How to sample independently with pymc3 - pymc3

Question

I am working with a simple bivariate normal model with a somewhat unconventional prior. The main issue I have is that my posteriors are inconsistent from one run to the next, which I'm guessing is related to an issue of high dependence between consecutive samples. Here are my specific questions.
What is the best way to get N independent samples? At the moment, I've been calling sample() to get a big chain (e.g. length 10,000) and then taking every 100th sample starting at 1,000. But looking now at an autocorrelation profile of one of the parameters, it looks like I need to take at least every 500th sample! (I could also use mutual information to get a better idea of dependence between lags.)
I've been following the fitting procedure described in the stochastic volatility example in the pymc3 tutorial. In particular I first find the MAP, then use it to generate a NUTS() object, then take a short sample, then use that to generate another NUTS() object, using gamma=0.25 (???), then finally get my big sample. I have no idea whether this is appropriate or whether I need the gamma=0.25.
Also, in that same example, there are testvals for the Exponential distribution. I don't know if I need these. (What is wrong with the default use of the mean?)
Here is the actual model I'm using.
import pymc3 as pymc
import numpy as np
import theano.tensor as th
from pymc3.distributions.continuous import Gamma, Uniform, Normal, Bounded
from pymc3.distributions.multivariate import MvNormal
from pymc3.model import Deterministic
data = np.random.randn(3000, 2) / 300 # I have actual data!
with pymc.Model():
tau = Gamma('tau', alpha=2, beta=1 / 20000)
sigma = Deterministic('sigma', 1 / th.sqrt(tau))
corr = Uniform('corr', lower=0, upper=1)
alpha_sig = Deterministic('alpha_sig', sigma / 50)
alpha_post = Normal('alpha_post', mu=0, sd=alpha_sig)
alpha_pre = Bounded(
'alpha_pre', Normal, alpha_post, np.Inf, mu=0, sd=alpha_sig)
corr_inv = th.stack([th.stack([1, -corr]),
th.stack([-corr, 1])]) / (1 - th.sqr(corr))
MvNormal(
'data', mu=th.stack([alpha_post, alpha_pre]),
tau=tau * corr_inv, observed=data)
map_ = pymc.find_MAP()
step1 = pymc.NUTS(scaling=map_)
trace1 = pymc.sample(1000, step=step1)
step2 = pymc.NUTS(scaling=trace1[-1], gamma=0.25)
trace2 = pymc.sample(10000, step=step2, start=trace1[-1])

Answer 1

I'm not sure what you're doing with the complex prior structure you have set up but I think there is something wrong there.
I simplified the model to:
import pymc3 as pymc
import numpy as np
import theano.tensor as th
from pymc3.distributions.continuous import Gamma, Uniform, Normal, Bounded
from pymc3.distributions.multivariate import MvNormal
from pymc3.model import Deterministic
data = np.random.randn(3000, 2) # I have actual data!
with pymc.Model():
corr = Uniform('corr', lower=0, upper=1)
corr_inv = th.stack([th.stack([1, -corr]),
th.stack([-corr, 1])]) / (1 - th.sqr(corr))
mu = Normal('mu', mu=0, sd=1, shape=2)
MvNormal('data',
mu=mu,
tau=corr_inv,
observed=data)
map_ = pymc.find_MAP()
step1 = pymc.NUTS(scaling=map_)
trace1 = pymc.sample(1000, step=step1)
step2 = pymc.NUTS(scaling=trace1[-1])
trace2 = pymc.sample(10000, step=step2, start=trace1[-1])
Which has great convergence. I think you can also just drop the gamma parameter.