replace Monte Carlo uncertainty estimation with Jacobian-based covariance

[moustakas]

@jdbuhler, below is a potential speed-up suggested by claude-code. It's probably a high-risk, high-reward change, but I welcome your thoughts before I proceed in this direction.

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The emission-line fitter computes parameter uncertainties by running nmonte (default 50) full optimizer passes on noise-perturbed flux realizations (emlines.emline_specfit.get_results). Each pass calls EMFitTools.optimize, which runs scipy.optimize.least_squares with tr_solver='lsmr'. On a typical object these 50 MC passes account for ~65–70% of emline_specfit wall-clock time.

Because optimize already supplies an analytical Jacobian (jac=obj.jacobian), least_squares computes and stores fit_info.jac at the best-fit solution at no extra cost. The parameter covariance matrix follows directly:

  Cov(θ) ≈ (J^T W J)^{-1}

where W is the diagonal inverse-variance weight matrix. Parameter uncertainties are sqrt(diag(Cov)). This replaces all MC passes with a single matrix inversion.

Tradeoffs to evaluate before adopting:

• Linear approximation: valid when the model is well-approximated as linear around the best fit, but may underestimate uncertainties near parameter bounds or for low-S/N detections.
• Bound constraints: parameters at their bounds have non-Gaussian posteriors; the Jacobian covariance will underestimate those uncertainties.
• Should be validated against the current MC results on a representative sample spanning a range of S/N and spectral types before replacing it in production.

[jdbuhler]

As far as I can tell, Claude is correct that calling the optimizer returns an estimate of the Jacobian at the solution point. This should use whatever Jacobian computation optimize() uses, which I believe would be my analytical one.

The question is whether the uncertainty estimates derived from the Jacobian are accurate. I think the caveats are spot-on. If it's reliable in practice, it would greatly reduce the cost of estimating uncertainties, but you are relying on the linearity property and on not hitting your parameter bounds. If it's only sort-of reliable, you could still use it as a "quick and dirty" mode if the user doesn't want to run full-on MC.