pymc-examples/examples/gaussian_processes /GP-Heteroskedastic.ipynb not working for PyMC v5

GP-Heteroskedastic.ipynb:
pymc-examples/examples/gaussian_processes /GP-Heteroskedastic.ipynb:

Issue description

The notebook has not been updated for PyMC v5.x.x

For example, in the "Sparse Heteroskedastic GP" the class SparseLatent will raise errors in v5.
Using support from PyMC Discourse I believe it could work replacing theano with pytensor and another couple minor issues (e.g. sigma instead of sd in pm. Normal)

class SparseLatent:
    def __init__(self, cov_func):
        self.cov = cov_func

    def prior(self, name, X, Xu): 
        Kuu = self.cov(Xu)
        self.L = pt.linalg.cholesky(pm.gp.util.stabilize(Kuu)) 

    # def prior(self, name, X, Xu):
    #     Kuu = self.cov(Xu)
    #     self.L = pm.gp.util.cholesky(pm.gp.util.stabilize(Kuu))

        self.v = pm.Normal(f"u_rotated_{name}", mu=0.0, sigma=1.0, shape=len(Xu))
        self.u = pm.Deterministic(f"u_{name}", pt.dot(self.L, self.v))

        Kfu = self.cov(X, Xu)
        # self.Kuiu = tt.slinalg.solve_upper_triangular(
        #     self.L.T, tt.slinalg.solve_lower_triangular(self.L, self.u)
        # )
        self.Kuiu = pt.slinalg.solve_triangular(
            self.L.T, pt.slinalg.solve_triangular(self.L, self.u, lower = True),
            lower = False
        )
        self.mu = pm.Deterministic(f"mu_{name}", pt.dot(Kfu, self.Kuiu))
        return self.mu

    def conditional(self, name, Xnew, Xu):
        Ksu = self.cov(Xnew, Xu)
        mus = pt.dot(Ksu, self.Kuiu)
        tmp = pt.slinalg.solve_triangular(self.L, Ksu.T, lower = True)
        Qss = pt.dot(tmp.T, tmp)  # Qss = tt.dot(tt.dot(Ksu, tt.nlinalg.pinv(Kuu)), Ksu.T)
        Kss = self.cov(Xnew)
        #Lss = pm.gp.util.cholesky(pm.gp.util.stabilize(Kss - Qss))
        Lss = pt.linalg.cholesky(pm.gp.util.stabilize(Kss - Qss)) 
        mu_pred = pm.MvNormal(name, mu=mus, chol=Lss, shape=len(Xnew))
        return mu_pred

# Explicitly specify inducing points by downsampling our input vector
Xu = X[1::2]

with pm.Model() as model_hts:
    ℓ = pm.InverseGamma("ℓ", mu=ℓ_μ, sigma=ℓ_σ)
    η = pm.Gamma("η", alpha=2, beta=1)
    cov = η**2 * pm.gp.cov.ExpQuad(input_dim=1, ls=ℓ)

    μ_gp = SparseLatent(cov)
    #μ_f = μ_gp.prior("μ", X_obs, Xu)
    μ_f = μ_gp.prior("μ", X_obs, Xu)

    σ_ℓ = pm.InverseGamma("σ_ℓ", mu=ℓ_μ, sigma=ℓ_σ)
    σ_η = pm.Gamma("σ_η", alpha=2, beta=1)
    σ_cov = σ_η**2 * pm.gp.cov.ExpQuad(input_dim=1, ls=σ_ℓ)

    lg_σ_gp = SparseLatent(σ_cov)
    lg_σ_f = lg_σ_gp.prior("lg_σ_f", X_obs, Xu)
    σ_f = pm.Deterministic("σ_f", pm.math.exp(lg_σ_f))

    lik_hts = pm.Normal("lik_hts", mu=μ_f, sigma=σ_f, observed=y_obs_)
    trace_hts = pm.sample(target_accept=0.95, return_inferencedata=True, random_seed=SEED, chains = 1)

with model_hts:
    μ_pred = μ_gp.conditional("μ_pred", Xnew, Xu)
    lg_σ_pred = lg_σ_gp.conditional("lg_σ_pred", Xnew, Xu)
    samples_hts = pm.sample_posterior_predictive(trace_hts, var_names=["μ_pred", "lg_σ_pred"])

Start with pymc-examples/examples/gaussian_processes/GP-Heteroskedastic.ipynb, especially the SparseLatent implementation and the PyMC v5 compatibility issues described in the report. Run the notebook with PyMC v5, update the outdated APIs and imports identified in the example, and verify that the sampling and posterior-predictive cells execute successfully.