# Consistent nonparametric Bayesian inference for discretely observed scalar diffusions

@article{Meulen2013ConsistentNB, title={Consistent nonparametric Bayesian inference for discretely observed scalar diffusions}, author={Frank van der Meulen and Harry van Zanten}, journal={Bernoulli}, year={2013}, volume={19}, pages={44-63} }

We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.

## 32 Citations

### Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations*

- Mathematics
- 2014

We consider nonparametric Bayesian estimation of the drift coefficient of a multidimensional stochastic differential equation from discrete-time observations on the solution of this equation. Under…

### Consistency of Bayesian nonparametric inference for discretely observed jump diffusions

- MathematicsBernoulli
- 2019

We introduce verifiable criteria for weak posterior consistency of identifiable Bayesian nonparametric inference for jump diffusions with unit diffusion coefficient and uniformly Lipschitz drift and…

### Bernoulli Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data

- Mathematics, Computer Science
- 2018

A general theorem detailing conditions under which Bayesian posteriors will contract in L–distance around the true drift function b0 at the frequentist minimax rate is proved.

### CONSISTENT NON-PARAMETRIC BAYESIAN ESTIMATION FOR A TIME-INHOMOGENEOUS BROWNIAN MOTION ∗

- Mathematics
- 2014

We establish posterior consistency for non-parametric Bayesian estimation of the dispersion coefficient of a time-inhomogeneous Brownian motion.

### Adaptive posterior contraction rates for empirical Bayesian drift estimation of a diffusion

- Mathematics
- 2019

Due to their conjugate posteriors, Gaussian process priors are attractive for estimating the drift of stochastic differential equations with continuous time observations. However, their performance…

### Nonparametric Bayesian Estimation of a Hölder Continuous Diffusion Coefficient

- MathematicsBrazilian Journal of Probability and Statistics
- 2020

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…

### Convergence of Bayesian Estimators for Diffusions in Genetics

- Mathematics
- 2020

The statistical properties of the Bayesian estimator for the selection coefficient in this model is analysed and it is shown that this estimator is uniformly consistent over compact sets, uniformly asymptotically normal, and displays uniform convergence of moments on compact sets.

### Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data

- Mathematics, Computer ScienceBernoulli
- 2019

The results show that the Bayesian method adapts both to an unknown sampling regime and to unknown smoothness.

### Nonparametric Bayesian Volatility Estimation

- Mathematics2017 MATRIX Annals
- 2019

Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate…

## References

SHOWING 1-10 OF 36 REFERENCES

### Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)

- Mathematics
- 2006

Monte Carlo methods are proposed, which build on recent advances on the exact simulation of diffusions, for performing maximum likelihood and Bayesian estimation for discretely observed diffusions.

### Convergence rates of posterior distributions

- Mathematics
- 2000

We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models. We give general results on the rate of convergence of the posterior…

### Likelihood INference for Discretely Observed Non-linear Diffusions

- Computer Science, Mathematics
- 1998

Bayesian estimation of non-linear stochastic differential equations when only discrete observations are available using a tuned MCMC method and by using the Euler-Maruyama discretisation scheme.

### Convergence rates of posterior distributions for non-i.i.d. observations

- Mathematics
- 2007

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general…

### Rates of contraction of posterior distributions based on Gaussian process priors

- Mathematics, Computer Science
- 2008

The rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process.

### On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm

- Mathematics, Computer Science
- 2001

A new Markov chain Monte Carlo approach to Bayesian analysis of discretely observed diffusion processes and shows that, because of full dependence between the missing paths and the volatility of the diffusion, the rate of convergence of basic algorithms can be arbitrarily slow if the amount of the augmentation is large.

### Rates of convergence of posterior distributions

- Mathematics
- 2001

We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are…

### New approaches to Bayesian consistency

- Computer Science, Psychology
- 2004

We use martingales to study Bayesian consistency. We derive sufficient conditions for both Hellinger and Kullback-Leibler consistency, which do not rely on the use of a sieve. Alternative sufficient…

### Bayesian inference for nonlinear multivariate diffusion models observed with error

- Computer ScienceComput. Stat. Data Anal.
- 2008

### Likelihood based inference for diffusion driven models

- Computer Science
- 2004

This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility…