Bug summary

HSGPPeriodic appears to apply periodic basis coefficients at the wrong scale, which can distort the GP prior variance and produce systematically incorrect prior draws.

This is not a cosmetic mismatch: it can change model behavior and inference quality whenever users rely on HSGPPeriodic as an approximation to gp.Latent with a periodic kernel.

Why this is high impact

• It affects prior correctness for a core GP approximation API.
• It can bias uncertainty and function amplitudes before seeing any data.
• It directly undermines the expected equivalence between HSGPPeriodic and full periodic GP priors.

Evidence in code

1) Coefficients are documented/returned as variance-level terms (q_j^2)

• pymc/gp/cov.py (Periodic.power_spectral_density_approx): docs and implementation indicate coefficients correspond to (\tilde q_j^2).

2) Same coefficients are used as if they were std-level terms

• pymc/gp/hsgp_approx.py (HSGPPeriodic.prior_linearized, prior, _build_conditional): coefficients are multiplied directly into beta ~ Normal(0,1) via terms like phi @ (coeff * beta).

If coeff = q_j^2, then variance contribution becomes (q_j^2)^2 instead of q_j^2.

Observable symptom

The periodic prior equivalence test is currently xfailed:

• tests/gp/test_hsgp_approx.py::TestHSGPPeriodic::test_prior

A scaling mismatch is a plausible root cause of this persistent mismatch.

Expected behavior

When beta ~ Normal(0,1), basis multipliers should be std-level coefficients.
If the covariance helper returns variance-level coefficients (q_j^2), HSGPPeriodic should use sqrt(q_j^2) at the multiplication site.

Suggested fix direction

• Option A (minimal): keep power_spectral_density_approx as variance-level output and apply pt.sqrt(...) in HSGPPeriodic where coefficients multiply beta.
• Option B: change semantics/API so the helper returns std-level coefficients consistently, then update naming/docs accordingly.

Either way, resolving this should help un-xfail periodic prior equivalence and restore expected approximation behavior.