 5,
Issue 1 (2008) p36-5
Standard imsets for undirected and chain graphical models
We derive standard imsets for undirected graphical models and chain graphical
models. Standard imsets for undirected graphical models are described in terms
of minimal triangulations for maximal prime subgraphs of the undirected graphs.
For describing standard imsets for chain graphical models, we first define a
triangulation of a chain graph. We then use the triangulation to generalize our
results for the undirected graphs to chain graphs.Comment: Published at http://dx.doi.org/10.3150/14-BEJ611 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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