AePPL and convolutions

What to do when the sum has no closed-form posterior?

import aeppl
import aesara.tensor as at
 
srng = at.random.RandomStream(0)
 
x_rv = srng.normal(0, 1)
y_rv = srng.lognormal(0, 1)
z = x_rv + y_rv
 
logprob, (x_vv, z_vv) = aeppl.joint_logprob(x_rv, z)

In this case we should be able to compute the density by binding $x_{vv}$ to the density of $x_{r}v$, and binding $z_{vv}-x_{vv}$ to the density of $y_{rv}$.

Now slightly more complicated. Is this defined without $x_{rv}$ or $y_{rv}$ being stochastic? (I don’t think so). And if it is not, then how do we proceed about tying the values taken by $x_{rv}$ and $y_{rv}$ together?

import aeppl
import aesara.tensor as at
 
srng = at.random.RandomStream(0)
 
mu_rv = srng.normal(0, 1)
sigma_rv = srng.halfcauchy(1)
 
x_rv = srng.normal(mu_rv, 1)
y_rv = srng.lognormal(0, sigma_rv)
 
z = x_rv + y_rv
 
logprob, (mu_vv, sigma_vv, z_vv) = aeppl.joint_logprob(x_rv, z)

Finally the following example:

import aeppl
import aesara.tensor as at
 
srng = at.random.RandomStream(0)
 
a = srng.normal(0, 1)
b = srng.normal(0, 1)
c = srng.normal(0, 1)
d = srng.normal(0, 1)
 
X_rv = at.uniform(a, b)
Y_rv = at.uniform(c, d)
Z_rv = X_rv + Y_rv
 
logprob, value_variables = aeppl.joint_logprob(a, b, c, d, Z_rv)

In this case $Z$‘s density has a closed-form expression; $Z_{rv}$ has a trapezoidal distribution. So we should be able to replace this sum with this distribution directly.

A list of cases where the convolution of two or more random variables has a closed-form solutions, or rather is a known measure, is available on Wikipedia.