Fix GaussianPosterior input whitening buffers to float64

[cweniger]

Summary

• _input_mean and _input_std in GaussianPosterior were initialised without an explicit dtype, defaulting to float32. This caused float64 condition inputs to be silently downcast before whitening statistics were computed — inconsistent with the output-side buffers (_output_mean, _output_std, _residual_cov, etc.) which are all explicitly float64.
• Both buffers are now dtype=torch.float64, and are added to the to() override so they survive .to(device) calls.
• The dtype cast is decoupled in _forward_mean: whitening runs in float64, and the explicit cast to the MLP's dtype (next(self.net.parameters()).dtype, i.e. float32) happens only immediately before self.net(...).

Precision pipeline before and after

Step	Before	After
Condition input	float64	float64
Cast before whitening stats	→ float32	→ float64 (no-op)
Whitening (c - mean) / std	float32	float64
Cast before MLP	(implicit, already float32)	explicit → float32
MLP forward	float32	float32
MLP output upcast	→ float64	→ float64
De-whitening	float64	float64

Why it matters

The _input_mean.dtype was implicitly serving two roles: statistics precision and the gate cast into the float32 MLP. Making it float32 meant any catastrophic cancellation in the whitening step (e.g. conditions with large values and small variance, as in x = exp(z) with z near 5) was amplified before the MLP could even see the data. The fix separates the two concerns.

Test plan

• Run examples/04_gaussian: falcon launch -o output/run && falcon sample posterior -o output/run
• Confirm training loss decreases and posterior samples are reasonable
• Verify no dtype mismatch errors on CPU and GPU

🤖 Generated with Claude Code

[codecov]

Codecov Report

❌ Patch coverage is 0% with 4 lines in your changes missing coverage. Please review.
✅ Project coverage is 9.72%. Comparing base (7974ab3) to head (3fd4c73).
⚠️ Report is 3 commits behind head on main.

Files with missing lines	Patch %	Lines
falcon/estimators/gaussian_fullcov.py	0.00%	4 Missing ⚠️

Additional details and impacted files

@@          Coverage Diff          @@
##            main     #88   +/-   ##
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  Coverage   9.72%   9.72%           
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  Files         33      33           
  Lines       4154    4154           
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  Hits         404     404           
  Misses      3750    3750           

Flag	Coverage Δ
unit	9.72% <0.00%> (ø)

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