pymc3：如何在多级线性回归中建模相关截距和斜率

Question

In the Pymc3 example for multilevel linear regression (the example is here , with the radon data set from Gelman et al.'s (2007)), the intercepts (for different counties) and slopes (for apartment with and without basement) each have a Normal prior. How can I model them together with a multivariate normal prior, so that I can examine the correlation between them?

The hierarchical model given in the example is like this:

with pm.Model() as hierarchical_model:
    # Hyperpriors for group nodes
    mu_a = pm.Normal('mu_a', mu=0., sd=100**2)
    sigma_a = pm.HalfCauchy('sigma_a', 5)
    mu_b = pm.Normal('mu_b', mu=0., sd=100**2)
    sigma_b = pm.HalfCauchy('sigma_b', 5)

    # Intercept for each county, distributed around group mean mu_a
    # Above we just set mu and sd to a fixed value while here we
    # plug in a common group distribution for all a and b (which are
    # vectors of length n_counties).
    a = pm.Normal('a', mu=mu_a, sd=sigma_a, shape=n_counties)
    # Intercept for each county, distributed around group mean mu_a
    b = pm.Normal('b', mu=mu_b, sd=sigma_b, shape=n_counties)

    # Model error
    eps = pm.HalfCauchy('eps', 5)

    radon_est = a[county_idx] + b[county_idx] * data.floor.values

    # Data likelihood
    radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps, observed=data.log_radon)
    hierarchical_trace = pm.sample(2000)

And I'm trying to make some change to the priors

    with pm.Model() as correlation_model:
        # Hyperpriors for group nodes
        mu_a = pm.Normal('mu_a', mu=0., sd=100**2)
        mu_b = pm.Normal('mu_b', mu=0., sd=100**2)

        # here I want to model a and b together
        # I borrowed some code from a multivariate normal model
        # but the code does not work
        sigma = pm.HalfCauchy('sigma', 5, shape=2)

        C_triu = pm.LKJCorr('C_triu', n=2, p=2)
        C = T.fill_diagonal(C_triu[np.zeros((2,2), 'int')], 1)
        cov = pm.Deterministic('cov', T.nlinalg.matrix_dot(sigma, C, sigma))
        tau = pm.Deterministic('tau', T.nlinalg.matrix_inverse(cov))

        a, b = pm.MvNormal('mu', mu=(mu_a, mu_b), tau=tau,
                           shape=(n_counties, n_counties))

        # Model error
        eps = pm.HalfCauchy('eps', 5)
        radon_est = a[county_idx] + b[county_idx] * data.floor.values

        # Data likelihood
        radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps, observed=data.log_radon)
        correlation_trace = pm.sample(2000)

Here is the error message I got:

  File "<ipython-input-108-ce400c54cc39>", line 14, in <module>
    tau = pm.Deterministic('tau', T.nlinalg.matrix_inverse(cov))

  File "/home/olivier/anaconda3/lib/python3.5/site-packages/theano/gof/op.py", line 611, in __call__
    node = self.make_node(*inputs, **kwargs)

  File "/home/olivier/anaconda3/lib/python3.5/site-packages/theano/tensor/nlinalg.py", line 73, in make_node
    assert x.ndim == 2

AssertionError

Clearly I've made some mistakes about the covariance matrix, but I'm new to pymc3 and completely new to theano so have no idea how to fix it. I gather this should be a rather common use case so maybe there have been some examples on it? I just can't find them.

The full replicable code and data can be seen on the example page (link given above). I didn't include it here because it's too long and also I thought those familiar with pymc3 are very likely already quite familiar with it:)

Answer 1

You forgot to add one line when creating the covariance matrix you miss-specified the shape of the MvNormal. Your model should look something like this:

with pm.Model() as correlation_model:
    mu = pm.Normal('mu', mu=0., sd=10, shape=2)
    sigma = pm.HalfCauchy('sigma', 5, shape=2)

    C_triu = pm.LKJCorr('C_triu', n=2, p=2)
    C = tt.fill_diagonal(C_triu[np.zeros((2,2), 'int')], 1.)
    sigma_diag = tt.nlinalg.diag(sigma) # this line
    cov = tt.nlinalg.matrix_dot(sigma_diag, C, sigma_diag)
    tau = tt.nlinalg.matrix_inverse(cov)

    ab = pm.MvNormal('ab', mu=mu, tau=tau, shape=(n_counties, 2))

    eps = pm.HalfCauchy('eps', 5)
    radon_est = ab[:,0][county_idx] + ab[:,1][county_idx] * data.floor.values

    radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps, observed=data.log_radon)
    trace = pm.sample(2000)

Notice that alternatively, you can evaluate the correlation of the intercept and the slope from the posterior of hierarchical_model . You can use a frequentist method or build another Bayesian model, that takes as the observed data the result of hierarchical_model . May be this could be faster.

EDIT

If you want to evaluate the correlation of two variables from the posterior you can do something like.

chain = hierarchical_trace[100:]
x_0 = chain['mu_a']
x_1 = chain['mu_b']
X = np.vstack((x_0, x_1)).T

and then you can run the following model:

with pm.Model() as correlation:
    mu = pm.Normal('mu', mu=0., sd=10, shape=2)
    sigma = pm.HalfCauchy('sigma', 5, shape=2)

    C_triu = pm.LKJCorr('C_triu', n=2, p=2)
    C = tt.fill_diagonal(C_triu[np.zeros((2,2), 'int')], 1.)
    sigma_diag = tt.nlinalg.diag(sigma)
    cov = tt.nlinalg.matrix_dot(sigma_diag, C, sigma_diag)
    tau = tt.nlinalg.matrix_inverse(cov)

    yl = pm.MvNormal('yl', mu=mu, tau=tau, shape=(2, 2), observed=X)
    trace = pm.sample(5000, pm.Metropolis())

You can replace x_0 and x_1 according to your needs. For example you may want to do:

x_0 = np.random.normal(chain['mu_a'], chain['sigma_a'])
x_1 = np.random.normal(chain['mu_b'], chain['sigma_b'])