Model evaluation for signals and autoregressive noise

Questions
1

I’m modeling data as a signal plus autoregessive noise- for example, this test data:

#test
nt = 500
noise = np.zeros((nt,))

# true stationarity:
true_theta = 0.3

# true standard deviation of the innovation:
true_sigma=1.0


for t in range(1, nt):
    noise [t] = true_theta * noise [t - 1] + np.random.normal(loc=0, scale=true_sigma)

x=np.linspace(0,1,nt)
true_signal= 2*x**2
true_beta=.7

#The [synthetic] observations are the scaled signal + AR(1) noise
obs=true_beta*true_signal+noise

#The true signal is quadratic. What if we looked for a linear signal instead?
other_signal=x

I have two models: one that with the “true” signal

with pm.Model() as true_signal_model:
    theta = pm.Uniform("theta", -1.0, 1.0)
    # precision of the innovation term
    tau = pm.Exponential("tau", 0.5)
    
    # scale the quadratic "true" signal
    beta=pm.Normal("beta",0,1)
    signal=pm.Deterministic("forced_response",beta*true_signal)

    likelihood_obs = pm.AR1("likelihood_obs", k=theta, tau_e=tau,observed=obs-signal)`Preformatted text`

    
    goodtrace = pm.sample(return_inferencedata=True)

and one with the linear signal


with pm.Model() as bad_signal_model:
    theta = pm.Uniform("theta", -1.0, 1.0)
    # precision of the innovation term
    tau = pm.Exponential("tau", 0.5)
    
    # scale the linear signal
    beta=pm.Normal("beta",0,1)
    signal=pm.Deterministic("forced_response",beta*other_signal)

    likelihood_obs = pm.AR1("likelihood_obs", k=theta, tau_e=tau,observed=obs-signal)
    
    
    badtrace = pm.sample(return_inferencedata=True)

It strikes me that I should be able to use LOO or WAIC to test which of these models best captures structure in the observations. And if I were doing standard regressions obs \sim \mathcal{N}(\beta \cdot signal,\sigma) I’d have log likelihoods of dimension (4,1000, 500) and could use az.loo. But the shape of the resulting log likelihoods is (4,1000,1) because they represent a single draw from the distribution. Basically, how do you do LOO when you have a nontrivial covariance structure in your data? Any advice for how to do model comparison in this case? Thanks!