9,869 research outputs found
Stable components in the parameter plane of transcendental functions of finite type
We study the parameter planes of certain one-dimensional, dynamically-defined
slices of holomorphic families of entire and meromorphic transcendental maps of
finite type. Our planes are defined by constraining the orbits of all but one
of the singular values, and leaving free one asymptotic value. We study the
structure of the regions of parameters, which we call {\em shell components},
for which the free asymptotic value tends to an attracting cycle of
non-constant multiplier. The exponential and the tangent families are examples
that have been studied in detail, and the hyperbolic components in those
parameter planes are shell components. Our results apply to slices of both
entire and meromorphic maps. We prove that shell components are simply
connected, have a locally connected boundary and have no center, i.e., no
parameter value for which the cycle is superattracting. Instead, there is a
unique parameter in the boundary, the {\em virtual center}, which plays the
same role. For entire slices, the virtual center is always at infinity, while
for meromorphic ones it maybe finite or infinite. In the dynamical plane we
prove, among other results, that the basins of attraction which contain only
one asymptotic value and no critical points are simply connected. Our dynamical
plane results apply without the restriction of finite type.Comment: 41 pages, 13 figure
Imprecise probability models for inference in exponential families
When considering sampling models described by a distribution from an exponential family, it is possible to create two types of imprecise probability models. One is based on the corresponding conjugate distribution and the other on the corresponding predictive distribution. In this paper, we show how these types of models can be constructed for any (regular, linear, canonical) exponential family, such as the centered normal distribution.
To illustrate the possible use of such models, we take a look at credal classification. We show that they are very natural and potentially promising candidates for describing the attributes of a credal classifier, also in the case of continuous attributes
Information Aggregation in Exponential Family Markets
We consider the design of prediction market mechanisms known as automated
market makers. We show that we can design these mechanisms via the mold of
\emph{exponential family distributions}, a popular and well-studied probability
distribution template used in statistics. We give a full development of this
relationship and explore a range of benefits. We draw connections between the
information aggregation of market prices and the belief aggregation of learning
agents that rely on exponential family distributions. We develop a very natural
analysis of the market behavior as well as the price equilibrium under the
assumption that the traders exhibit risk aversion according to exponential
utility. We also consider similar aspects under alternative models, such as
when traders are budget constrained
Statistical properties of unimodal maps: smooth families with negative Schwarzian derivative
We prove that there is a residual set of families of smooth or analytic
unimodal maps with quadratic critical point and negative Schwarzian derivative
such that almost every non-regular parameter is Collet-Eckmann with
subexponential recurrence of the critical orbit. Those conditions lead to a
detailed and robust statistical description of the dynamics. This proves the
Palis conjecture in this setting.Comment: 33 pages, no figures, third version, to appear in Ast\'erisqu
Statistical Geometry in Quantum Mechanics
A statistical model M is a family of probability distributions, characterised
by a set of continuous parameters known as the parameter space. This possesses
natural geometrical properties induced by the embedding of the family of
probability distributions into the Hilbert space H. By consideration of the
square-root density function we can regard M as a submanifold of the unit
sphere in H. Therefore, H embodies the `state space' of the probability
distributions, and the geometry of M can be described in terms of the embedding
of in H. The geometry in question is characterised by a natural Riemannian
metric (the Fisher-Rao metric), thus allowing us to formulate the principles of
classical statistical inference in a natural geometric setting. In particular,
we focus attention on the variance lower bounds for statistical estimation, and
establish generalisations of the classical Cramer-Rao and Bhattacharyya
inequalities. The statistical model M is then specialised to the case of a
submanifold of the state space of a quantum mechanical system. This is pursued
by introducing a compatible complex structure on the underlying real Hilbert
space, which allows the operations of ordinary quantum mechanics to be
reinterpreted in the language of real Hilbert space geometry. The application
of generalised variance bounds in the case of quantum statistical estimation
leads to a set of higher order corrections to the Heisenberg uncertainty
relations for canonically conjugate observables.Comment: 32 pages, LaTex file, Extended version to include quantum measurement
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