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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:
```bash
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.
