# 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

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

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

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`

:

### 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:

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

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

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:

## Overview of the solutions in BridgeSDEInference.jl

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

`BridgeSDEInference.mcmc`

— Function`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 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):
- Bayesian inference with guided proposals for diffusions observed exactly and discretely in time:
- 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].

- Frank van der Meulen, Moritz Schauer.
- 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].

- Joris Bierkens, Frank van der Meulen, Moritz Schauer.
- 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].

- Frank van der Meulen, Moritz Schauer.

### First passage time set_observations

None published

### Mixed-effect models

None published