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   Brown, Gavin; Gaboardi, Marco; Smith, Adam; Ullman, Jonathan; Zakynthinou, Lydia (January 2021, Advances in neural information processing systems)
2. Covariance-Aware Private Mean Estimation Without Private Covariance Estimation

   Brown, Gavin; Gaboardi, Marco; Smith, Adam; Ullman, Jonathan; Zakynthinou, Lydia (December 2021, Advances in Neural Information Processing Systems 34 (NeurIPS 2021))
3. Covariance-Aware Private Mean Estimation Without Private Covariance Estimation

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4. Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

   https://doi.org/10.5802/smai-jcm.62  

   Dunlop, Matthew M.; Helin, Tapio; Stuart, Andrew M. (April 2020, The SMAI journal of computational mathematics)

   null (Ed.)

   The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; it allows for principled learning of hyperparameters; and it can provide uncertainty quantification. The posterior distribution may in principle be sampled by means of MCMC or SMC methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization; it is important to study MAP estimators, however, because they provide a link with classical optimization approaches to inverse problems and the Bayesian link may be used to improve upon classical optimization approaches. The lack of invariance of MAP estimators under change of parameterization is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises naturally when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein–Uhlenbeck process. The applicability of the results is demonstrated concretely with the study of hierarchical Whittle–Matérn and ARD priors. 

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5. Hybrid Iterative Linear Quadratic Estimation: Optimal Estimation for Hybrid Systems

   https://doi.org/10.1109/LRA.2025.3540387  

   Payne, J Joe; Zhu, James; Kong, Nathan J; Johnson, Aaron M (April 2025, IEEE Robotics and Automation Letters)

   In this letter we present Hybrid iterative Linear Quadratic Estimation (HiLQE), an optimization based offline state estimation algorithm for hybrid dynamical systems. We utilize the saltation matrix, a first order approximation of the variational update through an event driven hybrid transition, to calculate gradient information through hybrid events in the backward pass of an iterative linear quadratic optimization over state estimates. This enables accurate computation of the value function approximation at each timestep. Additionally, the forward pass in the iterative algorithm is augmented with hybrid dynamics in the rollout. A reference extension method is used to account for varying impact times when comparing states for the feedback gain in noise calculation. The proposed method is demonstrated on an ASLIP hopper system with position measurements. In comparison to the Salted Kalman Filter (SKF), the algorithm presented here achieves a maximum of 63.55% reduction in estimation error magnitude over all state dimensions near impact events. 

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