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:
NormalUniformLogNormalLogUniformTrucatedNormalVonMises
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)
plx ~ truncated(Normal(50.0, 0.02), lower=0)
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 = 1.13 seconds
Compute duration = 1.13 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 1.2005 0.0975 0.0036 741.4254 712.0350 0.9 ⋯
plx 50.0005 0.0196 0.0006 964.8033 426.2000 1.0 ⋯
b_a 10.0067 1.0312 0.0316 1077.4768 583.4725 1.0 ⋯
b_e 0.5204 0.2867 0.0101 728.7875 360.8107 1.0 ⋯
b_i 1.6098 0.6779 0.0225 889.6782 587.9417 1.0 ⋯
b_ωy 0.0666 0.7067 0.0397 334.3524 756.0302 1.0 ⋯
b_ωx 0.0092 0.7155 0.0407 348.2050 738.9199 1.0 ⋯
b_Ωy 0.0343 0.7096 0.0393 359.8931 809.4000 1.0 ⋯
b_Ωx 0.0195 0.7101 0.0436 297.0492 693.5678 1.0 ⋯
b_θy -0.0294 0.7129 0.0363 424.8380 609.3305 1.0 ⋯
b_θx 0.0309 0.7155 0.0380 365.1478 548.9754 0.9 ⋯
b_ω 0.0562 1.7317 0.0938 380.4489 561.4524 1.0 ⋯
b_Ω 0.0168 1.7734 0.0848 517.8987 765.2233 1.0 ⋯
b_θ 0.0925 1.8411 0.0861 553.8887 653.7758 1.0 ⋯
b_tp 45044.9667 4344.3027 155.1877 783.9082 735.6916 1.0 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 1.0174 1.1295 1.2028 1.2656 1.3937
plx 49.9619 49.9878 50.0003 50.0141 50.0384
b_a 8.0136 9.3034 10.0371 10.7099 11.9658
b_e 0.0442 0.2681 0.5358 0.7692 0.9683
b_i 0.3246 1.0983 1.6358 2.1294 2.8362
b_ωy -1.0544 -0.6020 0.0935 0.7621 1.0722
b_ωx -1.0578 -0.7052 0.0404 0.6986 1.0510
b_Ωy -1.0703 -0.6652 0.0770 0.7296 1.0382
b_Ωx -1.0424 -0.6529 -0.0034 0.7242 1.0599
b_θy -1.0697 -0.7278 -0.0589 0.6704 1.0739
b_θx -1.0465 -0.6784 0.0614 0.7302 1.0637
b_ω -2.9263 -1.3696 0.0878 1.5657 2.9444
b_Ω -2.9460 -1.4805 -0.0058 1.5259 2.9608
b_θ -2.9772 -1.5660 0.1350 1.7206 2.9956
b_tp 37545.8190 41231.4528 45063.3941 49610.3265 50404.4166
We now examine the posterior and verify that it matches our KDE prior:
dat = chain[:b_a][:]
@show mean(dat) std(dat)mean(dat) = 10.00671813457469
std(dat) = 1.0312330429039755Observable 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)
end b