Two-stage Bayesian regression enforcing a fixed distribution (not Just Hierarchical regression)

In any case, @Ray_Kruger this is a proof of concept you can take a look:
Say you have model at first level as:

mu0, sigma0 = 10., 1.5
y0 = np.random.normal(mu0, sigma0, size=1000)

with pm.Model() as m:
    mu = pm.Normal('mu', 0, 100)
    sigma = pm.HalfNormal('sigma', 10)
    y = pm.Normal('y', mu, sigma, observed=y0)
    trace = pm.sample(1000)

and another model like:

beta0 = 2.
sigma_new = 4.
X = np.linspace(-10, 10, 100)
y1 = np.random.normal(X * beta0, sigma_new)

with pm.Model() as m2:
    beta = pm.Normal('beta', 0, 100)
    sigma = pm.HalfNormal('sigma', 10)
    y = pm.Normal('y', X * beta, sigma, observed=y1)

To have sigma fixed in m2 using the posterior from m1, you can do:

from pymc.step_methods.arraystep import ArrayStepShared
from pymc.blocking import RaveledVars

class NormalProposal:
    def __init__(self, loc, scale):
        self.loc = loc
        self.scale = scale

    def __call__(self, rng=None, size=()):
        if rng is None:
            rng = np.random
        return rng.normal(self.loc, scale=self.scale, size=size)

class FixedDistSample(ArrayStepShared):
    """Return sample from a fixed proposal distribution.

    name = "fixed_dist_sample"

    generates_stats = False

    def __init__(self, vars, proposal_kwarg_dict, model=None):
        model = pm.modelcontext(model)
        initial_values = model.initial_point()

        vars = [model.rvs_to_values.get(var, var) for var in vars]
        vars = pm.inputvars(vars)
        initial_values_shape = [initial_values[].shape for v in vars]
        self.ndim = int(sum( for ivs in initial_values_shape))
        self.proposal_dist = NormalProposal(**proposal_kwarg_dict)

        shared = pm.make_shared_replacements(initial_values, vars, model)
        super().__init__(vars, shared)

    def astep(self, q0: RaveledVars) -> RaveledVars:
        point_map_info = q0.point_map_info
        q0 =
        q = self.proposal_dist(size=self.ndim)

        return RaveledVars(q, point_map_info)

from scipy.stats import t, norm
sigma_log__ = m.rvs_to_values[m.free_RVs[1]].tag.transform.forward(np.ravel(trace.posterior['sigma'].values))
sigma_log__ = aesara.function([], sigma_log__)()
# df, loc, scale =
loc, scale =

with m2:
    fixed_step = FixedDistSample([sigma], {'loc': loc, 'scale': scale})
    trace3 = pm.sample(1000, step=fixed_step)


You will see that sigma in trace3 follow the posterior from m instead of m2.

You can find the complete notebook here: link