| 1. |
- Arnaudon, Alexis, et al.
(författare)
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Diffusion Bridges for Stochastic Hamiltonian Systems and Shape Evolutions
- 2022
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Ingår i: SIAM Journal on Imaging Sciences. - 1936-4954. ; 15:1, s. 293-323
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Tidskriftsartikel (refereegranskat)abstract
- Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modeling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalization as diffusion bridges in a sto-chastic setting. Simulation of such bridges is key to solving inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models, thereby substantially expanding the appli-cability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.
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| 2. |
- Arnaudon, A., et al.
(författare)
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Diffusion Bridges for Stochastic Hamiltonian Systems and Shape Evolutions\ast
- 2022
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Ingår i: Siam Journal on Imaging Sciences. - : Society for Industrial & Applied Mathematics (SIAM). - 1936-4954. ; 15:1, s. 293-323
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Tidskriftsartikel (refereegranskat)abstract
- Stochastically evolving geometric systems are studied in shape analysis and computational anatomy for modeling random evolutions of human organ shapes. The notion of geodesic paths between shapes is central to shape analysis and has a natural generalization as diffusion bridges in a sto-chastic setting. Simulation of such bridges is key to solving inference and registration problems in shape analysis. We demonstrate how to apply state-of-the-art diffusion bridge simulation methods to recently introduced stochastic shape deformation models, thereby substantially expanding the appli-cability of such models. We exemplify these methods by estimating template shapes from observed shape configurations while simultaneously learning model parameters.
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| 3. |
- Arya, Gaurav, et al.
(författare)
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Automatic Differentiation of Programs with Discrete Randomness
- 2022
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Ingår i: Advances in Neural Information Processing Systems. - 1049-5258. ; 35
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Konferensbidrag (refereegranskat)abstract
- Automatic differentiation (AD), a technique for constructing new programs which compute the derivative of an original program, has become ubiquitous throughout scientific computing and deep learning due to the improved performance afforded by gradient-based optimization. However, AD systems have been restricted to the subset of programs that have a continuous dependence on parameters. Programs that have discrete stochastic behaviors governed by distribution parameters, such as flipping a coin with probability p of being heads, pose a challenge to these systems because the connection between the result (heads vs tails) and the parameters (p) is fundamentally discrete. In this paper we develop a new reparameterization-based methodology that allows for generating programs whose expectation is the derivative of the expectation of the original program. We showcase how this method gives an unbiased and low-variance estimator which is as automated as traditional AD mechanisms. We demonstrate unbiased forward-mode AD of discrete-time Markov chains, agent-based models such as Conway's Game of Life, and unbiased reverse-mode AD of a particle filter. Our code package is available at https://github.com/gaurav-arya/StochasticAD.jl.
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| 4. |
- Belomestny, Denis, et al.
(författare)
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Nonparametric Bayesian inference for Gamma-type Lévy subordinators
- 2019
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Ingår i: Communications in Mathematical Sciences. - 1539-6746 .- 1945-0796. ; 17:3, s. 781-816
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Tidskriftsartikel (refereegranskat)abstract
- Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Levy density of a Levy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size n ->infinity, concentrates around the Levy density under which the data have been generated. Finally, we test our method on a classical insurance dataset.
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| 5. |
- Belomestny, Denis, et al.
(författare)
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Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
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Ingår i: Bernoulli. - 1350-7265. ; 28:4, s. 2151-2180
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Tidskriftsartikel (refereegranskat)abstract
- We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward procedure for posterior inference via an MCMC procedure. We give theoretical performance guarantees (minimax optimal contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.
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| 6. |
- Belomestny, D., et al.
(författare)
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Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations
- 2022
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Ingår i: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 28:4, s. 2151-2180
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Tidskriftsartikel (refereegranskat)abstract
- We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation driven by a gamma process. The volatility function is modelled a priori as piecewise constant, and we specify a gamma prior on its values. This leads to a straightforward procedure for posterior inference via an MCMC procedure. We give theoretical performance guarantees (minimax optimal contraction rates for the posterior) for the Bayesian estimate in terms of the regularity of the unknown volatility function. We illustrate the method on synthetic and real data examples.
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| 7. |
- Belomestny, Denis, et al.
(författare)
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Weak solutions to gamma-driven stochastic differential equations
- 2023
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Ingår i: Indagationes Mathematicae. - 0019-3577. ; 34:4, s. 820-829
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Tidskriftsartikel (refereegranskat)abstract
- We study a stochastic differential equation driven by a gamma process, for which we give results on the existence of weak solutions under conditions on the volatility function. To that end we provide results on the density process between the laws of solutions with different volatility functions.
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| 8. |
- Bierkens, Joris, et al.
(författare)
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A piecewise deterministic Monte Carlo method for diffusion bridges
- 2021
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Ingår i: Statistics and Computing. - : Springer Science and Business Media LLC. - 0960-3174 .- 1573-1375. ; 31:3
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Tidskriftsartikel (refereegranskat)abstract
- We introduce the use of the Zig-Zag sampler to the problem of sampling conditional diffusion processes (diffusion bridges). The Zig-Zag sampler is a rejection-free sampling scheme based on a non-reversible continuous piecewise deterministic Markov process. Similar to the Lévy–Ciesielski construction of a Brownian motion, we expand the diffusion path in a truncated Faber–Schauder basis. The coefficients within the basis are sampled using a Zig-Zag sampler. A key innovation is the use of the fully local algorithm for the Zig-Zag sampler that allows to exploit the sparsity structure implied by the dependency graph of the coefficients and by the subsampling technique to reduce the complexity of the algorithm. We illustrate the performance of the proposed methods in a number of examples.
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| 9. |
- Bierkens, Joris, et al.
(författare)
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Simulation of elliptic and hypo-elliptic conditional diffusions
- 2020
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Ingår i: Advances in Applied Probability. - : Cambridge University Press (CUP). - 0001-8678 .- 1475-6064. ; 52:1, s. 173-212
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Tidskriftsartikel (refereegranskat)abstract
- Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time 0$ ]]> only linear combinations of its components are observed. That is, one only observes the vector for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model.
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| 10. |
- Bierkens, Joris, et al.
(författare)
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Sticky PDMP samplers for sparse and local inference problems
- 2023
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Ingår i: Statistics and Computing. - : Springer Science and Business Media LLC. - 0960-3174 .- 1573-1375. ; 33
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Tidskriftsartikel (refereegranskat)abstract
- We construct a new class of efficient Monte Carlo methods based on continuous-time piecewise deterministic Markov processes (PDMPs) suitable for inference in high dimensional sparse models, i.e. models for which there is prior knowledge that many coordinates are likely to be exactly 0. This is achieved with the fairly simple idea of endowing existing PDMP samplers with “sticky” coordinate axes, coordinate planes etc. Upon hitting those subspaces, an event is triggered during which the process sticks to the subspace, this way spending some time in a sub-model. This results in non-reversible jumps between different (sub-)models. While we show that PDMP samplers in general can be made sticky, we mainly focus on the Zig-Zag sampler. Compared to the Gibbs sampler for variable selection, we heuristically derive favourable dependence of the Sticky Zig-Zag sampler on dimension and data size. The computational efficiency of the Sticky Zig-Zag sampler is further established through numerical experiments where both the sample size and the dimension of the parameter space are large.
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| 11. |
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| 12. |
- Gugushvili, S., et al.
(författare)
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Bayesian wavelet de-noising with the caravan prior
- 2020
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Ingår i: Esaim-Probability and Statistics. - : EDP Sciences. - 1292-8100 .- 1262-3318. ; 23, s. 947-978
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Tidskriftsartikel (refereegranskat)abstract
- According to both domain expert knowledge and empirical evidence, wavelet coefficients of real signals tend to exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). A wavelet de-noising approach that takes into account such a feature of the signal may in practice outperform other, more vanilla methods, both in terms of the estimation error and visual appearance of the estimates. Motivated by this observation, we present a Bayesian approach to wavelet de-noising, where dependencies between neighbouring wavelet coefficients are a priori modelled via a Markov chain-based prior, that we term the caravan prior. Posterior computations in our method are performed via the Gibbs sampler. Using representative synthetic and real data examples, we conduct a detailed comparison of our approach with a benchmark empirical Bayes de-noising method (due to Johnstone and Silverman). We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox.
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| 13. |
- Gugushvili, Shota, et al.
(författare)
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Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient
- 2020
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Ingår i: Brazilian Journal of Probability and Statistics. - 0103-0752. ; 34:3, s. 537-579
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Tidskriftsartikel (refereegranskat)abstract
- We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Hölder-continuous diffusion coefficient with smoothness parameter 0<λ≤1. Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.
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| 14. |
- Gugushvili, Shota, et al.
(författare)
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Nonparametric Bayesian volatility estimation
- 2019
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Ingår i: 2017 MATRIX Annals. - Cham : Springer International Publishing. - 9783030041601 ; , s. 279-302
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Bokkapitel (övrigt vetenskapligt/konstnärligt)
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| 15. |
- Gugushvili, S., et al.
(författare)
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Nonparametric Bayesian volatility learning under microstructure noise
- 2023
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Ingår i: Japanese Journal of Statistics and Data Science. - : Springer Science and Business Media LLC. - 2520-8756 .- 2520-8764. ; 6, s. 551-71
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we study the problem of learning the volatility under market microstructure noise. Specifically, we consider noisy discrete time observations from a stochastic differential equation and develop a novel computational method to learn the diffusion coefficient of the equation. We take a nonparametric Bayesian approach, where we a priori model the volatility function as piecewise constant. Its prior is specified via the inverse Gamma Markov chain. Sampling from the posterior is accomplished by incorporating the Forward Filtering Backward Simulation algorithm in the Gibbs sampler. Good performance of the method is demonstrated on two representative synthetic data examples. We also apply the method on a EUR/USD exchange rate dataset. Finally we present a limit result on the prior distribution.
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| 16. |
- Mider, Marcin, et al.
(författare)
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Continuous-discrete smoothing of diffusions
- 2021
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Ingår i: Electronic Journal of Statistics. - 1935-7524. ; 15:2, s. 4295-4342
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Tidskriftsartikel (refereegranskat)abstract
- Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the Backward Filtering Forward Guiding (BFFG) algorithm. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in [53]. We illustrate its efficiency in a number of challenging problems.
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| 17. |
- Scherrer, Chad, et al.
(författare)
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Applied measure theory for probabilistic modeling.
- 2022
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Ingår i: JuliaCon Proceedings. - : The Open Journal. - 2642-4029. ; 2022:1
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Konferensbidrag (refereegranskat)abstract
- Probabilistic programming and statistical computing are vibrant areas in the development of the Julia programming language, but the underlying infrastructure dramatically predates recent developments. The goal of MeasureTheory.jl is to provide Julia with the right vocabulary and tools for these tasks. In the package we introduce a well-chosen set of notions from the foundations of probability together with powerful combinators and transforms, giving a gentle introduction to the concepts in this article. The task is foremost achieved by recognizing measure as the central object. This enables us to develop a proper concept of densities as objects relating measures with each others. As densities provide local perspective on measures, they are the key to efficient implementations. The need to preserve this computationally so important locality leads to the new notion of locally-dominated measure, solving the so-called “base measure problem” and making work with densities and distributions in Julia easier and more flexible.
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| 18. |
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| 19. |
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