import numpy as np
import pymc3 as pm

true_mu = 1.0
true_sd = 0.05
n_obs = 20

np.random.seed(1234)
data = np.random.normal(loc=true_mu, scale=true_sd, size=n_obs)

assumed_sd = 0.1

with pm.Model() as model:
    mu = pm.Uniform('mu', lower=-10, upper=10)
    center = pm.Normal('obs', mu=mu, sd=assumed_sd, observed=data)
    start_MAP = pm.find_MAP()
    trace = pm.sample(3000, start=start_MAP, step=pm.NUTS())
    trace = trace[1000:]

with pm.Model() as model:
    mu = pm.Uniform('mu', lower=-10, upper=10)
    sd = pm.HalfCauchy('sigma', beta=10, testval=1.0)
    center = pm.Normal('obs', mu=mu, sd=sd, observed=data)
    start_MAP = pm.find_MAP()
    trace = pm.sample(3000 start=start_MAP, step=pm.NUTS())
    trace = trace[1000:]