1,287 research outputs found
Envelopes of conditional probabilities extending a strategy and a prior probability
Any strategy and prior probability together are a coherent conditional
probability that can be extended, generally not in a unique way, to a full
conditional probability. The corresponding class of extensions is studied and a
closed form expression for its envelopes is provided. Then a topological
characterization of the subclasses of extensions satisfying the further
properties of full disintegrability and full strong conglomerability is given
and their envelopes are studied.Comment: 2
Diffusive optical tomography in the Bayesian framework
Many naturally-occuring models in the sciences are well-approximated by
simplified models, using multiscale techniques. In such settings it is natural
to ask about the relationship between inverse problems defined by the original
problem and by the multiscale approximation. We develop an approach to this
problem and exemplify it in the context of optical tomographic imaging.
Optical tomographic imaging is a technique for infering the properties of
biological tissue via measurements of the incoming and outgoing light
intensity; it may be used as a medical imaging methodology. Mathematically,
light propagation is modeled by the radiative transfer equation (RTE), and
optical tomography amounts to reconstructing the scattering and the absorption
coefficients in the RTE from boundary measurements. We study this problem in
the Bayesian framework, focussing on the strong scattering regime. In this
regime the forward RTE is close to the diffusion equation (DE). We study the
RTE in the asymptotic regime where the forward problem approaches the DE, and
prove convergence of the inverse RTE to the inverse DE in both nonlinear and
linear settings. Convergence is proved by studying the distance between the two
posterior distributions using the Hellinger metric, and using Kullback-Leibler
divergence
Unforeseen Contingencies
We develop a model of unforeseen contingencies. These are contingencies that are understood by economic agents - their consequences and probabilities are known - but are such that every description of such events necessarily leaves out relevant features that have a non-negligible impact on the parties' expected utilities. Using a simple co-insurance problem as a backdrop, we introduce a model where states are described in terms of objective features, and the description of an event specifies a finite number of such features. In this setting, unforeseen contingencies are present in the co-insurance problem when the first-best risk-sharing contract varies with the states of nature in a complex way that makes it highly sensitive to the component features of the states. In this environment, although agents can compute expected pay-offs, they are unable to include in any ex-ante agreement a description of the relevant contingencies that captures (even approximately) the relevant complexity of the risky environment.Unforeseen contingencies, incomplete contracts, finite invariance, fine variability.
Confidence intervals for nonhomogeneous branching processes and polymerase chain reactions
We extend in two directions our previous results about the sampling and the
empirical measures of immortal branching Markov processes. Direct applications
to molecular biology are rigorous estimates of the mutation rates of polymerase
chain reactions from uniform samples of the population after the reaction.
First, we consider nonhomogeneous processes, which are more adapted to real
reactions. Second, recalling that the first moment estimator is analytically
known only in the infinite population limit, we provide rigorous confidence
intervals for this estimator that are valid for any finite population. Our
bounds are explicit, nonasymptotic and valid for a wide class of nonhomogeneous
branching Markov processes that we describe in detail. In the setting of
polymerase chain reactions, our results imply that enlarging the size of the
sample becomes useless for surprisingly small sizes. Establishing confidence
intervals requires precise estimates of the second moment of random samples.
The proof of these estimates is more involved than the proofs that allowed us,
in a previous paper, to deal with the first moment. On the other hand, our
method uses various, seemingly new, monotonicity properties of the harmonic
moments of sums of exchangeable random variables.Comment: Published at http://dx.doi.org/10.1214/009117904000000775 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonparametric Bayes modeling of count processes
Data on count processes arise in a variety of applications, including
longitudinal, spatial and imaging studies measuring count responses. The
literature on statistical models for dependent count data is dominated by
models built from hierarchical Poisson components. The Poisson assumption is
not warranted in many applications, and hierarchical Poisson models make
restrictive assumptions about over-dispersion in marginal distributions. This
article proposes a class of nonparametric Bayes count process models, which are
constructed through rounding real-valued underlying processes. The proposed
class of models accommodates applications in which one observes separate
count-valued functional data for each subject under study. Theoretical results
on large support and posterior consistency are established, and computational
algorithms are developed using Markov chain Monte Carlo. The methods are
evaluated via simulation studies and illustrated through application to
longitudinal tumor counts and asthma inhaler usage
Improper vs finitely additive distributions as limits of countably additive probabilities
In Bayesian statistics, improper distributions and finitely additive probabilities (FAPs) are the two main alternatives to proper distributions, i.e. countably additive probabilities. Both of them can be seen as limits of proper distribution sequences w.r.t. to some specific convergence modes. Therefore, some authors attempt to link these two notions by this means, partly using heuristic arguments. The aim of the paper is to compare these two kinds of limits. We show that improper distributions and FAPs represent two distinct characteristics of a sequence of proper distributions and therefore, surprisingly, cannot be connected by the mean of proper distribution sequences. More specifically, for a sequence of proper distribution which converge to both an improper distribution and a set of FAPs, we show that another sequence of proper distributions can be constructed having the same FAP limits and converging to any given improper distribution. This result can be mainly explained by the fact that improper distributions describe the behavior of the sequence inside the domain after rescaling, whereas FAP limits describe how the mass concentrates on the boundary of the domain. We illustrate our results with several examples and we show the difficulty to define properly a uniform FAP distribution on the natural numbers as an equivalent of the improper flat prior. MSC 2010 subject classifications: Primary 62F15; secondary 62E17,60B10
Geometrically stopped Markovian random growth processes and Pareto tails
Many empirical studies document power law behavior in size distributions of
economic interest such as cities, firms, income, and wealth. One mechanism for
generating such behavior combines independent and identically distributed
Gaussian additive shocks to log-size with a geometric age distribution. We
generalize this mechanism by allowing the shocks to be non-Gaussian (but
light-tailed) and dependent upon a Markov state variable. Our main results
provide sharp bounds on tail probabilities, a simple equation determining
Pareto exponents, and comparative statics. We present two applications: we show
that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic
general equilibrium model with idiosyncratic investment risk are Paretian, and
(ii) a random growth model for the population dynamics of Japanese
municipalities is consistent with the observed Pareto exponent but only after
allowing for Markovian dynamics
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