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fix LinAlgError in HSGP-Basic.ipynb, drop find_constrained_prior #801

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fix LinAlgError in HSGP-Basic.ipynb, drop find_constrained_prior #801

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663 changes: 288 additions & 375 deletions examples/gaussian_processes/HSGP-Basic.ipynb
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24 changes: 11 additions & 13 deletions examples/gaussian_processes/HSGP-Basic.myst.md
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Original file line number Diff line number Diff line change
Expand Up @@ -5,7 +5,7 @@ jupytext:
format_name: myst
format_version: 0.13
kernelspec:
display_name: preliz
display_name: Python 3 (ipykernel)
language: python
name: python3
---
Expand Down Expand Up @@ -87,7 +87,8 @@ def simulate_1d(x, ell_true, eta_true, sigma_true):
# Draw one sample from the underlying GP.
n = len(x)
cov_func = eta_true**2 * pm.gp.cov.Matern52(1, ell_true)
gp_true = pm.MvNormal.dist(mu=np.zeros(n), cov=cov_func(x[:, None]))
cov_mat_jitted = pm.gp.util.stabilize(cov_func(x[:, None]))
gp_true = pm.MvNormal.dist(mu=np.zeros(n), cov=cov_mat_jitted)
f_true = pm.draw(gp_true, draws=1, random_seed=rng)

# The observed data is the latent function plus a small amount
Expand Down Expand Up @@ -153,7 +154,7 @@ with pm.Model(coords={"basis_coeffs": np.arange(200), "obs_id": np.arange(len(y_
sigma = pm.Exponential("sigma", scale=1.0)
pm.Normal("y_obs", mu=f, sigma=sigma, observed=y_obs, dims="obs_id")

idata = pm.sample()
idata = pm.sample(chains=4)
idata.extend(pm.sample_posterior_predictive(idata, random_seed=rng))
```

Expand Down Expand Up @@ -188,7 +189,7 @@ ax.scatter(x, y_obs, marker="o", color="grey", s=15, label="Observed data")
ax.plot(x, f_true, color="#FBE64D", lw=2, label="True underlying GP 'f'")

ax.set(title="The HSGP Fit", xlabel="x", ylabel="y")
ax.legend(frameon=True, fontsize=11, ncol=2);
ax.legend(frameon=True, fontsize=11, ncol=2, loc="lower left");
```

The inferred underlying GP (in bordeaux) accurately matches the true underlying GP (in yellow). We also see that the posterior predictive samples (in light blue) fit the observed data really well.
Expand Down Expand Up @@ -360,7 +361,7 @@ def calculate_Kapprox(Xs, L, m):
```

```{code-cell} ipython3
fig, axs = plt.subplots(2, 4, figsize=(14, 7), sharey=True)
fig, axs = plt.subplots(2, 4, figsize=(14, 7), sharey=True, layout="tight")

axs[0, 0].imshow(K, cmap="inferno", vmin=0, vmax=1)
axs[0, 0].set(xlabel="x1", ylabel="x2", title=f"True Gram matrix\nTrue $\\ell$ = {chosen_ell}")
Expand Down Expand Up @@ -413,7 +414,6 @@ axs[1, 3].set_title(f"m = {m}, c = {c}")

for ax in axs.flatten():
ax.grid(False)
plt.tight_layout();
```

The plots above compare the approximate Gram matrices to the unapproximated Gram matrix in the top left panel. The goal is to compare the approximated Gram matrices to the true one (upper left). Qualitatively, **the more similar they look the better the approximation**. Also, these results are **only relevant to the context of the particular domain defined by `X` and the chosen lengthscale**, $\ell = 3$ -- just because it looks good for $\ell = 3$ doesn't mean it will look good for, for instance, $\ell = 10$.
Expand Down Expand Up @@ -499,7 +499,6 @@ y_tr, y_te = y_obs[ix_tr], y_obs[ix_te]

```{code-cell} ipython3
fig = plt.figure(figsize=(13, 4))
plt.subplots_adjust(wspace=0.02)

ax1 = plt.subplot(131)
ax1.scatter(X_tr[:, 0], X_tr[:, 1], c=mu[ix_tr] - f_true[ix_tr])
Expand Down Expand Up @@ -539,10 +538,9 @@ with pm.Model() as model:

# Prior on the HSGP
eta = pm.Exponential("eta", scale=2.0)
ell_params = pm.find_constrained_prior(
pm.Lognormal, lower=0.5, upper=5.0, mass=0.9, init_guess={"mu": 1.0, "sigma": 1.0}
)
ell = pm.Lognormal("ell", **ell_params)
ell_dist = pz.maxent(pz.LogNormal(), lower=0.5, upper=5.0, mass=0.9, plot=False)
ell = ell_dist.to_pymc("ell")

cov_func = eta**2 * pm.gp.cov.Matern52(input_dim=2, ls=ell)

# m and c control the fidelity of the approximation
Expand Down Expand Up @@ -608,7 +606,7 @@ Now, let's sample the model and quickly check the results:

```{code-cell} ipython3
with model:
idata.extend(pm.sample(nuts_sampler="numpyro", random_seed=rng))
idata.extend(pm.sample(chains=4, nuts_sampler="numpyro", random_seed=rng))
```

```{code-cell} ipython3
Expand Down Expand Up @@ -654,7 +652,6 @@ pm.model_to_graphviz(model)

```{code-cell} ipython3
fig = plt.figure(figsize=(13, 4))
plt.subplots_adjust(wspace=0.02)

ax1 = plt.subplot(131)
ax1.scatter(X[:, 0], X[:, 1], c=f_true)
Expand Down Expand Up @@ -687,6 +684,7 @@ Sampling diagnostics all look good, and we can see that the underlying GP was in

* Created by [Bill Engels](https://github.com/bwengals) and [Alexandre Andorra](https://github.com/AlexAndorra) in 2024 ([pymc-examples#647](https://github.com/pymc-devs/pymc-examples/pull/647))
* Use `pz.maxent` instead of `pm.find_constrained_prior`, and add random seed. [Osvaldo Martin](https://aloctavodia.github.io/). August 2024
* Use `pm.gp.util.stabilize` in `simulate_1d`. Use `pz.maxent` rather than `pm.find_constrained_prior` in linearized HSGP model. [Alexander Armstrong](https://github.com/Armatron44), July 2025.

+++

Expand Down