Priors

All parameters to your model must have a prior defined. You may provide any continuous, univariate distribution from the Distributions.jl. A few useful distributions include:

  • Normal
  • Uniform
  • LogNormal
  • LogUniform
  • TrucatedNormal
  • VonMises

This pacakge also defines the Sine() distribution for e.g. inclination priors and UniformCircular() for periodic variables. Internally, UniformCircular() creates two standard normal variables and finds the angle between them using arctan. This allows the sampler to smoothly cycle past the ends of the domain. You can specify a different circular domain than (0,2pi) by passing the size of the domain e.g. τ = UniformCircular(1.0).

The VonMise distribution is notable but not commonly used. It is the analog of a normal distribution defined on a circular domain (-π, +π). If you have a Gaussian prior on an angular parameter, a Von Mises distribution is probably more appropriate.

Behind the scenes, Octofitter remaps your parameters to unconstrained domains using the Bijectors.jl (and corrects the priors accordingly). This is essential for good sampling efficiency with HMC based samplers.

This means that e.g. if you define the eccentricity prior as e=Uniform(0,0.5), the sampler will actually generate values across the whole real line and transform them back into the [0,0.5] range before evaluating the orbit. It is therefore essential that your priors do not include invalid domains.

For example, setting a=Normal(3,2) will result in poor sampling efficiency as sometimes negative values for semi-major axis will be drawn (especially if you're using the parallel tempered sampler).

Instead, for parameters like semi-major axis, eccentricity, parallax, and masses, you should truncate any distributions that have negative tails. This can easily be accomplished with TrauncatedNormal or Trunacted(dist, low, high) for any arbitrary distribution.

Kernel Density Estimate Priors

Octofitter has support for sampling from smoothed kernel density estimate priors. These are non-parametric distributions fit to a 1D dataset consisting of random draws. This is one way to include the output of a different model as the prior to a new model. That said, it's usually best to try and incorporate the model directly into the code. There are a few examples on GitHub of this, including atmosphere model grids, cooling tracks, etc.

Using a KDE

First, we will generate some data. In the real world, you would load this data eg. from a CSV file.

using Octofitter, Distributions

# create a smoothed KDE estimate of the samples from a 10+-1 gaussian
kde = Octofitter.KDEDist(randn(1000).+10)
KDEDist kernel density estimate distribution

Note that in Octofitter the KDE will have its support truncated to the minimum and maximum values that occur in your dataset, ie. it doesn't allow for infinite long tails.

Now add it to your model as a prior:

@planet b Visual{KepOrbit} begin
    a ~ kde # Sample from the KDE here
    e ~ Uniform(0.0, 0.99)
    i ~ Sine()
    ω ~ UniformCircular()
    Ω ~ UniformCircular()
    θ ~ UniformCircular()
    tp = θ_at_epoch_to_tperi(system,b,50000)
end
@system Tutoria begin
    M ~ truncated(Normal(1.2, 0.1), lower=0.1)
    plx ~ truncated(Normal(50.0, 0.02), lower=0.1)
end b
model = Octofitter.LogDensityModel(Tutoria)
chain = octofit(model)
Chains MCMC chain (1000×28×1 Array{Float64, 3}):

Iterations        = 1:1:1000
Number of chains  = 1
Samples per chain = 1000
Wall duration     = 0.73 seconds
Compute duration  = 0.73 seconds
parameters        = M, plx, b_a, b_e, b_i, b_ωy, b_ωx, b_Ωy, b_Ωx, b_θy, b_θx, b_ω, b_Ω, b_θ, b_tp
internals         = n_steps, is_accept, acceptance_rate, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt, loglike, logpost, tree_depth, numerical_error

Summary Statistics
  parameters         mean         std       mcse    ess_bulk   ess_tail      r ⋯
      Symbol      Float64     Float64    Float64     Float64    Float64   Floa ⋯

           M       1.1979      0.1037     0.0031   1145.2492   482.4900    0.9 ⋯
         plx      49.9996      0.0199     0.0006   1083.2175   712.4667    1.0 ⋯
         b_a      10.0690      0.9636     0.0289   1120.0647   496.0030    1.0 ⋯
         b_e       0.4998      0.2759     0.0082   1071.8186   549.0207    1.0 ⋯
         b_i       1.5318      0.7030     0.0248    739.9494   359.8334    1.0 ⋯
        b_ωy      -0.0143      0.7142     0.0356    455.8501   720.8847    0.9 ⋯
        b_ωx      -0.0230      0.7112     0.0410    342.8093   840.4579    1.0 ⋯
        b_Ωy       0.0533      0.7157     0.0337    492.5810   929.5566    0.9 ⋯
        b_Ωx       0.0405      0.7106     0.0333    495.6578   732.6694    1.0 ⋯
        b_θy      -0.0234      0.7169     0.0371    417.2007   841.9656    1.0 ⋯
        b_θx       0.0010      0.7004     0.0355    434.2525   783.5202    1.0 ⋯
         b_ω      -0.0245      1.8307     0.0934    453.4350   811.6041    0.9 ⋯
         b_Ω       0.1040      1.7525     0.0762    582.9501   655.3153    0.9 ⋯
         b_θ      -0.0370      1.8457     0.0853    563.8540   837.9178    1.0 ⋯
        b_tp   45123.4441   4220.0653   153.5460    814.9355   843.5914    0.9 ⋯
                                                               2 columns omitted

Quantiles
  parameters         2.5%        25.0%        50.0%        75.0%        97.5%
      Symbol      Float64      Float64      Float64      Float64      Float64

           M       0.9960       1.1316       1.1985       1.2659       1.3955
         plx      49.9630      49.9854      49.9992      50.0135      50.0389
         b_a       8.2241       9.3851      10.0664      10.7009      11.9726
         b_e       0.0286       0.2677       0.5011       0.7272       0.9634
         b_i       0.2621       0.9946       1.5035       2.0776       2.8005
        b_ωy      -1.0700      -0.7035      -0.0298       0.6846       1.0647
        b_ωx      -1.0531      -0.6970      -0.0636       0.6749       1.0568
        b_Ωy      -1.0728      -0.6373       0.1056       0.7374       1.0654
        b_Ωx      -1.0468      -0.6679       0.0865       0.7353       1.0591
        b_θy      -1.0752      -0.7226      -0.0088       0.6686       1.0599
        b_θx      -1.0332      -0.6846      -0.0295       0.6963       1.0435
         b_ω      -2.9839      -1.5834      -0.1366       1.6166       3.0291
         b_Ω      -2.9482      -1.3196       0.1584       1.5980       2.9908
         b_θ      -3.0146      -1.5971      -0.0466       1.5489       3.0011
        b_tp   37926.2456   41249.8087   45349.0263   49438.5626   50403.3448

We now examine the posterior and verify that it matches our KDE prior:

dat = chain[:b_a][:]
@show mean(dat) std(dat)
mean(dat) = 10.069043774384285
std(dat) = 0.9636174036592088

Observable Based Priors

Octofitter implements observable-based priors from O'Neil 2019 for relative astrometry. You can fit a model to astrometry using observable-based priors using the following recipe:

using Octofitter, Distributions

astrom_like = PlanetRelAstromLikelihood(
    (;epoch=mjd("2020-12-20"), ra=400.0, σ_ra=5.0, dec=400.0, σ_dec=5.0)
)

@planet b Visual{KepOrbit} begin
    # For using with ObsPriors:
    P ~ Uniform(0.001, 1000)
    a = cbrt(system.M * b.P^2)

    e ~ Uniform(0.0, 1.0)
    i ~ Sine()
    ω ~ UniformCircular()
    Ω ~ UniformCircular()
    mass ~ LogUniform(0.01, 100)

    τ ~ UniformCircular(1.0)
    tp =  b.τ*b.P*365.25 + 58849 # reference epoch for τ. Choose an MJD date near your data.
end astrom_like ObsPriorAstromONeil2019(astrom_like);

@system System1 begin
    plx ~ Normal(21.219, 0.060)
	M ~ truncated(Normal(1.1, 0.2),lower=0.1)
end b