PyMC

Per-target obligations of Transpilation correctness for \(\mathsf{T} = \mathrm{PyMC}\).

Semantics

PyMC's denotational semantics is the factor-graph semantics of Koller and Friedman 2009 Chapter 4 implemented via the pymc.Model context: each pymc.<Distribution>(name, ..., dims=..., observed=...) constructor inside the with pymc.Model() scope registers a random variable; the joint log-density is computed by pymc.Model.compile_logp. The reference is Salvatier, Wiecki, and Fonnesbeck (2016) for PyMC3 and the PyMC v5 documentation for the current API.

Unconstrained-space change of variables

PyMC applies per-distribution unconstrained-space transforms during inference (e.g. pymc.distributions.transforms.log for <lower=0>-style constraints) and adds the Jacobian automatically via the same change-of-variables principle as Stan. The constrained-space density that compile_logp returns at a constrained-space point is the QVR reference; no model-side emit-time Jacobian is needed. \(\Psi_{\mathsf{PyMC}} = \mathrm{id}\) at the constrained-space level.

Family parameterizations

PyMC families use the pymc.distributions.* classes with PyMC-specific keyword arguments. The QVR ↔ PyMC arg mapping requires non-trivial aliases:

QVR arg	PyMC arg
loc	mu
scale	sigma
concentration (Dirichlet)	a
concentration1 (Beta)	alpha
concentration0 (Beta)	beta
probs	p
total_count (Binomial)	n
df (StudentT)	nu

These live in FAMILY_META[F].arg_aliases["pymc"] per the SSoT rule. The renamings are syntactic; the density \(f_{\mathsf{PyMC}}(v \mid \text{renamed args})\) equals \(f_{\mathrm{QVR}}(v \mid \text{original args})\) term-by-term, so \(c_{F, \mathsf{PyMC}} = 0\) for every family with native PyMC support.

Per-construct emit

Sample / observe. <name> = pymc.<Family>(<name>, **args, dims=(<plate axes>), observed=<obs>) inside the with pymc.Model() as model: scope. The dims= declaration carries the batch-axis names; PyMC's coords (in the pymc.Model(coords=...) constructor) declares each axis with its size.

Plate. dims= names axes and coords supplies their labels or sizes. Independence comes from the distribution's batch shape and parameter broadcasting, not from the dimension label by itself.

Marginalize. Explicit-latent rewrite. PyMC's pm.Mixture also supports analytic marginalization for finite mixtures; the QVR renderer chooses the explicit rewrite for uniformity across backends.

Score / let / return. pymc.Potential("name", expr) for score (Plummer 2003 zero-trick is not needed; PyMC has Potential as a first-class log-density contribution); deterministic assignment for let; the function returns the pymc.Model instance.

Acceptance

• Tier 1 structural. Every emit has a def build_model(...) with a with pymc.Model(coords=...) as model: body containing one pymc.<Family>(...) call per IR sample / observe step plus return model.
• Tier 1 pipeline composition. Direct and composed pipeline calls agree.
• Tier 2 external syntax. Python's AST parser accepts the emitted source.
• Tier 3 numeric equivalence. pymc.Model.compile_logp() is compared with the QVR reference on the selected fixture grids.