Reparametizing Gelman's rat tumour example

Questions
3

Actually, the problem is because you didnt model a vector of theta:

with pm.Model() as model:
    # Uninformative prior for alpha and beta
    u = pm.Uniform('u', 0, 1)
    v = pm.Uniform('v', 0, 1)
    
    alpha = pm.Deterministic('a', u / tt.pow(v, 2))
    beta = pm.Deterministic('b', (1 - u) / tt.pow(v, 2))

    # theta_i with i=1,2,...,J
    theta = pm.Beta('theta', alpha=alpha, beta=beta, shape=len(n_rat))

    p = pm.Binomial('y', p=theta, observed=n_tumour, n=n_rat)
    trace = pm.sample(1000, tune=1000)

pm.traceplot(trace);

Alternatively, following the equation (5.9) in BDA 3 you can directly specify the prior of \alpha, \beta using a DensityDist:

import pymc3.distributions.transforms as tr

def logp_ab(value):
    return tt.log(tt.pow(tt.sum(value), -5/2))

with pm.Model() as model:
    # Uninformative prior for alpha and beta
    ab = pm.DensityDist('ab', logp_ab, transform=tr.log, 
                        shape=2, 
                        testval=np.asarray([1., 1.]))
    
    theta = pm.Beta('theta', alpha=ab[0], beta=ab[1], shape=len(n_rat))

    p = pm.Binomial('y', p=theta, observed=n_tumour, n=n_rat)