Prior Parameters with MCMC + Extra Optimization Step

[jejjohnson]

I wanted to know where/how you put prior parameters for the distribution using the emcee algorithms? I could not find anything there except a note of a uninformative prior.

In addition, I notice you run the emcee sampler and then you perform some optimization scheme using the parameters with kde kerne l and a derivative-free optimization scheme? I’m curious if there is some reason for this? I’ve tried fitting GEVD/GPD distributions before with numpyro but I’ve always noticed that the fits with emcee seem to better follow the return period observations. I wonder if it has something to do with this extra step… Any thoughts/comments?

[georgebv]

Priors are set here:

pyextremes/src/pyextremes/models/model_emcee.py

Line 73 in ce530c4

initial_state=self.distribution.get_initial_state(n_walkers=n_walkers),

Where it calls this:

pyextremes/src/pyextremes/models/distribution.py

Lines 225 to 252 in ce530c4

	def get_initial_state(self, n_walkers: int) -> np.ndarray:
	"""
	Get initial positions for the ensemble sampler walkers.
	This method is used by the 'Emcee' model.
	Positions are sampled from a normal distribution
	for each of the free distribution parameters (e.g. c, loc, scale)
	with location for each of them taken from scipy.stats MLE fit
	and standard deviation being 0.01.
	Parameters
	----------
	n_walkers : int
	Number of walkers used by the sampler.
	Returns
	-------
	initial_positions : numpy.ndarray
	Array with initial positions of the ensemble sampler walkers.
	"""
	logger.debug("getting initial positions for %s walkers", n_walkers)
	mle_parameters = [self.mle_parameters[key] for key in self.free_parameters]
	return scipy.stats.norm.rvs(
	loc=mle_parameters,
	scale=0.01,
	size=(n_walkers, self.number_of_parameters),
	)

KDE kernel is used to find maximum aposteriori (MAP) for each parameter. Once sampler is done, we have a large number of estimates for each of the distribution parameters (shape, loc, scale) - we need to get a representative number ("central fit") for those. I use gaussian KDE to find the mode for each of those. These are shown as orange lines on the corner plot:

https://raw.githubusercontent.com/georgebv/pyextremes-notebooks/master/notebooks/documentation/readme%20figures/corner.png

[jejjohnson]

Thank you! Makes sense now. Do you have any experience fitting specific priors? If so, I know it's data specific, but perhaps you've found some general good principals to follow. Or any resources at all would be helpful.

[georgebv]

Not really, these choices were purely emprirical - I chose what worked better. I tried different priors (including truly random) and found that the sampler will converge to solution regardless, you'll just have longer burn-in. Fitting GEVD/GPD is not a complex problem where you can get stuck in local maxima and miss the true global maxima.