TuringLang / JuliaBUGS.jl / 11174363245
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Add some benchmarking code with Stan and Nimble (#214)

The scripts can be used to run examples from Volume 1 of WinBUGS
examples.

Stan examples are more optimized (vectorized computation, human effort
went into transformed data and generated quantities, etc.).
~~Regardless, on these smaller problems, we should be able to match it's
performance when compilation works.~~ With `ReverseDiff` compiled tape,
JuliaBUGS gradient and density computation is several times (5- 10x)
slower than Stan. `ReverseDiff` is quite optimized, but it still has
some fundamental overhead with the tape. To get comparable performance
as Stan, I think we need to generate Julia code and use `Mooncake` or
`Enzyme`.

Nimble's interface is a bit more tricky, the logp and gradient
computation code is from [Chapter
17](https://r-nimble.org/html_manual/example-maximum-likelihood-estimation-using-optim-with-gradients-from-nimderivs..html).
(code from [Chapter
16](https://r-nimble.org/html_manual/cha-AD.html#derivatives-involving-model-calculations)
runs slowly). I am a bit confused by how to write `nimblefunction`. But
the speed is good and it seems to work. Nimble's BUGS examples are a bit
out-of-date, it looks to me some examples are from WinBUGS 0.5 era. But
the models are generally the same.

I think moving forward, use Stan as the target is good, easy to setup
and also a good performance aim.

- [x] integrate Stan into the current benchmark code

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Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>

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