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name: apt_posterior_transformation description: Compute APT posterior-to-proposal-posterior transformations for Gaussian and finite normalized density settings without importance weighting.

APT Posterior Transformation

Use this skill when a recovery or implementation needs the central APT transformation from a true-posterior estimator q(theta | x) to the proposal-posterior density used for proposal-drawn training data. Do not use it for SNPE-B-style importance weighting or SNPE-A-style post-hoc posterior correction.

Inputs

• Posterior estimate density values or Gaussian parameters.
• Prior density values or Gaussian parameters.
• Proposal density values or Gaussian parameters.
• Optional finite grid or atom scores for normalization checks.

Outputs

• Gaussian transformed proposal-posterior mean and variance.
• Finite normalized probabilities proportional to q(theta | x) * proposal(theta) / prior(theta).
• Diagnostics for invalid transformed precision, nonfinite scores, or support mismatches.

Workflow

1. For generic finite points, compute log_q + log_proposal - log_prior.
2. Normalize with log-sum-exp to get proposal-posterior probabilities.
3. For Gaussian inputs, compute transformed precision: posterior_precision + proposal_precision - prior_precision.
4. Compute transformed mean from the transformed natural mean.
5. Reject or flag invalid transforms where precision is not positive.

Validation

Run:

python tests/test_transformation.py

The tests compare Gaussian closed form against finite-grid normalization and check numerical stability. They also assert that invalid Gaussian precision is reported rather than converted into a nonsensical variance.

Limitations

• The script implements scalar Gaussian transforms and finite-score normalization.
• Mixture-of-Gaussians formulas are described in the paper but not fully implemented here because the reduced recovery only needs scalar Gaussian checks.
• This skill provides transformation math; it does not run simulator rounds or optimizer steps.