2,960 research outputs found
Equivalence of weak and strong modes of measures on topological vector spaces
A strong mode of a probability measure on a normed space can be defined
as a point such that the mass of the ball centred at uniformly
dominates the mass of all other balls in the small-radius limit. Helin and
Burger weakened this definition by considering only pairwise comparisons with
balls whose centres differ by vectors in a dense, proper linear subspace of
, and posed the question of when these two types of modes coincide. We show
that, in a more general setting of metrisable vector spaces equipped with
measures that are finite on bounded sets, the density of and a uniformity
condition suffice for the equivalence of these two types of modes. We
accomplish this by introducing a new, intermediate type of mode. We also show
that these modes can be inequivalent if the uniformity condition fails. Our
results shed light on the relationships between among various notions of
maximum a posteriori estimator in non-parametric Bayesian inference.Comment: 22 pages, 3 figure
Quasi-invariance of countable products of Cauchy measures under non-unitary dilations
Consider an infinite sequence (Un)nâN of independent Cauchy random variables, defined by a sequence (ÎŽn)nâN of location parameters and a sequence (Îłn)nâN of scale parameters. Let (Wn)nâN be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (ÏnÎłn)nâN of scale parameters, with Ïnâ 0 for all nâN. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)nâN and (Wn)nâN are equivalent if and only if the sequence (|Ïn|â1)nâN is square-summable
FrĂ©chet differentiable drift dependence of PerronâFrobenius and Koopman operators for non-deterministic dynamics
We prove the FrĂ©chet differentiability with respect to the drift of PerronâFrobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic PerronâFrobenius and Koopman operators
Choosing observation operators to mitigate model error in Bayesian inverse problems
In statistical inference, a discrepancy between the parameter-to-observable
map that generates the data and the parameter-to-observable map that is used
for inference can lead to misspecified likelihoods and thus to incorrect
estimates. In many inverse problems, the parameter-to-observable map is the
composition of a linear state-to-observable map called an `observation
operator' and a possibly nonlinear parameter-to-state map called the `model'.
We consider such Bayesian inverse problems where the discrepancy in the
parameter-to-observable map is due to the use of an approximate model that
differs from the best model, i.e. to nonzero `model error'. Multiple approaches
have been proposed to address such discrepancies, each leading to a specific
posterior. We show how to use local Lipschitz stability estimates of posteriors
with respect to likelihood perturbations to bound the Kullback--Leibler
divergence of the posterior of each approach with respect to the posterior
associated to the best model. Our bounds lead to criteria for choosing
observation operators that mitigate the effect of model error for Bayesian
inverse problems of this type. We illustrate the feasibility of one such
criterion on an advection-diffusion-reaction PDE inverse problem, and use this
example to discuss the importance and challenges of model error-aware
inference.Comment: 33 pages, 5 figure
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