Consistency of Bayesian nonparametric inference for discretely observed jump diffusions

Koskela, Jere; Spanò, Dario; Jenkins, Paul A.

doi:10.3150/18-bej1050

Cited by 9 publications

(8 citation statements)

References 49 publications

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“…Lemma 1 of [Koskela et al, 2017] yields that the topology generated by U φ f,ε is Hausdorff, and hence separates points.…”

Section: By Fubini's Theoremmentioning

confidence: 99%

“…one which does not assign full mass to the truth), including parametric families. In Theorem 2 we adapt a result of [Koskela et al, 2017] to provide verifiable criteria on the prior for posterior consistency when time series data is available. We also show in Section 5 that the popular Dirichlet process mixture model prior [Lo, 1984] satisfies these conditions.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

Koskela¹,

Jenkins²,

Spanò³

2018

Bernoulli

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We investigate Bayesian non-parametric inference of the Λ-measure of Λ-coalescent processes with recurrent mutation, parametrised by probability measures on the unit interval. We give verifiable criteria on the prior for posterior consistency when observations form a time series, and prove that any non-trivial prior is inconsistent when all observations are contemporaneous. We then show that the likelihood given a data set of size n ∈ N is constant across Λ-measures whose leading n−2 moments agree, and focus on inferring truncated sequences of moments. We provide a large class of functionals which can be extremised using finite computation given a credible region of posterior truncated moment sequences, and a pseudo-marginal Metropolis-Hastings algorithm for sampling the posterior. Finally, we compare the efficiency of the exact and noisy pseudo-marginal algorithms with and without delayed acceptance acceleration using a simulation study. 1 arXiv:1512.00982v5 [stat.ME] 23 Jan 2017 and incorrect inference. Consequently, likelihood-based inference for Λ-coalescents has also been an active area of research [Birkner and Blath, 2008, Birkner et al., 2011, Koskela et al., 2015. A review of Λ-coalescents and their use in population genetic inference can be found in [Birkner and Blath, 2009], and Steinrücken et al. [2013] present a review of Beta-coalescent models for marine species. In this paper we make the following contributions:1. We provide the first non-parametric analysis of inferring Λ-measures from genetic data.Our method is also the first instance of inferring Λ using the Bayesian paradigm.2. We prove inconsistency of the posterior in full generality when data is contemporaneous, and give verifiable criteria for consistency when the data set forms a time series.3. We present an implementable parametrisation of the non-parametric inference problem by quotienting the infinite dimensional space M 1 ([0, 1]) in a suitable, data-driven way. We believe this quotienting approach to have utility in infinite dimensional inference beyond the context of this work.4. We implement a pseudo-marginal MCMC algorithm for sampling the posterior, and provide an illustrative simulation study which demonstrates the feasibility of the algorithm for inference.

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“…Lemma 1 of [Koskela et al, 2017] yields that the topology generated by U φ f,ε is Hausdorff, and hence separates points.…”

Section: By Fubini's Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

Koskela¹,

Jenkins²,

Spanò³

2018

Bernoulli

Self Cite

View full text Add to dashboard Cite

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“…where L 2 is the usual space of square integrable functions with respect to Lebesgue measure. Occasionally it will be convenient to replace the L 2 -inner product above by the L 2 (µ)-inner product, where µ is the invariant measure of (X t : t ≥ 0), which by (24) induces a norm which is equivalent to the norm induced by (16). We will also use the fractional Sobolev spaces H s for real s ≥ 0, which are obtained by interpolation, see [17].…”

Section: S Wangmentioning

confidence: 99%

“…In other sampling schemes, various methods have been studied, see e.g. [15] for a frequentist approach, [13,16,28,1,22] for recent posterior consistency and contraction rate results for Bayesian methods as well as [25,26] for MCMC methodology for the computation of the Bayesian posterior.…”

Section: Introductionmentioning

confidence: 99%

The nonparametric LAN expansion for discretely observed diffusions

Wang¹

2019

Electron. J. Statist.

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Consider a scalar reflected diffusion (Xt : t ≥ 0), where the unknown drift function b is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of (X 0 , X ∆ , ..., X n∆ ) for some fixed sampling distance ∆ > 0, the model satisfies the local asymptotic normality (LAN) property, assuming that b satisfies some mild regularity assumptions. This is established by using the connections of diffusion processes to elliptic and parabolic PDEs. The key tools used are regularity estimates for certain parabolic PDEs as well as a detailed analysis of the spectral properties of the elliptic differential operator related to (Xt : t ≥ 0). MSC 2010 subject classifications: 62M99. 1329 1330 S. Wang LAN expansion for discretely observed diffusions 1331• The second main ingredient consists of two well known limit theorems, the central limit theorem for martingale difference sequences [4] and the ergodic theorem, which ensure the right limits for the first and second order terms in the Taylor expansion respectively.

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“…While such Bayesian methods are attractive in applications [15], [27], [35], particularly since they provide associated uncertainty quantification procedures ('credible regions'), our understanding of their frequentist sampling performance is extremely limited. This is particularly so in the 'low frequency' regime when ∆ > 0 is thought to be fixed: the only references we are aware of are the consistency results in [31], [16], [18], which only hold under the very restrictive assumption that σ is constant and known, and only in a weak topology. As pointed out by Stuart [27] and van Zanten [35], obtaining theoretical performance guarantees for Bayesian algorithms in nonlinear inverse problems is, however, of key importance if such methods are to be used in scientific applications.…”

mentioning

confidence: 99%

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

Nickl¹,

Söhl²

2017

Ann. Statist.

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We consider nonparametric Bayesian inference in a reflected diffusion model dXt = b(Xt)dt+σ(Xt)dWt, with discretely sampled observations X0, X∆, . . . , Xn∆. We analyse the nonlinear inverse problem corresponding to the 'low frequency sampling' regime where ∆ > 0 is fixed and n → ∞. A general theorem is proved that gives conditions for prior distributions Π on the diffusion coefficient σ and the drift function b that ensure minimax optimal contraction rates of the posterior distribution over Hölder-Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest. Primary 62G05; secondary 60J60 62F15 62G20

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Consistency of Bayesian nonparametric inference for discretely observed jump diffusions

Cited by 9 publications

References 49 publications

Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

Bayesian non-parametric inference for $\Lambda$-coalescents: Posterior consistency and a parametric method

The nonparametric LAN expansion for discretely observed diffusions

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

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