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[MCMC] Add diagnostics and a potential sequal #335

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@HumphreyYang

These are summerized from a discussion with @jstac:

Some minor suggestions that we could add to the lecture on how to diagnose MCMC and where the Metropolis–Hastings algorithm may not perform well:

  • Add trace plots for the JAX sampler under different proposal scales.
  • Add autocorrelation plots for the same chains.
  • Add a 2D correlated posterior example with contours and the chain trajectory overlaid.
  • Use the 2D example to illustrate where random-walk Metropolis struggles in finite samples.

In the sequel, we could:

  • Study how these issues motivate MALA and HMC/NUTS.
  • Connect these algorithms to what is used in numpyro.

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