Retrieval Likelihood

After fitting a transmission spectrum with ExoIris, the next step is often atmospheric retrieval: comparing the measured spectrum to theoretical models to infer atmospheric properties. This step requires a likelihood function that properly accounts for the statistical structure of the ExoIris posterior, and a naive approach fails in a subtle but important way.

The Rank-Deficiency Problem

ExoIris parameterises the transmission spectrum with \(K\) radius-ratio knots, which are then interpolated to \(M\) data wavelengths. Since \(M \gg K\), the posterior samples live in a \(K\)-dimensional subspace of the \(M\)-dimensional wavelength space. The full \(M \times M\) covariance matrix is therefore singular: it has only \(K\) non-zero eigenvalues. A standard multivariate Gaussian likelihood cannot be evaluated because the covariance matrix is not invertible.

Karhunen-Loeve Decomposition

ExoIris solves this using a Karhunen-Loeve (principal component) decomposition of the posterior covariance. The eigendecomposition identifies the \(K\) significant principal components and discards the near-zero eigenvalues that correspond to directions with no posterior variance. The resulting reduced-rank likelihood is:

\[\ln \mathcal{L} = -\frac{1}{2} \left( \sum_{i=1}^{K} \frac{p_i^2}{\lambda_i} + \ln \prod_{i=1}^{K} \lambda_i + K \ln 2\pi \right)\]

where \(p_i\) are the projections of the residuals onto the eigenvectors and \(\lambda_i\) are the corresponding eigenvalues. This gives a statistically rigorous likelihood that lives in the correct subspace, ready to be plugged into any atmospheric retrieval framework.