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Hierarchicial Interferometric Bayesian Imaging (HIBI) ​

In this tutorial, we will demonstrate how to use Comrade to utilize Bayesian hierarchical modeling to reconstruct an image of M87. Regular imaging tries to make minimal assumptions about the source structure. We will demonstrate how a use can incorporate their knowledge of the source structure into the model to potentially improve the image reconstruction.

Explicitly, we will take advantage of the fact that optically thin accretion flows from a black hole that is face on is expected to produce a ring-like structure. To take advantage of this we will use a hierarchical imaging approach where we can easily incorporate this domain model into the image reconstruction.

This approach is called Hierarchical Interferometric Bayesian Imaging (HIBI) and is described in detail in the paper Hierarchical Interferometric Bayesian Imaging (in prep). For our image model we will use a raster image similar to Stokes I Simultaneous Image and Instrument Modeling. We will decompose the image raster components Fij as follows

Fij=F0μ(xi,yj)exp⁡(σrij)∑ijμ(xi,yj)exp⁡(σrij)

where F0 is the total flux of the image, μ(xi,yj) is a mean image and rij is a is the multiplicative fluctuations about this mean image.

However, before we continue let's load the packages and tutorials we will need.

Loading the Data ​

To get started we will load Comrade

julia
using Comrade

Load the Data ​

julia
using Pyehtim
    CondaPkg Found dependencies: /home/runner/.julia/packages/DimensionalData/hv9KC/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/CondaPkg/0UqYV/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/PythonCall/wkBj7/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/Pyehtim/bQtHC/CondaPkg.toml
    CondaPkg Resolving changes
             + ehtim (pip)
             + libstdcxx
             + libstdcxx-ng
             + numpy
             + numpy (pip)
             + openssl
             + pandas
             + python
             + setuptools (pip)
             + uv
             + xarray
    CondaPkg Initialising pixi
             │ /home/runner/.julia/artifacts/cefba4912c2b400756d043a2563ef77a0088866b/bin/pixi
             │ init
             │ --format pixi
             └ /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/Hibi/.CondaPkg
✔ Created /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/Hibi/.CondaPkg/pixi.toml
    CondaPkg Wrote /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/Hibi/.CondaPkg/pixi.toml
             │ [dependencies]
             │ openssl = ">=3, <3.6, >=3, <3.6"
             │ libstdcxx = ">=3.4,<15.0"
             │ uv = ">=0.4"
             │ libstdcxx-ng = ">=3.4,<15.0"
             │ pandas = "<2"
             │ xarray = "*"
             │ numpy = ">=1.24, <2.0"
             │ 
             │     [dependencies.python]
             │     channel = "conda-forge"
             │     build = "*cp*"
             │     version = ">=3.10,<3.14, >=3.6,<=3.12"
             │ 
             │ [project]
             │ name = ".CondaPkg"
             │ platforms = ["linux-64"]
             │ channels = ["conda-forge"]
             │ channel-priority = "strict"
             │ description = "automatically generated by CondaPkg.jl"
             │ 
             │ [pypi-dependencies]
             │ ehtim = ">=1.2.10, <2.0"
             │ numpy = ">=1.24, <2.0"
             └ setuptools = "*"
    CondaPkg Installing packages
             │ /home/runner/.julia/artifacts/cefba4912c2b400756d043a2563ef77a0088866b/bin/pixi
             │ install
             └ --manifest-path /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/Hibi/.CondaPkg/pixi.toml
✔ The default environment has been installed.
/home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/Hibi/.CondaPkg/.pixi/envs/default/lib/python3.11/site-packages/ehtim/__init__.py:58: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81.
  import pkg_resources

For reproducibility we use a stable random number genreator

julia
using StableRNGs
rng = StableRNG(42)
StableRNGs.LehmerRNG(state=0x00000000000000000000000000000055)

To download the data visit https://doi.org/10.25739/g85n-f134 To load the eht-imaging obsdata object we do:

julia
obs = ehtim.obsdata.load_uvfits(joinpath(__DIR, "..", "..", "Data", "SR1_M87_2017_096_lo_hops_netcal_StokesI.uvfits"))
Python: <ehtim.obsdata.Obsdata object at 0x7fc72508d810>

Now we do some minor preprocessing:

  • Scan average the data since the data have been preprocessed so that the gain phases coherent.
julia
obs = scan_average(obs)
Python: <ehtim.obsdata.Obsdata object at 0x7fc720d82910>

For this tutorial we will once again fit complex visibilities since they provide the most information once the telescope/instrument model are taken into account. Additionally, we will preprocess the data to remove effects we don't want to model. Specifically we will remove the

  • Baselines shorter than 0.1Gλ since this represents overresolved structure

  • Add 1% systematic uncertainty to handle residual calibration errors such as leakage.

julia
dvis = add_fractional_noise(flag(x -> uvdist(x) < 0.1e9, extract_table(obs, Visibilities())), 0.01)
EHTObservationTable{Comrade.EHTVisibilityDatum{:I}}
  source:      M87
  mjd:         57849
  bandwidth:   1.856e9
  sites:       [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples:    242

Building the Model/Posterior ​

Now let's construct our model using the decomposition described above. For this we will need to define

julia
function sky(θ, metadata)
    (; c, σimg, r, ain, aout) = θ
    (; ftot, grid) = metadata
    # Form the image model
    # First transform to simplex space first applying the non-centered transform
    mb = RingTemplate(RadialDblPower(ain, aout), AzimuthalUniform())
    mr = modify(mb, Stretch(r))
    mimg = intensitymap(mr, grid)
    rast = apply_fluctuations(CenteredLR(), mimg, σimg * c.params)
    rast .*= ftot
    mimg = ContinuousImage(rast, grid, BSplinePulse{3}())
    return mimg
end
sky (generic function with 1 method)

Unlike other imaging examples (e.g., Imaging a Black Hole using only Closure Quantities) we also need to include a model for the instrument, i.e., gains as well. The gains will be broken into two components

  • Gain amplitudes which are typically known to 10-20%, except for LMT, which has amplitudes closer to 50-100%.

  • Gain phases which are more difficult to constrain and can shift rapidly.

julia
using VLBIImagePriors
using Distributions
fgain(x) = exp(x.lg + 1im * x.gp)
G = SingleStokesGain(fgain)

intpr = (
    lg = ArrayPrior(IIDSitePrior(ScanSeg(), Normal(0.0, 0.2)); LM = IIDSitePrior(ScanSeg(), Normal(0.0, 1.0))),
    gp = ArrayPrior(IIDSitePrior(ScanSeg(), DiagonalVonMises(0.0, inv(π^2))); refant = SEFDReference(0.0)),
)
intmodel = InstrumentModel(G, intpr)
InstrumentModel
  with Jones: SingleStokesGain
  with reference basis: PolarizedTypes.CirBasis()

Before we move on, let's go into the model function a bit. This function takes two arguments θ and metadata. The θ argument is a named tuple of parameters that are fit to the data. The metadata argument is all the ancillary information we need to construct the model. For our hybrid model, we will need two variables for the metadata, a grid that specifies the locations of the image pixels and a cache that defines the algorithm used to calculate the visibilities given the image model. This is required since ContinuousImage is most easily computed using number Fourier transforms like the NFFT or FFT. To combine the models, we use Comrade's overloaded + operators, which will combine the images such that their intensities and visibilities are added pointwise.

Now let's define our metadata. First we will define the cache for the image. This is required to compute the numerical Fourier transform.

julia
fovxy = μas2rad(150.0)
npix = 48
g = imagepixels(fovxy, fovxy, npix, npix)
RectiGrid(
executor: ComradeBase.Serial()
Dimensions: 
(↓ X Sampled{Float64} LinRange{Float64}(-3.560350470648155e-10, 3.560350470648155e-10, 48) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-3.560350470648155e-10, 3.560350470648155e-10, 48) ForwardOrdered Regular Points)
)

Part of hybrid imaging is to force a scale separation between the different model components to make them identifiable. To enforce this we will set the raster component to have a correlation length of 5 times the beam size.

julia
cprior = corr_image_prior(g, dvis)
HierarchicalPrior(
	map: 
	ConditionalMarkov(
Random Field: VLBIImagePriors.GaussMarkovRandomField
Graph: MarkovRandomFieldGraph{1}(
dims: (48, 48)
)
)	hyper prior: 
	Truncated(Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.027179060796761068)
θ: 36.79303002696731
)
; lower=1.0, upper=96.0)

)

For the other parameters we use a uniform priors for the ring fractional flux f ring radius r, ring width σ, and the flux fraction of the Gaussian component fg and the amplitude for the ring brightness modes. For the angular variables ξτ and ξ we use the von Mises prior with concentration parameter inv(π^2) which is essentially a uniform prior on the circle. Finally for the standard deviation of the MRF we use a half-normal distribution. This is to ensure that the MRF has small differences from the mean image.

julia
skyprior = (
    c = cprior,
    σimg = truncated(Normal(0.0, 0.5); lower = 0.0),
    r = Uniform(μas2rad(10.0), μas2rad(40.0)),
    ain = Uniform(1.0, 20.0),
    aout = Uniform(1.0, 20.0),
)
(c = HierarchicalPrior(
	map: 
	ConditionalMarkov(
Random Field: VLBIImagePriors.GaussMarkovRandomField
Graph: MarkovRandomFieldGraph{1}(
dims: (48, 48)
)
)	hyper prior: 
	Truncated(Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.027179060796761068)
θ: 36.79303002696731
)
; lower=1.0, upper=96.0)

)
, σimg = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0), r = Distributions.Uniform{Float64}(a=4.84813681109536e-11, b=1.939254724438144e-10), ain = Distributions.Uniform{Float64}(a=1.0, b=20.0), aout = Distributions.Uniform{Float64}(a=1.0, b=20.0))

Now we form the metadata

julia
skymetadata = (; ftot = 1.1, grid = g)
skym = SkyModel(sky, skyprior, g; metadata = skymetadata)
SkyModel
  with map: sky
   on grid: 
RectiGrid(
executor: ComradeBase.Serial()
Dimensions: 
(↓ X Sampled{Float64} LinRange{Float64}(-3.560350470648155e-10, 3.560350470648155e-10, 48) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-3.560350470648155e-10, 3.560350470648155e-10, 48) ForwardOrdered Regular Points)
)
   )

This is everything we need to specify our posterior distribution, which our is the main object of interest in image reconstructions when using Bayesian inference.

julia
using Enzyme
post = VLBIPosterior(skym, intmodel, dvis)
VLBIPosterior
ObservedSkyModel
  with map: sky
   on grid: 
FourierDualDomain(
Algorithm: VLBISkyModels.NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}(1, 1.0e-9, AbstractNFFTs.TENSOR, 0x00000000)
Image Domain: RectiGrid(
executor: ComradeBase.Serial()
Dimensions: 
(↓ X Sampled{Float64} LinRange{Float64}(-3.560350470648155e-10, 3.560350470648155e-10, 48) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-3.560350470648155e-10, 3.560350470648155e-10, 48) ForwardOrdered Regular Points)
)
Visibility Domain: UnstructredDomain(
executor: ComradeBase.Serial()
Dimensions: 
242-element StructArray(::Vector{Float64}, ::Vector{Float64}, ::Vector{Float64}, ::Vector{Float64}) with eltype @NamedTuple{U::Float64, V::Float64, Ti::Float64, Fr::Float64}:
 (U = -4.405690154666661e9, V = -4.523017159111106e9, Ti = 0.9166666567325592, Fr = 2.27070703125e11)
 (U = -4.444299918222218e9, V = -4.597825294222218e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 1.337045162666665e9, V = -3.765300401777774e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = -1.336260540444443e9, V = 3.763616127999996e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 4.445088654222218e9, V = 4.596145080888884e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 5.781345607111105e9, V = 8.325259893333325e8, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 1.4806382151111097e9, V = -3.741479615999996e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = -4.455366328888884e9, V = -4.673060451555551e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = -1.4798758791111097e9, V = 3.739809735111107e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = 4.456123861333328e9, V = 4.671391715555551e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = 5.936013027555549e9, V = 9.315912497777768e8, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = -4.438811278222218e9, V = -4.748261176888884e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 1.615060401777776e9, V = -3.715306943999996e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = -1.6143345528888872e9, V = 3.713649343999996e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 4.439536184888884e9, V = 4.746598001777773e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 6.053865329777771e9, V = 1.0329465084444433e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = -4.394744988444439e9, V = -4.822934485333328e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = 1.7394626631111093e9, V = -3.686947740444441e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = -1.7387777884444425e9, V = 3.6853017173333297e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = 4.395427669333329e9, V = 4.821289443555551e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
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 (U = -1.8643712995555537e9, V = 3.651452543999996e9, Ti = 2.449999988079071, Fr = 2.27070703125e11)
 (U = 6.178850375111104e9, V = 1.2516714417777765e9, Ti = 2.449999988079071, Fr = 2.27070703125e11)
 (U = 4.314477112888885e9, V = 4.903116785777773e9, Ti = 2.449999988079071, Fr = 2.27070703125e11)
 (U = 1.9658264106666646e9, V = -3.6206990791111073e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = -4.212778346666662e9, V = -4.976839239111106e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = -1.965238759111109e9, V = 3.6190756053333297e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 6.178607487999993e9, V = 1.356139594666665e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 4.213364039111107e9, V = 4.975216568888884e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 2.0544588266666646e9, V = -3.586710200888885e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = -4.085601863111107e9, V = -5.046995640888884e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = -2.0539194026666646e9, V = 3.5850976213333297e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 4.0861423288888845e9, V = 5.045379299555551e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 6.140063729777771e9, V = 1.4602817422222207e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 2.130349845333331e9, V = -3.5513328995555515e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = -3.933104519111107e9, V = -5.114784767999995e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = -2.1298682808888867e9, V = 3.5497276302222185e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = 6.063457521777771e9, V = 1.5634511359999983e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = 3.9335861404444404e9, V = 5.113178993777773e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = 2.578938929777775e9, V = -4.734788238222218e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -3.735105663999996e9, V = -5.186825713777772e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 2.1991734328888865e9, V = -3.5106569031111073e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -2.5785191964444413e9, V = 4.733192191999995e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -2.198756124444442e9, V = 3.509060273777774e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
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 (U = 3.735526620444441e9, V = 5.185227192888883e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 6.314048611555549e9, V = 4.520333351111106e8, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 5.934280931555549e9, V = 1.6761668835555537e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -3.5324296035555515e9, V = -5.248264732444439e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 2.246733795555553e9, V = -3.4730721351111073e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 2.7040352142222195e9, V = -4.690125752888884e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = -2.687901513142854e9, V = 4.694663167999995e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = -2.2408562468571405e9, V = 3.476569380571425e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 4.573011991111106e8, V = -1.2170540408888876e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 3.561408036571425e9, V = 5.238641298285708e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 5.779150734222217e9, V = 1.7751998862222204e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 6.236464440888882e9, V = 5.581377528888882e8, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 2.81236712533333e9, V = -4.643490716444439e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 2.280368668444442e9, V = -3.43479938133333e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = -3.307854179555552e9, V = -5.306092316444439e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = -2.812399111529409e9, V = 4.641748931764702e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = -2.280159088941174e9, V = 3.43308967905882e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 5.3200019644444394e8, V = -1.2086904924444432e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 3.307383679999996e9, V = 5.304687194352936e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 6.120220771555549e9, V = 6.626024835555549e8, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 5.588211185777772e9, V = 1.8712975182222202e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 5.092881095111105e9, V = -4.199598421333329e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.903268209777775e9, V = -4.595170062222218e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.299867178666664e9, V = -3.396077226666663e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = -3.062767004444441e9, V = -5.359950862222217e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.050181503999994e9, V = -4.210810559999995e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = -2.8907034879999967e9, V = 4.600886271999995e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.189606492444442e9, V = 3.955690115555551e8, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = -2.2976930559999976e9, V = 3.400280575999996e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.79301371733333e9, V = -8.035218719999992e8, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 6.034046399999994e8, V = -1.199091349333332e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 3.1008046079999967e9, V = 5.350639359999994e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 8.155639423999991e9, V = 1.1603480675555544e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.966032455111105e9, V = 7.64783736888888e8, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.36262916266666e9, V = 1.963875015111109e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.388079544888883e9, V = -4.101135303111107e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.3048084479999976e9, V = -3.3528182257777743e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -2.768253631999997e9, V = -5.414730879999994e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.388055679999994e9, V = -4.101249678222218e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.9831417315555525e9, V = -4.539868501333328e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.356877311999994e9, V = -4.110922695111107e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -2.975202047999997e9, V = 4.544589937777773e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.4049356302222195e9, V = 4.387324826666662e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.4049011413333306e9, V = 4.3861474933333284e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -2.305032220444442e9, V = 3.356104334222219e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 3.083277411555552e9, V = -7.48315953777777e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 3.0832264319999967e9, V = -7.484370453333325e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 6.783317208888881e8, V = -1.187050453333332e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.8027259164444413e9, V = 5.407289457777772e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 8.156328149333324e9, V = 1.3135985066666653e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 8.156297400888881e9, V = 1.3134812622222207e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.751382712888883e9, V = 8.74867279999999e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.073054606222218e9, V = 2.0619144959999979e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -5.356795278222217e9, V = 4.1110577777777734e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.59465437866666e9, V = -4.018611114666662e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 3.030624810666663e9, V = -4.494683207111106e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.2960602239999976e9, V = -3.3182473599999967e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.594626830222217e9, V = -4.0187286115555515e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -2.5172720142222195e9, V = -5.45444669155555e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.579493814857137e9, V = -4.023605065142853e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -3.0274068479999967e9, V = 4.496662674285709e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.5640353279999976e9, V = 4.7607165511111057e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.564000689777775e9, V = 4.7595243111111057e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -2.2970926933333306e9, V = 3.3193651078095202e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 3.2985927679999967e9, V = -7.003643235555549e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 7.345665155555549e8, V = -1.1764336924444432e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 3.2985603128888855e9, V = -7.004829137777771e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.5374800944761877e9, V = 5.44992148723809e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 8.111924807111103e9, V = 1.4358354488888874e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 4.813340984888884e9, V = 2.136196949333331e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.547893447111105e9, V = 9.597677279999989e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 8.111898709333324e9, V = 1.4357162488888874e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -5.579455049142851e9, V = 4.0237272990476146e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.773825464888883e9, V = -3.933187214222218e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.2760630257777753e9, V = -3.283890659555552e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 3.0632664319999967e9, V = -4.448890581333328e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.773791857777772e9, V = -3.9333102648888845e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -2.253957283555553e9, V = -5.490298367999994e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.744240127999994e9, V = -3.946954495999996e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -3.0586252799999967e9, V = 4.455449599999995e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.710558015999997e9, V = 5.1570311111111057e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.7105363982222195e9, V = 5.1558591999999946e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -2.2804378879999976e9, V = 3.2883723519999967e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 3.4977656675555515e9, V = -6.492943324444438e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 7.872002471111102e8, V = -1.164999438222221e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 3.497740145777774e9, V = -6.494127164444437e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.3015257599999976e9, V = 5.482690303999994e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 8.027784391111103e9, V = 1.5571086684444427e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.317234005333327e9, V = 1.0414041991111101e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 8.027758634666658e9, V = 1.556993443555554e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 4.530021788444439e9, V = 2.206406378666664e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -5.744238079999993e9, V = 3.9470612479999957e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.924709319111105e9, V = -3.8452848639999957e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 5.924688497777772e9, V = -3.845403854222218e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.08089890133333e9, V = -4.402725802666661e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 2.244918698666664e9, V = -3.2499217208888855e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 2.8438056746666636e9, V = 5.57437258666666e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 2.8437909475555525e9, V = 5.573208195555549e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.6797841706666627e9, V = -5.953657653333327e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.679770524444441e9, V = -5.954820355555549e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 8.35980113777777e8, V = -1.1528026026666653e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.081715143111108e9, V = -4.344833365333329e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 6.07242564266666e9, V = -3.732578396444441e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 2.190544647111109e9, V = -3.208251676444441e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 6.05836305066666e9, V = -3.7438256071111073e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = -3.0829770239999967e9, V = 4.34976796444444e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 2.99071138133333e9, V = 6.122509119999994e8, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = -2.1976609848888865e9, V = 3.211308942222219e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 3.881882190222218e9, V = -5.2432489066666615e8, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 8.911678026666657e8, V = -1.1365769386666653e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 6.157184056888882e9, V = -3.6406887111111073e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.157173617777771e9, V = -3.6408075306666627e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 2.1349524657777758e9, V = -3.175750087111108e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 3.065380067555552e9, V = -4.298637105777774e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.14621098666666e9, V = -3.6535959466666627e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = -3.0691708159999967e9, V = 4.304277162666662e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 3.091809799111108e9, V = 6.579570684444437e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 3.091795150222219e9, V = 6.578363697777771e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = -2.1445705386666644e9, V = 3.1792032426666636e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 4.0222341617777734e9, V = -4.6493393599999946e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 4.0222294044444404e9, V = -4.650499119999995e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 9.304277457777768e8, V = -1.1228893119999988e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = -6.146197247999993e9, V = 3.653717759999996e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.211784632888882e9, V = -3.5477483875555515e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 6.211780195555549e9, V = -3.547863480888885e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 2.068902503111109e9, V = -3.1441626808888855e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 3.034031331555552e9, V = -4.2528075377777734e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 3.177752227555552e9, V = 7.050634364444437e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 3.177752810666663e9, V = 7.049488924444437e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 4.1428866915555515e9, V = -4.035797448888885e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 9.651305439999989e8, V = -1.108645550222221e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 4.1428811519999957e9, V = -4.036979662222218e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 6.235954645333326e9, V = -3.4543274026666627e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 2.987818652444441e9, V = -4.207558826666662e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 6.235956423111104e9, V = -3.454211199999996e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 1.9927185599999979e9, V = -3.1136442239999967e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 6.23440708266666e9, V = -3.468812543999996e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -2.9972223146666636e9, V = 4.213749674666662e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 3.2481352177777743e9, V = 7.533482702222215e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 3.2481338097777743e9, V = 7.532299875555547e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -2.006915669333331e9, V = 3.1172445866666636e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 4.243231630222218e9, V = -3.4069073333333296e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 4.243239466666662e9, V = -3.4056780622222185e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 9.951025973333323e8, V = -1.0939171839999988e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -6.23440452266666e9, V = 3.4689294506666627e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 6.223212003555549e9, V = -3.337163527111108e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 6.223215416888882e9, V = -3.337282766222219e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 2.909506687999997e9, V = -4.152156949333329e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 1.8838133048888867e9, V = -3.07722722133333e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 6.225276682239994e9, V = -3.340931624959996e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -2.913978071039997e9, V = 4.1530716876799955e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 3.313704135111108e9, V = 8.14994561777777e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 3.3137085368888855e9, V = 8.148773351111102e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -1.889491799039998e9, V = 3.077254901759997e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 4.339403889777773e9, V = -2.5993809599999973e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 4.339408696888885e9, V = -2.6005522088888863e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 1.0256997831111101e9, V = -1.0749325617777767e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -6.225278402559994e9, V = 3.3410462617599964e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 6.178709461333326e9, V = -3.243972359111108e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 2.8307861475555525e9, V = -4.1090220657777734e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 6.178720483555549e9, V = -3.244096682666663e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 1.786251743999998e9, V = -3.049647466666663e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 6.179113016888882e9, V = -3.242380138666663e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 3.347918890666663e9, V = 8.650494186666657e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -2.8311915662222195e9, V = 4.107428693333329e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 3.347926620444441e9, V = 8.649315555555546e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -1.7866596017777758e9, V = 3.048053795555552e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 4.392453788444439e9, V = -1.9432878977777755e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 4.392462890666661e9, V = -1.944448759999998e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 1.0445360586666656e9, V = -1.0593764586666656e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -6.179119544888882e9, V = 3.2424975217777743e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 2.7382116835555525e9, V = -4.0671804444444404e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 6.103944163555549e9, V = -3.151683946666663e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 1.6799544319999983e9, V = -3.023603527111108e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 6.103956750222216e9, V = -3.1518036479999967e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 6.10439825066666e9, V = -3.150082410666663e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 3.3657344497777743e9, V = 9.154982524444435e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -2.7386646755555525e9, V = 4.065578346666662e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 3.365744974222219e9, V = 9.153749564444435e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -1.6804040142222204e9, V = 3.02200078933333e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 4.423988807111106e9, V = -1.2808080399999987e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 4.424001663999995e9, V = -1.2819521311111099e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 1.0582553617777767e9, V = -1.0435755999999989e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -6.104410581333326e9, V = 3.150201891555552e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 5.999277468444439e9, V = -3.060741980444441e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 2.6322108302222195e9, V = -4.0268253795555515e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 5.999288348444438e9, V = -3.060856782222219e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 1.565423818666665e9, V = -2.999217870222219e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 5.999780252444438e9, V = -3.0591319537777743e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -2.6327225742222195e9, V = 4.025219150222218e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 3.3670615039999967e9, V = 9.660856764444433e8, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 3.367077162666663e9, V = 9.659668408888879e8, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 4.433867007999995e9, V = -6.164012344444438e7, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -1.5659301759999983e9, V = 2.99760936533333e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 4.433851406222218e9, V = -6.1523151111111045e7, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 1.0667900266666656e9, V = -1.0276084337777767e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -5.999801315555549e9, V = 3.059254300444441e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
)
   )
ObservedInstrumentModel
  with Jones: SingleStokesGain
  with reference basis: PolarizedTypes.CirBasis()Data Products: Comrade.EHTVisibilityDatum

We can sample from the prior to see what the model looks like

julia
using CairoMakie
xrand = prior_sample(rng, post)
gpl = refinespatial(g, 3)
fig = imageviz(intensitymap(skymodel(post, xrand), gpl));

Reconstructing the Image ​

To find the image we will demonstrate two methods:

  • Optimization to find the MAP (fast but often a poor estimator)

  • Sampling to find the posterior (slow but provides a substantially better estimator)

For optimization we will use the Optimization.jl package and the LBFGS optimizer. To use this we use the comrade_opt function

julia
using Optimization, OptimizationLBFGSB
xopt, sol = comrade_opt(
    post, LBFGSB();
    initial_params = xrand, maxiters = 2000, g_tol = 1.0e0
);

First we will evaluate our fit by plotting the residuals

julia
res = residuals(post, xopt);
fig = plotfields(res[1], :uvdist, :res);

These residuals suggest that we are substantially overfitting the data. This is a common side effect of MAP imaging. As a result if we plot the image we see that there is substantial high-frequency structure in the image that isn't supported by the data.

julia
fig = imageviz(intensitymap(skymodel(post, xopt), gpl), figure = (; resolution = (500, 400)));

To improve our results we will now move to Posterior sampling. This is the main method we recommend for all inference problems in Comrade. While it is slower the results are often substantially better. To sample we will use the AdvancedHMC package.

julia
using AdvancedHMC
chain = sample(rng, post, NUTS(0.8), 700; n_adapts = 500, progress = false, initial_params = xopt);
[ Info: Found initial step size 0.0015625

chain = load_samples(out) We then remove the adaptation/warmup phase from our chain

julia
chain = chain[501:end]
PosteriorSamples
  Samples size: (200,)
  sampler used: AHMC
Mean
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┬────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky                                                                                                                                                                                                                                                                                               │ instrument                                                                                                                                                                                                                                                                                                                                                                                                                                     │
│ @NamedTuple{c::@NamedTuple{params::Matrix{Float64}, hyperparams::Float64}, σimg::Float64, r::Float64, ain::Float64, aout::Float64}                                                                                                                                                                │ @NamedTuple{lg::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}, gp::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}}                                                                                      │
├───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = (params = [-0.00501787 -0.0681767 … 0.00565649 -0.0196176; 0.019559 -0.0114447 … -0.0483644 -0.0385897; … ; -0.0441325 -0.0730607 … 0.207624 0.0881275; -0.084905 -0.0805285 … 0.102907 0.0316352], hyperparams = 35.9796), σimg = 0.931426, r = 8.06635e-11, ain = 14.5097, aout = 1.42732) │ (lg = [-0.00457581, 0.0103134, -0.0444081, -0.0157358, -0.399716, 0.152378, -0.0272762, -0.00608473, -0.230104, 0.103801  …  -0.0896535, -0.0122102, -0.918155, -0.0163118, -0.0022846, 0.0238679, -0.159109, -0.0242234, -0.972276, -0.0284988], gp = [0.0, 0.321752, 0.0, -2.29875, 1.65232, 0.158985, 0.0, -2.29873, 1.71132, -0.0168472  …  -1.56593, 0.132133, -2.49471, -1.21391, 0.0, -1.84576, -1.69262, 0.150563, -2.64837, -1.3446]) │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
Std. Dev.
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┬────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky                                                                                                                                                                                                                                                                       │ instrument                                                                                                                                                                                                                                                                                                                                                                                                                         │
│ @NamedTuple{c::@NamedTuple{params::Matrix{Float64}, hyperparams::Float64}, σimg::Float64, r::Float64, ain::Float64, aout::Float64}                                                                                                                                        │ @NamedTuple{lg::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}, gp::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}}                                                                          │
├───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = (params = [0.563853 0.520023 … 0.641234 0.535703; 0.550565 0.611435 … 0.683031 0.568034; … ; 0.543594 0.611478 … 0.644462 0.647095; 0.514136 0.570125 … 0.614897 0.521904], hyperparams = 21.8316), σimg = 0.112137, r = 4.65685e-12, ain = 3.5915, aout = 0.199179) │ (lg = [0.159713, 0.148516, 0.0590305, 0.0649566, 0.0884423, 0.068159, 0.0533819, 0.0582349, 0.0770241, 0.064203  …  0.063978, 0.0664528, 0.0608703, 0.0653915, 0.0546429, 0.0601864, 0.0620556, 0.0721245, 0.0643015, 0.0720776], gp = [0.0, 0.162463, 0.0, 0.0314039, 0.155051, 0.169011, 0.0, 0.0348432, 0.148204, 0.176303  …  0.118345, 0.193979, 0.0808804, 2.70197, 0.0, 0.0376671, 0.115113, 0.191672, 0.0806203, 2.63066]) │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

Warning

This should be run for 4-5x more steps to properly estimate expectations of the posterior

Now lets plot the mean image and standard deviation images. To do this we first clip the first 250 MCMC steps since that is during tuning and so the posterior is not sampling from the correct sitesary distribution.

julia
using StatsBase
msamples = skymodel.(Ref(post), chain[begin:5:end]);
WARNING: using StatsBase.residuals in module ##225 conflicts with an existing identifier.

The mean image is then given by

julia
imgs = intensitymap.(msamples, Ref(gpl))
fig = imageviz(mean(imgs), size = (400, 300));

julia
fig = imageviz(std(imgs), colormap = :batlow, size = (400, 300));

Finally, let's take a look at some of the ring parameters

julia
figd = Figure(; resolution = (650, 200));
p1 = density(figd[1, 1], rad2μas(chain.sky.r) * 2, axis = (xlabel = "Ring Pwr Law Transition (μas)",))
p2 = density(figd[1, 2], chain.sky.ain, axis = (xlabel = "Inner Radial Index",))
p3 = density(figd[1, 3], chain.sky.aout, axis = (xlabel = "Outer Radial Index",))

Now let's check the residuals using draws from the posterior

julia
fig = Figure(; size = (600, 400))
resch = residuals(post, chain[end])
ax, = plotfields!(fig[1, 1], resch[1], :uvdist, :res, scatter_kwargs = (; label = "MAP", color = :blue, colorim = :red, marker = :circle), legend = false)
for s in sample(chain, 10)
    baselineplot!(ax, residuals(post, s)[1], :uvdist, :res, alpha = 0.2, label = "Draw")
end
axislegend(ax, merge = true)

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