Modeling a categorical that depends on another

Questions
1

I have a simple m-by-n table of categoricals:

   x  y
0  a  A
1  b  B
2  c  C
3  a  C

I want to model that the y column depends on the x column.

I made an attempt, and I expected that posterior-predictive values would have each row showing which categorical y value was predicted for the given x. But the predicted rows actually have four columns, and don’t indicate which of the y values is predicted.

I’m not sure if my model is wrong or if I’m using sample_posterior_predictive() incorrectly. How can I get y predictions for given values of x?

Here’s my attempt.

import arviz
import pandas as pd
import pymc3
import numpy as np
import theano
import theano.tensor

data = pd.DataFrame(
    {
        'x': pd.Categorical(['a', 'b', 'c', 'a']),
        'y': pd.Categorical(['A', 'B', 'C', 'C']),
    }
)


def build(x, y, ncats_x):
    with pymc3.Model() as mod:
        p = pymc3.Dirichlet("p", a=np.array([1] * ncats_x))
        out = pymc3.Categorical("out", p=p[x], observed=y)
    return mod


np.set_printoptions(threshold=np.inf)

cats_x = len(set(data["x"]))
x_shared = theano.shared(data["x"].cat.codes.values)
model = build(x=x_shared, y=data["y"].cat.codes.values, ncats_x=cats_x)
trace = pymc3.sample(model=model)
inference = arviz.from_pymc3(trace)
print(arviz.summary(inference))

out_of_sample_x = np.random.choice(data["x"].cat.codes.values, size=2)
x_shared.set_value(out_of_sample_x)
pred = pymc3.sample_posterior_predictive(trace=trace, model=model,)

print(pred)



#        mean     sd  hpd_3%  hpd_97%  ...  ess_sd  ess_bulk  ess_tail  r_hat
# p[0]  0.219  0.143   0.009    0.484  ...  1632.0    1620.0    1184.0    1.0
# p[1]  0.311  0.168   0.030    0.608  ...  1953.0    1901.0    1033.0    1.0
# p[2]  0.470  0.180   0.156    0.803  ...  1624.0    1683.0    1168.0    1.0

# [3 rows x 11 columns]
# {'out': array([[0, 0, 0, 0],
#        [1, 0, 0, 1],
#        [0, 0, 1, 1],
#        [1, 0, 0, 1],
#        [0, 1, 0, 1], ...