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Parameter Estimation for Partially Observed McKean-Vlasov Diffusions

Ajay Jasra, Mohamed Maama, Raul Tempone

TL;DR

It is proved, under assumptions, that the expectation of the estimator is biased, but with expected small and controllable bias, and a new randomized multilevel Monte Carlo method is developed based upon Markovian stochastic approximation methodology for estimating the parameters.

Abstract

In this article we consider likelihood-based estimation of static parameters for a class of partially observed McKean-Vlasov (POMV) diffusion process with discrete-time observations over a fixed time interval. In particular, using the framework of [5] we develop a new randomized multilevel Monte Carlo method for estimating the parameters, based upon Markovian stochastic approximation methodology. New Markov chain Monte Carlo algorithms for the POMV model are introduced facilitating the application of [5]. We prove, under assumptions, that the expectation of our estimator is biased, but with expected small and controllable bias. Our approach is implemented on several examples.

Parameter Estimation for Partially Observed McKean-Vlasov Diffusions

TL;DR

It is proved, under assumptions, that the expectation of the estimator is biased, but with expected small and controllable bias, and a new randomized multilevel Monte Carlo method is developed based upon Markovian stochastic approximation methodology for estimating the parameters.

Abstract

In this article we consider likelihood-based estimation of static parameters for a class of partially observed McKean-Vlasov (POMV) diffusion process with discrete-time observations over a fixed time interval. In particular, using the framework of [5] we develop a new randomized multilevel Monte Carlo method for estimating the parameters, based upon Markovian stochastic approximation methodology. New Markov chain Monte Carlo algorithms for the POMV model are introduced facilitating the application of [5]. We prove, under assumptions, that the expectation of our estimator is biased, but with expected small and controllable bias. Our approach is implemented on several examples.

Paper Structure

This paper contains 29 sections, 5 theorems, 88 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Model
  4. Time Discretization
  5. Methodology and Theory
  6. Introduction
  7. Approximating the Laws
  8. Approximating the Smoother
  9. Modified Objective Function
  10. MCMC Method
  11. Approximating Pairs of Smoothers
  12. Stochastic Approximation
  13. Computation of $H_l^N$
  14. Final Algorithm
  15. Discussion of Methodology
  16. ...and 14 more sections

Key Result

Theorem 3.1

Assume (Aass:1_new-ass:4). Then we have that $\mathbb{E}[\widehat{\theta}_{\star}]=\theta_{\star}^L$ .

Figures (4)

  • Figure 1: Outputs of the stochastic Kuramoto model. (a) Phases of the oscillators over time. (b) Evolution of one oscillator's phase over time.
  • Figure 2: Outputs of the training neural networks with McKean-Vlasov stochastic differential equations. (a) Evolution of one neuron's activation over time $X_1$. (b) Histograms of weights and biases.
  • Figure 3: Log-log scale of MSE versus Cost for parameter estimation in the Kuramoto and neural network models, evaluated with $L=4$. (a) Estimation of parameter $\theta$, which governs synchronization dynamics in the Kuramoto model; (b) Estimation of neural network parameter $\alpha$; (c) Estimation of neural network parameter $\beta$.
  • Figure 4: Log-log scale of MSE versus Cost for parameter estimation in the Kuramoto and neural network models, evaluated with $L=8$. (a) Estimation of parameter $\theta$; (b) Estimation of neural network parameter $\alpha$; (c) Estimation of neural network parameter $\beta$.

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • Lemma A.4
  • proof
  • proof