(This is mostly for Stan math devs)

@James_Savage sent me the tweet thread below

https://twitter.com/DavidDuvenaud/status/1215347970159382534?s=19

Var doesnt have any templates so we would need a new type, but could we just store the gaussian noise in a var and keep doing something similar to what we do now?

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*When* we move up to C++17 adding this would be a lot easier because class templates are automatically deduced by the constructor so

```
template <typename Arith, typename NoiseArith = double>
class var {
using Scalar = Arith;
using Noise = NoiseArith;
// stuff here
explicit var(Arith x) {
// ...
}
explicit var(Arith x, Noise x) {
// ...
}
};
var<double> a(10.0); // fine!
var a(10.0); // fine!
var<double, float> a(10.0 , -1.96); // also fine!
var a(10.0 , -1.96); // also fine!
```

Gives us full backwards compatability!

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That looks really interestring. Iâ€™ve been reading through how autograd works in Dougal Maclaurinâ€™s thesis.

I think this stuff is now getting implemented in JAX. Hereâ€™s their Autodiff cookbook

Also, this is very relevant for the kinds of econometric equilibrium models we want to fit.

Neat! Iâ€™ll have to give that a read-through

The main bummer is the C++17 wait for this, but once R bumps up we can look at how Python can handle 17 and go from there

Iâ€™m not sure what the need for augmenting `var`

s is. The SDE method is basically an adjoint ODE method with implicit marginalization over the diffusion, and it can be implemented in a similar was to our current ODE solvers. On the forward pass the Brownian bridge realization reconstructor is set up and stored, and then on the backwards pass the Jacobian-vector product is computed using a reverse-time SDE solve and the fixed Brownian bridge realization.

Oh, it needs to be Jim to get your attention (cf. Adjoint sensitivity method for stochastic differential equations)

I met David Duvenaud before Xmas and he said he would be happy to help to get this in Stan.

although it seems Mike already knows what to do.

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Coincidentally @charlesm93, @vianeylb, and I have been working through old adjoint methods (and a shiny new, albeit somewhat obscure one). The first step is to expose CVODES adjoint method which will require storing enough of the checkpointing and interpolation during the forward solve to support the reverse solve. Charles has expressed interest in working this out with me.

Once thatâ€™s been demonstrated we can then think about this SDE method. The real novelty of the adjoint method is that it defines a well-posed derivative of the SDE solution, allowing us to have SDEs at all let alone with efficient Jacobian-vector products.

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For some reason I missed your previous post on this, too.

We want to solve SDEs for the Sloan project for equilibrium in econometric models, so itâ€™d be great if we could get this into Stan.

He was super patient in explaining to me how what I was calling â€ślazy chainingâ€ť related to the adjoint-Jacobian formulation of reverse mode. @bbbales2 then figured out how to code the adjoint-Jacobian products in Stan neatly to encapsulate all stack fiddling.

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