BUG: pymc.sample raises SamplingError when using ExpQuad with HSGP.

[Armatron44]

Describe the issue:

Hi all,

I've just updated my pymc to 5.25.1 and I'm finding that attempting to sample a HSGP with the ExpQuad covariance function is raising a SamplingError: Initial evaluation of model at starting point failed!. This appears to relate to the logp value of the HSGP at initialisation, which is inf according to the traceback (see below). There doesn't appear to be an issue when I swap out ExpQuad for something like Matern32. I've recreated this issue on google colab with pymc 5.25.1. I was working with pymc 5.23 recently on a different computer, which ran ExpQuad HSGPs fine, although I can't test this now.

Reproduceable code example:

import pymc as pm
import numpy as np

# fake some data
x = np.sort(np.random.uniform(-1, 1, 101))
y = 3*np.cos(x*0.9) - 1
y += np.random.normal(scale=0.05, size=101)

with pm.Model(coords={"basis_coeffs": np.arange(200), "obs_id": np.arange(y.size)}) as model:
    ell = pm.Exponential("ell", scale=1) # dont @ me for these priors...
    eta = pm.Exponential("eta", scale=1.0)
    cov_func = eta**2 * pm.gp.cov.ExpQuad(input_dim=1, ls=ell) # fails with pymc sampler, nutpie, numpyro, blackjax (so possibly not to do with nuts_sampler kwarg in pm.sample...)
    #cov_func = eta**2 * pm.gp.cov.Matern32(input_dim=1, ls=ell) # this works with all

    m, c = 200, 1.5
    gp = pm.gp.HSGP(m=[m], c=c, parametrization="centered", cov_func=cov_func)
    f = gp.prior("f", X=x[:, None], hsgp_coeffs_dims="basis_coeffs", gp_dims="obs_id")

    sigma = pm.Exponential("sigma", scale=1.0)
    pm.Normal("y_obs", mu=f, sigma=sigma, observed=y, dims="obs_id")

    idata = pm.sample()

Error message:

---------------------------------------------------------------------------
SamplingError                             Traceback (most recent call last)
Cell In[16], line 16
     13 sigma = pm.Exponential("sigma", scale=1.0)
     14 pm.Normal("y_obs", mu=f, sigma=sigma, observed=y, dims="obs_id")
---> 16 idata = pm.sample()

File [~\.local\share\mamba\envs\pymc_dev\Lib\site-packages\pymc\sampling\mcmc.py:825](http://localhost:8888/~/.local/share/mamba/envs/pymc_dev/Lib/site-packages/pymc/sampling/mcmc.py#line=824), in sample(draws, tune, chains, cores, random_seed, progressbar, progressbar_theme, step, var_names, nuts_sampler, initvals, init, jitter_max_retries, n_init, trace, discard_tuned_samples, compute_convergence_checks, keep_warning_stat, return_inferencedata, idata_kwargs, nuts_sampler_kwargs, callback, mp_ctx, blas_cores, model, compile_kwargs, **kwargs)
    823         [kwargs.setdefault(k, v) for k, v in nuts_kwargs.items()]
    824     with joined_blas_limiter():
--> 825         initial_points, step = init_nuts(
    826             init=init,
    827             chains=chains,
    828             n_init=n_init,
    829             model=model,
    830             random_seed=random_seed_list,
    831             progressbar=progress_bool,
    832             jitter_max_retries=jitter_max_retries,
    833             tune=tune,
    834             initvals=initvals,
    835             compile_kwargs=compile_kwargs,
    836             **kwargs,
    837         )
    838 else:
    839     # Get initial points
    840     ipfns = make_initial_point_fns_per_chain(
    841         model=model,
    842         overrides=initvals,
    843         jitter_rvs=set(),
    844         chains=chains,
    845     )

File [~\.local\share\mamba\envs\pymc_dev\Lib\site-packages\pymc\sampling\mcmc.py:1598](http://localhost:8888/~/.local/share/mamba/envs/pymc_dev/Lib/site-packages/pymc/sampling/mcmc.py#line=1597), in init_nuts(init, chains, n_init, model, random_seed, progressbar, jitter_max_retries, tune, initvals, compile_kwargs, **kwargs)
   1595     q, _ = DictToArrayBijection.map(ip)
   1596     return logp_dlogp_func([q], extra_vars={})[0]
-> 1598 initial_points = _init_jitter(
   1599     model,
   1600     initvals,
   1601     seeds=random_seed_list,
   1602     jitter="jitter" in init,
   1603     jitter_max_retries=jitter_max_retries,
   1604     logp_fn=model_logp_fn,
   1605 )
   1607 apoints = [DictToArrayBijection.map(point) for point in initial_points]
   1608 apoints_data = [apoint.data for apoint in apoints]

File [~\.local\share\mamba\envs\pymc_dev\Lib\site-packages\pymc\sampling\mcmc.py:1479](http://localhost:8888/~/.local/share/mamba/envs/pymc_dev/Lib/site-packages/pymc/sampling/mcmc.py#line=1478), in _init_jitter(model, initvals, seeds, jitter, jitter_max_retries, logp_fn)
   1476 if not np.isfinite(point_logp):
   1477     if i == jitter_max_retries:
   1478         # Print informative message on last attempted point
-> 1479         model.check_start_vals(point)
   1480     # Retry with a new seed
   1481     seed = rng.integers(2**30, dtype=np.int64)

File [~\.local\share\mamba\envs\pymc_dev\Lib\site-packages\pymc\model\core.py:1761](http://localhost:8888/~/.local/share/mamba/envs/pymc_dev/Lib/site-packages/pymc/model/core.py#line=1760), in Model.check_start_vals(self, start, **kwargs)
   1758 initial_eval = self.point_logps(point=elem, **kwargs)
   1760 if not all(np.isfinite(v) for v in initial_eval.values()):
-> 1761     raise SamplingError(
   1762         "Initial evaluation of model at starting point failed!\n"
   1763         f"Starting value[s:\n](file:///S:/n){elem}\n\n"
   1764         f"Logp initial evaluation result[s:\n](file:///S:/n){initial_eval}\n"
   1765         "You can call `model.debug()` for more details."
   1766     )

SamplingError: Initial evaluation of model at starting point failed!
Starting values:
{'ell_log__': array(0.06154412), 'eta_log__': array(-0.07390129), 'f_hsgp_coeffs': array([ 0.8964528 ,  0.49850679,  0.77766117, -0.21469201, -0.83229329,
       -0.2429649 , -0.1551994 , -0.16094197, -0.66134098,  0.08932747,
        0.85003664, -0.96698846, -0.24095418,  0.95928666,  0.40584951,
        0.65621531,  0.34217628, -0.13498463,  0.6735691 , -0.91265536,
       -0.05148727,  0.37550022, -0.66848646, -0.237498  ,  0.13401424,
        0.07616318, -0.76062642,  0.42537257, -0.78619399,  0.71017075,
        0.62232671,  0.76061207, -0.25878416, -0.71957469,  0.75449224,
       -0.2458921 , -0.64380881, -0.88398595, -0.73363227, -0.72695346,
       -0.26828684,  0.64891776, -0.68961931,  0.72908515, -0.42343627,
        0.24523088, -0.50362676, -0.80204453, -0.47411123,  0.06919655,
       -0.85278136, -0.6872726 , -0.7074239 , -0.97904535, -0.2096503 ,
        0.41902197,  0.25750279,  0.16304053, -0.37161017, -0.36869419,
        0.87463671, -0.99804548, -0.2472362 , -0.99437107,  0.17233818,
       -0.53704303, -0.70933562, -0.6216585 , -0.74211035, -0.11780913,
       -0.33046545, -0.10765366,  0.09696944, -0.68235125, -0.78363395,
       -0.53045928, -0.17417613,  0.691059  , -0.05228136, -0.38724882,
        0.35066208, -0.5149922 , -0.77655213,  0.45167872,  0.96291537,
       -0.74180878,  0.47324007,  0.07420529,  0.45694168, -0.19554454,
       -0.08631478,  0.40328765, -0.82952522,  0.33224662,  0.06260759,
       -0.54895729,  0.75930369, -0.3085233 , -0.79609509,  0.82898824,
        0.53739623,  0.30328473, -0.90124674, -0.64246727, -0.21607528,
       -0.04892372,  0.90662235,  0.71510085, -0.22509855, -0.26623875,
       -0.641338  , -0.75124308, -0.8214267 ,  0.5451419 ,  0.02570617,
        0.4018908 , -0.1126687 , -0.31593296,  0.0362656 ,  0.76238948,
        0.3919529 , -0.27760741, -0.10068226, -0.04583653,  0.74203014,
        0.75065354,  0.54871431, -0.64430454,  0.53359048, -0.97495406,
       -0.73663779,  0.33514719,  0.69741655, -0.53137909,  0.78693164,
        0.17234047,  0.74777694, -0.1744733 ,  0.7607344 , -0.86257238,
       -0.17365085, -0.82280093,  0.7484344 ,  0.88597422, -0.91898113,
       -0.77001598, -0.95169786, -0.38264231, -0.81062648, -0.47486829,
        0.64959473,  0.24822373, -0.02866922,  0.4009877 ,  0.95769761,
        0.49782551, -0.99525773, -0.53563913,  0.77753391, -0.09806273,
       -0.20661118,  0.65396797, -0.97223146,  0.96095015, -0.66162998,
        0.07860522, -0.2993302 ,  0.03866622,  0.18522487,  0.54755709,
       -0.08540239,  0.18920188,  0.55557113,  0.26725513,  0.54615184,
       -0.18610811, -0.38184134, -0.4754358 , -0.54684329, -0.50546527,
        0.85735608, -0.89882684,  0.16445413, -0.86161286, -0.13556939,
       -0.8191506 ,  0.95764276,  0.55347055, -0.23243567, -0.35112965,
        0.34498401,  0.93611154,  0.51646511, -0.95735037,  0.28280915,
        0.18386817, -0.54182348,  0.48228643, -0.53230718, -0.82028976]), 'sigma_log__': array(0.69110407)}

Logp initial evaluation results:
{'ell': np.float64(-1.0), 'eta': np.float64(-1.0), 'f_hsgp_coeffs': np.float64(-inf), 'sigma': np.float64(-1.3), 'y_obs': np.float64(-534.46)}
You can call `model.debug()` for more details.

PyMC version information:

python 3.13.5
pymc 5.25.1
pytensor 2.31.7
numpy 2.2.6

Context for the issue:

RBF is the text book covariance function and is a go-to for exploring modelling data with GPs.

[ricardoV94]

Here is a more minimal example, removing parametrization="centered" seems to avoid the -inf:

import pymc as pm
import numpy as np

x = np.random.uniform(-1, 1, 10)

with pm.Model(coords={"basis_coeffs": np.arange(5)}) as model:
    cov_func = pm.gp.cov.ExpQuad(input_dim=1, ls=1)
    gp = pm.gp.HSGP(m=[200], c=1.5, parametrization="centered", cov_func=cov_func)
    f = gp.prior("f", X=x[:, None], hsgp_coeffs_dims="basis_coeffs")

model.point_logps()  # {'f_hsgp_coeffs': -inf}

[ricardoV94]

Issue is that the sqrt_psd is returning values that underflow to zero, and when you use a centered parametrization those are the sigma of a Normal variate, which must be >0.

import pymc as pm
import numpy as np

x = np.random.uniform(-1, 1, 10)

with pm.Model(coords={"basis_coeffs": np.arange(5)}) as model:
    cov_func = pm.gp.cov.ExpQuad(input_dim=1, ls=1)
    gp = pm.gp.HSGP(m=[200], c=1.5, parametrization="centered", cov_func=cov_func)
    _, sqrt_psd = gp.prior_linearized(x[:, None])

sqrt_psd.eval().min()  # 0.0

[ricardoV94]

I see the same problem with version 5.23: https://colab.research.google.com/drive/1gAk6dFHiT3w3JTrHMfkJlfP9GiXtqsg3?usp=sharing so I don't think it's a recent change.

It sounds to me the centered approach is sensitive to numerical precision or if these zeros are expected, then it is not valid or needs some artificial noise.

CC @bwengals

[bwengals]

Oh good catch! That does make sense actually, the Matern covariances have longer tails in the spectral domain. In practice, Materns are almost always better than RBF / ExpQuad, which is only common in textbooks because it has a simple form. It's a crappy default though.

I think the fix is to add a tiny bit to sigma so its numerically stable. Or, just decrease m. Your lengthscale values are so small that sigma goes to zero so there are zero of those basis functions that need to be included.

Edit: I think adding a bit to sigma isnt a good solution anymore. The point of the HSGP approx is to be fast, so the answer is to decrease m. A hack to make the code work in a suboptimal way, while sacrificing the point of the method, I dont think is a good solution.

[Armatron44]

Thanks both for the quick response. I've had a quick go with your solution @bwengals and I can now happily sample with a lower m. I think if I require a high m for small length scales I'll swap the kernel to the Matern family.

[ricardoV94]

Maybe we could mention something about this in the HSGP notebook as a way of closing this issue?

[Armatron44]

Good idea, if I have time this week I'll draft a section on this in the basic notebook.

[Armatron44]

Added a brief admonition section to the HSGP basic on this topic, so will close this topic. Thanks all.