Corner plots: distinct labels per component, one plot per galaxy, and a latent-parameter corner plot

[samlange04]

While modelling SLACS lenses I found that corner_anesthetic had silently stopped producing a plot for any model
with two components of one class (most lens models): anesthetic keys its columns by label, every Gaussian of an MGE
basis is labelled $\epsilon_1^{\rm g}$, plot_2d raises KeyError, and @log_plot_exception reports it as
"posterior estimate not yet sufficient", on converged fits, at every update. The one-line fix is to key the columns
by position and pass the labels as display labels. But it leaves the plot unreadable for the same reason it crashed:
a 19-parameter source-light fit has three pairs of (y, x) and four pairs of (\epsilon_1, \epsilon_2) with
identical labels, and its 171 panels do not fit on a screen. I would like to propose three connected changes, which I
have implemented and run on real fits, and get your view before opening the PRs.

1. Superscripts are the component's path of names. A parameter's superscript becomes the dot-joined names its
   owning components were given in the model: y^{\rm lens.bulge}, \epsilon_1^{\rm source.bulge},
   \gamma_1^{\rm lens.shear}. Collection containers (galaxies), tuple names (centre) and the list index of a
   basis whose profiles share one prior are dropped; an index that distinguishes independent priors is kept. A prior
   shared between differently named components (a power law's ellipticity linked into a multipole) takes the path its
   samples.csv column is named after. The notation config's per-class abbreviation (Gaussian: g) only applies to a
   prior directly on a bare Model with no name. A path three or more levels deep keeps its first name whole and
   cuts the rest to three characters (source.pix.reg) so it fits a corner-plot axis. This changes the text of
   model.results superscripts too, which I think is an improvement (\gamma_1^{\rm lens.shear} rather than
   \gamma^{\rm ext} twice), but it is a visible change.

2. One corner plot per top-level component. Alongside the full corner_anesthetic.png, a search writes
   corner_anesthetic_<component>.png for each top-level component of the model (each galaxy of a lens model) when
   there is more than one, from a new AbstractPriorModel.parameter_indices_by_component and a parameter_indices
   kwarg on corner_anesthetic. Config key nest.corner_anesthetic_components, default on; a workspace config
   that predates the key falls back to the default rather than raising.

3. A latent-parameter corner plot. Components are sampled in one parameterisation and read in another: a shear
   as (gamma_1, gamma_2) but read as a magnitude and angle, an ellipse as elliptical components but read as an axis
   ratio and position angle, a multipole as components but read as a strength and phase. A component class declares
   the conversion as a latent_parameters class attribute ({("gamma_1", "gamma_2"): convert.latent_shear_from}),
   the function taking every parameter of the component by name (free ones as arrays over the samples, fixed ones as
   floats, so a multipole's m is available) and returning the latent values by name; declarations are inherited
   down the MRO. PyAutoFit applies them vectorised over the samples (AbstractPriorModel.latent_parameters_from), and:

   • writes corner_anesthetic_latent.png (plus _<component> for the components that have a latent parameter)
     when nest.corner_anesthetic_latent is on (default off), independently of the two plots above;
   • adds the latent parameters to the existing latent output (latent/samples.csv, latent.results,
     latent_summary.json) under their model path (galaxies.lens.shear.magnitude), beside whatever the
     analysis's Latent class computes, with no Analysis code needed;
   • makes the latent plot on the latent-output schedule (after the fit; during it only if latent_during_fit),
     through a new plot_latent_results hook on the search, so the per-update cadence is unchanged.
     An angle is periodic and a conversion fixes its branch cut, so a shear angle posterior peaking at 135 deg was
     drawn as two peaks (at 135 and -45). A conversion returns an angle as af.Periodic(angle, period) and PyAutoFit
     recentres the samples on their weighted circular mean before plotting or writing them.

PyAutoGalaxy's side is small: the latent_parameters declarations on EllProfile (inherited by every light and
mass profile), ExternalShear, PowerLawMultipole, EllipseMultipole(Scaled) and the light-profile multipole
mixin, five convert.latent_*_from functions wrapping the existing conversions, and notation labels for the new
names (axis_ratio: q, angle: \phi, magnitude: \gamma, k_m, phi_m, k_3...). I also fixed the notation
bug that labels both gamma_1 and gamma_2 as \gamma.

Everything is on branches of my forks (PyAutoFit fix/corner-anesthetic-duplicate-labels, PyAutoGalaxy
feature/latent-parameters, and feature/corner-plot-labels in autofit/autogalaxy/autolens_workspace, HowToLens
and HowToGalaxy for the config keys and the script text). Full suites pass locally (PyAutoFit 3720, PyAutoGalaxy
1344 on the branch). The plots below are a SLACS lens's 19-parameter source-light stage (lens bulge, disk and point MGEs, isothermal mass, shear, source bulge).

Before: the plot the fixed function makes with the old labels (every MGE is g, both shear components are \gamma^{\rm ext})

After: full corner plot with component-path labels

After: corner_anesthetic_lens.png

After: corner_anesthetic_source.png

Latent: corner_anesthetic_latent.png (axis ratios, angles, shear magnitude and angle)

Latent: corner_anesthetic_latent_lens.png; the shear angle is one peak after recentring

Latent: corner_anesthetic_latent_source.png

Two things I would value a view on: (a) whether the superscript change to model.results text is acceptable or
should be plot-only; (b) whether corner_anesthetic_latent should default on once PyAutoGalaxy declares the
conversions, given it only costs a plot after the fit.

[samlange04]

PR for the PyAutoFit side: PyAutoLabs/PyAutoFit#1709. The PyAutoGalaxy declarations and the workspace config/script updates will follow as separate PRs once this is released, since they import af.Periodic and name the new config keys.

[Jammy2211]

Accepted, with both answers on the PR: superscripts change everywhere, and the latent corner plot stays a separate config-switched plot in PyAutoFit. Review is on PyAutoLabs/PyAutoFit#1709; I will apply the remaining changes on your branch and merge, then follow with the PyAutoGalaxy and workspace sides. Thanks @samlange04.

[Jammy2211]

Update: the review changes are pushed onto PyAutoLabs/PyAutoFit#1709 (summary in the PR comment). It merges once CI is green and goes out in the next release; the PyAutoGalaxy declarations and workspace config/docs follow after that.