virgil.observables

Spectro-interferometric observables beside V² and closure phase.

OIData keeps its visibilities (vis) and phases (phi) as before. The observables here are further blocks of the same data vector, in OIData.extras, read with read_oifits(..., extras=...) (see read_oifits). Each block predicts its data from the model and whitens its own residuals, so the likelihood keeps one whitened residual vector:

• "flux": OI_FLUX as a spectrum known up to a grey scale, F(λ) = k Σ fᵢ(λ) from total_spectrum (FluxSpectrum);
• "nflux": OI_FLUX normalized to its continuum, F(λ)/F_c(λ), the same with the model normalized over the continuum channels;
• "corrflux": correlated fluxes (VISAMP with AMPTYP = 'correlated flux'), k |Σ fᵢ(λ) Vᵢ|, up to a grey scale;
• "visamp": |V| (VISAMP, AMPTYP = 'absolute') beside V² (VisibilityAmplitude);
• "t3amp": triple amplitudes |V_ab V_bc V_ac| (TripleAmplitude);
• "visphi": differential phases, the exact arg V(λ) with the pipeline's continuum normalization applied as a linear operator (DifferentialPhase).

Grey scales are marginalized, not fitted. A spectrum's scale k (and an optional polynomial in λ times the spectrum) enters linearly, so with a Gaussian prior it integrates out in closed form (Luger, Foreman-Mackey & Hogg 2017; virgil._linear): the data are Gaussian with covariance D + A Λ Aᵀ about A μ, whitened by the same successive rank-one steps as the 6d gains, with the log-determinant (which depends on the model's spectral shape) kept in the effective errors. The conditional posterior of k is flux_scale_posterior.

The prior on k is stated, never taken from the data. Give it as scale=(mean, sd) in the data's units (e.g. Jy), with OIData.with_flux_scale. Normalized spectra ("nflux") default to (1, 0.1). The Gaussian is a proposal: k is a positive scale, whose Jeffreys prior is 1/k on stated bounds. The two differ by about σ_k/k, which is negligible for a well-measured spectrum. For the Jeffreys posterior, reweight samples of k from the conditional posterior by 1 / (k N(k; mean, sd²)) within the bounds, or sample log k under a log-uniform prior directly, as a model parameter rather than marginalized. Make no evidence claims that depend on the Gaussian's width.

Differential phases. A pipeline's differential phase is arg V minus a fit of a + b/λ (an offset and a delay) over continuum channels, per baseline and frame. That fit is a linear operator N = I − L on the phases of one row (continuum_operator), so virgil applies the same N to the exact model phase, never the photocentre approximation, which fails for resolved structure. If our basis contains the pipeline's, N N_pipe = N, so the data may be re-normalized safely. Their covariance is N D Nᵀ, which is whitened as a dense block per frame. Projecting out a linear basis is the flat-prior limit of marginalizing those nuisances: the likelihood of N d with covariance N D Nᵀ is the same whichever projection with that null space is used (restricted maximum likelihood).

Without double counting closure phases (the default of the design note, S §2.3): with closure phases in the data, each frame's baseline phases are also projected onto the telescope-differenced subspace φ_ab = a_a − a_b, orthogonal to every closure, and only channels in the line windows are kept. The closure phases (everywhere) and this closure-free part of the differential phase (in the lines) then measure different things. The cross-covariance between them is neglected as an approximation: it is exactly zero only when the baseline errors of a frame and channel are equal. For unequal errors it is nonzero, and the independent-block likelihood is not exact. The joint covariance is preferred when available.

mᵢ = Σ_j w_j tᵢ xᵢʲ,   w ~ N((μ, 0, ...), diag(s, τ_1 μ, ...)²),

Y = Qᵀ Φ Wᵀ,