Unified determinants via a single equation.
problem Defining determinants with all known properties.
method Proposing a single equation implying all known properties of determinants.
result Unified definition of determinants with all properties.
The paper generalizes Bayesian Cramér-Rao inequality using information geometry of relative α-entropy.
problem Establishing a lower bound for the variance of an unbiased estimator for the α-escort distribution.
method Proposes a general Riemannian metric based on relative α-entropy to derive a generalized Bayesian Cramér-Rao inequality.
result Establishes a lower bound for the variance of an unbiased estimator for the α-escort distribution.
Paper introduces CWDAE for better synthetic data generation.
problem Measuring discrepancy between generative and ground-truth distributions.
method Introduces mixture Cramer-Wold distance for joint and marginal distributional learning.
result CWDAE shows remarkable performance in generating synthetic data.
This work connects Cramér distance to QR-DQN for DRL.
problem Improving performance in DRL by capturing full distribution of returns.
method Proves Cramér distance's equivalence to 1-Wasserstein distance and proposes a low-complexity algorithm to compute Cramér distance.
result Cramér distance and quantile regression losses yield collinear gradients under non-crossing constraints.
We propose a new generative model, Cramer-Wold Autoencoder (CWAE). Following WAE, we directly encourage normality of the latent space. Our paper uses also the recent idea from Sliced WAE (SWAE) model, which uses one-dimensional projections as a method of verifying closeness of two distributions. The crucial new ingredi…
New PAC-Bayes bounds for unbounded losses using Cramér-Chernoff techniques.
problem Developing bounds for unbounded losses in PAC-Bayesian settings.
method Introducing a new PAC-Bayes oracle bound using Cramér-Chernoff bounds and controlling random variable tails.
result Our bounds generalize and improve upon previous results, providing more informative and potentially tighter bounds.
The Wasserstein probability metric has received much attention from the machine learning community. Unlike the Kullback-Leibler divergence, which strictly measures change in probability, the Wasserstein metric reflects the underlying geometry between outcomes. The value of being sensitive to this geometry has been demo…
Study on conditions for achieving optimal robustness in statistical estimators.
problem Achieving the optimal robustness of estimators in statistical models.
method Developed a Wasserstein analogue of the Cramer-Rao inequality and investigated conditions for achieving the Wasserstein-Cramer-Rao lower bound.
result Conditions for the existence of asymptotically efficient estimators in one-parameter models and location-scale families.
Paper extends Bayesian Cramér-Rao bound with geometric considerations.
problem Estimation of covariance matrices with geometric structures.
method Intrinsic Bayesian Cramér-Rao bound with Riemannian geometry.
result Performance bounds for covariance matrix estimation.
Despite many algorithmic advances, our theoretical understanding of practical distributional reinforcement learning methods remains limited. One exception is Rowland et al. (2018)'s analysis of the C51 algorithm in terms of the Cramér distance, but their results only apply to the tabular setting and ignore C51's use of…
We study power expansions of the characteristic function of a linear operator A in a p∣q-dimensional superspace V. We show that traces of exterior powers of A satisfy universal recurrence relations of period q. `Underlying' recurrence relations hold in the Grothendieck ring of representations of $\GL(V)$. The…
Proposes a learned Bayesian Cramér-Rao bound for unknown measurement models.
problem Computing the Bayesian Cramér-Rao bound requires full knowledge of priors and measurement distributions.
method Introduces a Physics-encoded score neural network to learn priors and measurements.
result Demonstrates improved sample complexity and interpretability through domain knowledge incorporation.
New theory of sensitivity for unbiased estimators using Wasserstein geometry.
problem Estimating the instability of estimators under small perturbations.
method Developed a new theory based on Wasserstein geometry, analogous to classical Cramér-Rao theory.
result Wasserstein-Cramér-Rao lower bound for sensitivity of unbiased estimators.
We consider the problem of maximizing the discounted utility of dividend payments of an insurance company whose reserves are modeled as a classical Cramér-Lundberg risk process. We investigate this optimization problem under the constraint that dividend rate is bounded. We prove that the value function fulfills the Ham…
Non-linear source separation is a challenging open problem with many applications. We extend a recently proposed Adversarial Non-linear ICA (ANICA) model, and introduce Cramer-Wold ICA (CW-ICA). In contrast to ANICA we use a simple, closed--form optimization target instead of a discriminator--based independence measure…
The paper explores how information geometry impacts classical CR inequalities.
problem Deriving and generalizing CR inequalities using information geometry.
method Examining Eguchi's theory and applying Amari-Nagoaka's theory to KL-divergence, and then extending to other divergences.
result Generalized CR inequalities derived from various divergences.
Optimal dividend strategy with ratcheting and capital injection under Cramér-Lundberg model.
problem Optimal dividend payout for an insurance company with ratcheting constraints and capital injections.
method Systematic probabilistic and PDE-based approach to solve HJB equation, constructing strong solution and optimal strategy.
result Existence and uniqueness of strong solution, explicit optimal feedback control strategy.
Study shows subordinated Cramér-Lundberg model increases ruin probability.
problem Analyzing the impact of subordinated time-changed claims on insurance ruin probability.
method Examined a compound Poisson process modified by a Lévy subordinator.
result Probability of ruin decreases slowly with initial capital, despite unchanged total claim amount.
Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.
problem Improving estimator efficiency in non-asymptotic settings.
method Incorporates curvature-aware corrections based on extrinsic geometry of statistical model manifold.
result Meaningful tightening of estimator variance bounds.
This work introduces a new data-driven estimator for the Bayesian Cramér-Rao bound using score matching.
problem Benchmarking the performance of statistical estimators and providing a principled metric for system design and optimization.
method Score matching technique to estimate the Bayesian Cramér-Rao bound from training data.
result Developed novel non-asymptotic bounds on the score matching error and the Bayesian CRB estimator.
Abstract: Geometrically reformulates estimation theory for finite-dimensional C*-algebras.
problem Estimation theory for finite-dimensional C*-algebras.
method Geometrical formulation of estimation theory.
result Derivation of Cramer-Rao and Helstrom bounds.
In this communication, we describe some interrelations between generalized q-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of en…
In this paper, we discuss the Cramér-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift a and volatility σ>0. By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/σ^2 …
Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber-Shiu functions) in a Cramer-Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modelled as time dependant control functions, which leads to a setting from the theory of opti…
In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramer-Lundberg model introducing a stochastic factor which affects …
Improves joint distribution learning for high-dimensional datasets with complex correlations.
problem Conditional independence assumption limitations in VAE decoders for high-dimensional datasets.
method Cramer-Wold distance regularization and two-step learning method for flexible prior modeling.
result Effective joint distributional learning for high-dimensional datasets with multiple categorical variables.
We study the geometry of probability distributions with respect to a generalized family of Csiszár f-divergences. A member of this family is the relative α-entropy which is also a Rényi analog of relative entropy in information theory and known as logarithmic or projective power divergence in statistics. We apply E…
The paper derives Cramer-Rao bounds for Laplacian matrix estimation under various constraints.
problem Estimating Laplacian matrices with structural constraints and sparsity.
method Linear reparametrization and closed-form expressions for Cramer-Rao bounds tailored to Laplacian matrix estimation.
result The derived CRBs provide performance limits for Laplacian matrix estimation and are validated in various applications.
We propose a modified χβ-divergence, give some of its properties, and show that this leads to the definition of a generalized Fisher information. We give generalized Cramér-Rao inequalities, involving this Fisher information, an extension of the Fisher information matrix, and arbitrary norms and power of the estimat…
Study optimal investment-reinsurance strategy for insurers under random coefficients and jumps.
problem Optimal investment-reinsurance strategy for insurers with random coefficients and jumps.
method Solves backward stochastic differential equations with jumps under a convex cone constraint.
result Optimal strategy and value remain the same even with random coefficients and jumps.
Study non-asymptotic estimation bounds for LTI models with Gaussian noise.
problem Estimating parameters of LTI models with non-asymptotic error bounds.
method Sharp non-asymptotic lower bounds using Cramér-Rao and van Trees inequalities, concentration results, and differential geometric constructions.
result Sharp and rate-optimal lower bounds for mean square estimation risk.
We provide a direct proof of Cramér's theorem for geodesic random walks in a complete Riemannian manifold (M,g). We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in M. Furthermore, we reveal the geometric obstructions one runs into …
Study optimal investment and reinsurance strategy for insurers under random coefficients.
problem Optimal mean-variance investment-reinsurance problem for insurers under Cramér-Lundberg model with random coefficients.
method Reduced to a constrained stochastic linear-quadratic control problem with jumps, solved using BSDE techniques and SREs.
result Explicit efficient investment-reinsurance strategy and mean-variance frontier.
CD algorithm achieves near-optimal convergence rate for unnormalized models.
problem Training unnormalized models with high efficiency.
method Non-asymptotic analysis of contrastive divergence algorithm.
result CD can achieve O(n−1/2) convergence rate under regularity assumptions. Paper improves off-policy evaluation for reinforcement learning with asymptotically efficient estimators.
problem Estimating target policy performance using offline data collected by a different policy.
method Developed a modified marginalized importance sampling (MIS) estimator that achieves asymptotically efficient error bounds.
result Proved that a simple modification to the MIS estimator can achieve a Cramer-Rao lower bound in mean square error.
New CRB derived for curved models using extrinsic geometry.
problem Estimate curved statistical families accurately.
method Vector generalization of CRB with curvature correction using SDP and SOS relaxations.
result Directional curvature correction provides more accurate estimation.
The Fisher-Rao geometry is applied to elliptical distributions for optimization and classification.
problem Optimizing and classifying covariance matrices using geometric tools.
method Riemannian optimization and intrinsic Cramér-Rao bounds.
result Geometric tools enhance covariance matrix estimation and classification.
Study optimal investment strategies for an insurer in two currency markets.
problem Maximizing expected exponential utility of terminal wealth for an insurer in two currency markets.
method Dynamic programming method applied to solve Hamilton-Jacobi-Bellman equations.
result Optimal investment strategies and value functions are derived.
Optimal reinsurance strategy found to minimize financial risk.
problem Minimizing financial risk in insurance companies through optimal reinsurance.
method Solving the problem via neural networks and a Cramér-Lundberg model.
result Optimal reinsurance strategy found to control terminal wealth and ruin probability.
Estimates chirp signal frequencies using probabilistic models.
problem Estimating instantaneous frequencies of chirp signals when true forms are unknown.
method Non-linear Gaussian processes and stochastic filters/smothers for posterior estimation.
result The method outperforms state-of-the-art methods on synthetic and real-world datasets.
We improve maximum likelihood for location estimation in finite samples.
problem Estimating a parameter from samples with unknown or varying distribution.
method Use smoothed Fisher information for finite sample size and varying distributions.
result Recover optimal estimation theory for finite n and arbitrary f. Unified geometric interpretation of statistical estimation inequalities.
problem Curvature corrections in parametric statistical estimation.
method Cartan-geometric jet bundle formulation and jet prolongations.
result Unified geometric interpretation of higher-order information inequalities.
A new method for state estimation in state-space models using incomplete data.
problem State estimation in nonlinear state-space models with incomplete observations.
method Statistical analysis of incomplete observations, score function, observed information matrices, EM-gradient-particle filtering.
result Maximum likelihood estimation of state-vector with explicit form of observed information matrix.
Optimal reinsurance strategy with fixed cost and exponential preferences.
problem Maximizing expected utility of terminal wealth with fixed reinsurance cost.
method Two-step procedure: stochastic control and optimal stopping problem.
result Deterministic optimal strategy depends on model parameters.
Novel method uses Bayesian filters and PCRLB for state estimation of option prices.
problem Estimating unobserved latent variables from option prices.
method Posterior Cramer-Rao Lower Bound (PCRLB) based adaptive state estimation using various Bayesian filters.
result Proposed method outperforms individual filters and improves forecasting.
Geometric approach to thermodynamics of chemical reaction networks.
problem Thermodynamics of chemical reaction networks with non-ideal behavior.
method Information geometry, Riemannian geometry, Cramer-Rao bound, absolute sensitivity.
result Absolute sensitivity is a projection operator onto the tangent bundle of the equilibrium manifold.
Typical dimensionality reduction methods focus on directly reducing the number of random variables while retaining maximal variations in the data. In this paper, we consider the dimensionality reduction in parameter spaces of binary multivariate distributions. We propose a general Confident-Information-First (CIF) prin…
Study optimizes insurance investment to maximize utility across all capital levels.
problem Maximizing expected utility across all capital levels in an insurance company's investment strategy.
method Dynamic Programming Principle and Hamilton-Jacobi-Bellman (HJB) equation to prove existence of optimal strategy.
result Existence of optimal investment strategy proven under certain conditions.