Can't use GaussianRandomWalk output as sigma for another GaussianRandomWalk

Questions v5
1

Hi all,

I’m trying to build a model that uses two GaussianRandomWalk distributions:

  • One to model a time-varying log_sigma
  • Another to model a latent state (predictors_noise) where the innovation scale (sigma) is defined as softplus(log_sigma)

However, when I try to use the output of one GaussianRandomWalk as the sigma for another, I get shape errors, even though the dimensions seem to align as expected. Below is a minimal example that reproduces the issue:

import pymc as pm
import pytensor.tensor as pt
import numpy as np
T = 100
N = 2
predictors_noise_0 = np.zeros(N)
initval_array = np.zeros((T, N))
initval_array[0, :] = predictors_noise_0
with pm.Model() as model:
    log_sigma = pm.GaussianRandomWalk("log_sigma", mu=0.0, sigma=0.1, shape=T)
    sigma = pt.softplus(log_sigma)  # Ensures positivity
    sigma_matrix = pt.repeat(sigma[:, None], N, axis=1)
    predictors_noise = pm.GaussianRandomWalk("predictors_noise",
                                             mu=0.0,sigma=sigma_matrix,shape=(T, N))
    idata = pm.sample(draws=100, tune=100, chains=2, cores=2,
                      initvals={"predictors_noise": initval_array})

When log_sigma.shape == T:
ValueError: Alloc static input type and target shape are incompatible: Matrix(float32, shape=(100, 2)) vs (1, 100)

When log_sigma.shape == T - 1:
ValueError: Alloc static input type and target shape are incompatible: Matrix(float32, shape=(99, 2)) vs (1, 100)

Am I doing something wrong here? Is there a workaround or recommended way to achieve this kind of structure, where sigma_t is itself a random walk?

Thank you

2

Couple of things. GRW expects T on the right, and more importantly it doesn’t allow sigma to vary over steps.

You can use a CustomDist for your purpose or if the variable is not observed use the expression in the dist directly (defining the latent Normal as a model variable instead of using .dist):

def noise_varying_grw_dist(sigma, size=None):
  return pm.Normal.dist(0, sigma, size=size).cumsum(-1)  # or wherever time is

with pm.Model() as m:
  ...
  predictors_noise = pm.CustomDist(
    "predictors_noise",                
    sigma, 
    dist=noise_varying_grw_dist,
    shape=(N, T),
)

Disclaimer: untested code

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