# Differential Equations as Generative Models

I wonder, if we can say stochastic differential equations are generative models. I usually think about the Kalman filtering for example, we fix a discrete-time evolution equation of a certain object, which is a differential equation in continuous domain, and then use this model as a generative model to derive inference algorithms in a Bayesian setting.

Then, can we say that all differential equations involving noise terms are generative models of some underlying phenomenon?

PS (December 7, 2012): Although it is discussed in the comments, I push my luck once more by adding a concrete example.

Consider a stochastic difference equation, $$x_{t+1} = x_t + w_t$$ where $w_t \sim \mathcal{N}(0,1)$. Then, we can write in the probabilistic terms, $$p(x_{t+1}|x_t) = \mathcal{N}(0,x_t)$$ and this is a generative model. Of course, we can write another equation, such as an observation model, and this picture completely define a discrete-time stochastic dynamical system as well as a generative model. Then I reask my question: is there a specific reference that promotes this relationship? Or is it quite trivial and obvious for researchers from both sides?

Thanks.

• This sounds more like a matter of speculation best reserved for a discussion forum or chat room. Is there a particular problem of statistics, data analysis, or machine learning this question might help solve?
– whuber
Nov 23, 2012 at 16:03
• I do not agree that it is a matter of speculation. Well defined motion models in Hamiltonian notation can be used for probabilistic inference problems therefore I wonder about whether these particular models can be represented in a some kind of unified framework. Actually, once this connection established in my mind, it could solve a plethora of machine learning problems. Thanks.
– user16776
Nov 24, 2012 at 17:19

If you know the differential equation and its boundary conditions, and if it's linear, I would imagine that you could generate a Gaussian process therefrom using the Green's function of the linear operator. L of the transformed variates would be white noise.

This would be similar to smoothing normal IID variates by taking their inner product with the square root of a covariance matrix S, say (i.e. take the eigenfunctions times the root of the eigenvalues). The transformed variables would be multivariate normal with covariance S.

See Ramsay and Silverman, *Functional Data

To be specific, if $G(s,t)$ is the Green's function, then a Gaussian process can be formed from $$\int G(s,t)\, dW(t)$$

where $W$ is a Wiener process and integration is over an appropriate interval. I give a simple example of this on my blog.

• Thanks for the reference (I can't upvote (reputation issues)). When a stochastic differential equation is converted to a difference equation, in a probabilistic framework, it immediately defines a generative model. For instance define an equation such that $$x_{t+1} = x_t + u_t + w_t$$ where $w_t$ is an arbitrary noise term with associated probability distribution. I think, this is a stochastic difference equation and also is a generative model. I wonder about, is there a specific reference which promotes this obvious connection?
– user16776
Nov 24, 2012 at 17:22
• In the discrete case, the difference equation gives you an algorithm for producing data that fit the model. In a continuous time situation, it is less clear how you would do that. When Jim Ramsay and I worked on this, our focus was in defining classes of spline functions that related naturally to a linear equation. The spline function would be an estimate of the actual Gaussian process, assuming one is thinking in those terms. In what context are you studying these processes? Nov 24, 2012 at 18:50
• I am working on inference problems in dynamical systems which involve models like above mentioned model. The motivation is: since these models can be represented as a generative model, inference algorithms can be immediately applied, therefore they can use for inference problems. To make concrete, suppose simpler model, $$x_{t+1} = x_t + w_t$$ where $w_t$ is a zero mean, unit variance Gaussian random variable and $x, w \in \mathbb{R}$. Then, this model will correspond to the following generative model:$$p(x_{t+1}|x_t) = \mathcal{N}(x_t,1)$$ or $$x_{t+1}|x_t \sim \mathcal{N}(x_t,1)$$
– user16776
Nov 24, 2012 at 19:19
• By the way, I note this reference which can be useful to interested ones: era.lib.ed.ac.uk/bitstream/1842/4157/2/Murray2009.pdf
– user16776
Nov 24, 2012 at 19:33