# BridgeSDEInference.jl

MCMC sampler for inference for diffusion processes with the use of Guided Proposals using the package Bridge.jl. Currently under development.

## Problem statements

### Partially observed diffusion

Consider a stochastic process $X$, a solution to the stochastic differential equation

$$$\label{eq:mainSDE} d X_t = b_{\theta}(t,X_t)dt + \sigma_{\theta}(t,X_t)dW_t,\quad t\in[0,T],\quad X_0=x_0,$$$

where $b_{\theta}:R^d\to R^d$ is the drift function, $\sigma_{\theta}:R^d\to R^{d\times d'}$ is the volatility coefficient and $W$ is a $d'$-dimensional standard Brownian motion. We refer to $X$ as a diffusion. $\theta\in R^p$ is some unknown parameter. Suppose that a linearly transformed $X$, perturbed by Gaussian noise is observed at some collection of time points, i.e. that

$$$\label{eq:partObsData} D:=\{V_{t_i}; i=1,\dots N\},$$$

is observed for some $t_i, i=1,\dots N$, where

$$$\label{eq:partialObservations} V_{t_i}=L_i X_{t_i} + \xi_i,$$$

with $L_i\in R^{d_i\times d}$ and independent $\xi_i\sim Gsn(0,\Sigma_i)$, $(i=1,\dots,d)$. Suppose further that $\theta$ is equipped with some prior distribution $\pi(\theta)$. The aim is to estimate the posterior distribution over $\theta$, given the observation set D:

$$$\label{eq:posterior} \pi(\theta|D)\propto \pi(\theta)\pi(D|\theta).$$$

### First passage time observations

Consider a stochastic differential equation (\ref{eq:mainSDE}) and suppose that the first coordinate of the drift term $b_{\theta}^{[1]}(t,x):R^d\to R$ is linear in $x$, whereas the first row of the volatility coefficient is identically equal to a zero vector $\sigma_{\theta}^{[1,1:d']}=0$. Suppose further that instead of partial observation scheme as in (\ref{eq:partObsData}), the process $X$ is observed at a collection of stopping times:

$D:=\{\tau^\star_i, i=1,\dots,N\},$

where $\tau^\star_i$'s are defined by

\begin{align*} \tau_{\star,0}&:=0\\ \tau^\star_{i+1}&:=\inf\{t\geq \tau_{\star,i}: X_t^{[1]}\geq l^\star_{i+1}\},\quad i=0,\dots\\ \tau_{\star,i}&:=\inf\{t\geq \tau^\star_{i}: X_t^{[1]}\leq l_{\star,i}\},\quad i=1,\dots, \end{align*}

for some known constants $l^\star_i$, $l_{\star,i}$, $i=1,\dots$. Note that $\tau^\star_i$'s are the first passage times of the first coordinate process to some thresholds $l^\star_i$, whereas $\tau_{\star,i}$ are the (latent) renewal times. The aim is to find a posterior (\ref{eq:posterior}) from such first passage time data.

### Mixed-effects models

Consider $M$ stochastic differential equations

$$$d X^{(i)}_t = b_{\theta,\eta_i}(t,X^{(i)}_t)dt + \sigma_{\theta,\eta_i}(t,X^{(i)}_t)dW^{(i)}_t,\quad t\in[0,T],\quad X^{(i)}_0=x^{(i)}_0,\quad i=1,\dots,M.$$$

where $\theta\in R^{p}$ denotes the parameter that is shared among all of $X^{(i)}$'s, whereas $\eta_i\in R^{p_i}$, $i=1,\dots,M$ are the parameters specific to a given $X^{(i)}$. Suppose that the observations of the type (\ref{eq:partObsData}) are given for each trajectory $X^{(i)}$ (let's denote a joint set with $D^\star:=\cup_{i=1}^M D_i$). The aim is to find the posterior:

$\pi(\theta,\{\eta_i,i=1,\dots M\}|D^{\star})\propto \pi(\theta)\prod_{i=1}^M\pi(\eta_i)\pi(D_i|\theta,\eta_i).$

## Overview of the solutions in BridgeSDEInference.jl

The first two problems from Problem statements are addressed by the function

BridgeSDEInference.mcmcFunction
mcmc(setup_mcmc::MCMCSetup, schedule::MCMCSchedule, setup)  <: ModelSetup

function for running the mcmc. receives as imput MCMCSetup, MCMCschedule and setup. See MCMCSetup, MCMCSchedule. setup typically is DiffusionSetup making additional setup choices when using the MCMC infrastructure to sample diffusion processes. See DiffusionSetup

The third problem (Mixed-effects models) is addressed by the function (TODO fix so that points to a function in repeated.jl)

Missing docstring.

Missing docstring for mcmc. Check Documenter's build log for details.

Please see the Tutorial section to see how to appropriately initialise setup, run the mcmc function and query the results.

## References

These are only the references which describe the algorithms implemented in this package. Note in particular that the list of references which treat the same problems as addressed by this package, but use methods which are not based on guided proposals is much, much longer. We refer to the bibliography sections of the papers listed below for references to other approaches.

### Partial observations of a diffusion

• Guided proposals for diffusion bridges (no-noise setting):
• Moritz Schauer, Frank van der Meulen, Harry van Zanten. Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli, 23(4A), 2017, pp. 2917–2950. [Bernoulli], [arXiv].
• Bayesian inference with guided proposals for diffusions observed exactly and discretely in time:
• Frank van der Meulen, Moritz Schauer. Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Electronic Journal of Statistics 11 (1), 2017. [EJoS], [arXiv].
• Bayesian inference with guided proposals for diffusions observed with noise (according to (\ref{eq:partialObservations})):
• Frank van der Meulen, Moritz Schauer. Bayesian estimation of incompletely observed diffusions. Stochastics 90 (5), 2018, pp. 641–662. [Stochastics], [arXiv].
• Simulation of hypo-elliptic diffusion bridges:
• Joris Bierkens, Frank van der Meulen, Moritz Schauer. Simulation of elliptic and hypo-elliptic conditional diffusions. arXiv, 2018. [arXiv].
• Efficient schemes for computing all the necessary term for a fully generic implementation of guided proposals:
• Frank van der Meulen, Moritz Schauer. Continuous-discrete smoothing of diffusions. arXiv, 2017. [arXiv].

None published

None published