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name: apt_atomic_loss description: Evaluate equation-6 atomic APT categorical probabilities and density-ratio losses for finite parameter atom sets.

APT Atomic Loss

Use this skill when APT cannot evaluate the transformed-density normalizer in closed form and a recovery can instead evaluate finite atom sets. The skill computes equation-6 probabilities and the cross-entropy loss for the generating parameter.

Inputs

• A finite atom set Theta.
• Posterior-estimator log densities log q(theta_i | x).
• Prior log densities log p(theta_i).
• Index of the atom that generated the observation.

Outputs

• Normalized atom probabilities.
• Atomic negative log likelihood.
• Ratio diagnostics showing whether probability ratios match posterior/prior score ratios.

Workflow

1. Validate matching array lengths, a valid true index, and finite prior density for the true atom.
2. Compute scores log_q_i - log_prior_i.
3. Normalize scores with log-sum-exp.
4. Return -log(probability[true_index]).
5. Use ratio diagnostics for mechanism checks in reduced recovery.

Validation

Run:

python tests/test_atomic_loss.py

The tests verify normalization, constant-shift invariance, score ordering, loss reduction after a targeted update, and rejection of a true atom outside prior support.

Limitations

• This helper is deterministic and framework-free. It does not backpropagate through a neural network.
• Full APT with MAFs or MDNs should use the same probability semantics but would rely on a deep learning framework for gradients.