Negative binomial regression

I have some count data which may follower a NB distribution. My goal is to get the posterior distribution of the mean (mu) for the NB distribution. I just started to learn Pyro as I need to sale the job. I really got stuck for several days and need some help.

# Simulate some data
mu = 0.3
variance = 1.6
phi = mu**2 / (variance-mu)
p = phi/(mu + phi)
y = np.random.negative_binomial(n=phi, p=p, size=10000)

y.mean(), y.var()
(0.309, 1.6999190000000002)

def model(y):
    b0 = pyro.sample("b0", dist.Normal(0, 5))
    phi = pyro.sample("phi", dist.HalfCauchy(2))
    # phi = pyro.sample("phi", dist.Uniform(0., 10.))
    mu = torch.exp(b0)
    beta = phi/mu
    alpha = phi
    with pyro.plate("data", len(y)):
        pyro.sample("obs", dist.GammaPoisson(alpha, beta), obs=y)
def guide(y):
    a_loc = pyro.param('a_loc', torch.tensor(0.))
    a_scale = pyro.param('a_scale', torch.tensor(1.),
    phi_loc = pyro.param('phi_loc', torch.tensor(1.),
    b0 = pyro.sample("b0", dist.Normal(a_loc, a_scale))
    phi = pyro.sample("phi", dist.Normal(phi_loc, torch.tensor(1.0)))

    mu = torch.exp(b0)
    beta = phi/mu
    alpha = phi

y_tensor = torch.tensor(y, dtype=torch.float)
from pyro.infer import SVI, Trace_ELBO

svi = SVI(model,
          optim.Adam({"lr": .0005}),

num_iters = 10000
loss = []
for i in range(num_iters):
    elbo = svi.step(y_tensor)
    if i % 500 == 0:
        print("Elbo loss: {}".format(elbo))

/home/ec2-user/anaconda3/envs/pytorch_p36/lib/python3.6/site-packages/pyro/infer/ UserWarning: Encountered NaN: loss
  warn_if_nan(loss, "loss")

Basically I want to model log(mu) through some linear model (just intercept right now). The phi in the code is the over dispersion parameter, then I mapped it to alpha and beta using the GammaPoisson format. Since phi is positive, I assigned a HCauchy distribution. In the end, the SVI runs but outputs are all NAs.

Can anyone give me some guidance? The only source I can find is here, but the way the over-dispersion parameter specified is confusing to me.

the phi distribution in the guide needs to have the same support as in the model (the positive real line)

You could also simply use an autoguide:

guide = AutoNormal(model)