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 = 1.21 seconds
Compute duration = 1.21 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 1.1991 0.0973 0.0033 879.7753 534.8883 0.9 ⋯
plx 50.0000 0.0207 0.0007 998.7142 608.3330 1.0 ⋯
b_a 10.0879 0.9853 0.0243 1658.5992 538.1545 1.0 ⋯
b_e 0.4915 0.2837 0.0101 710.4447 508.8646 1.0 ⋯
b_i 1.5671 0.6452 0.0169 1386.6076 466.0071 1.0 ⋯
b_ωy 0.0560 0.7254 0.0375 414.9906 826.9773 0.9 ⋯
b_ωx -0.0421 0.6979 0.0359 411.9189 873.7012 1.0 ⋯
b_Ωy -0.0382 0.6986 0.0366 400.3099 728.4223 1.0 ⋯
b_Ωx 0.0060 0.7284 0.0368 414.2744 740.0617 1.0 ⋯
b_θy -0.0377 0.7230 0.0398 385.3020 817.5571 1.0 ⋯
b_θx 0.0308 0.7021 0.0382 377.8799 756.0302 0.9 ⋯
b_ω -0.1123 1.7563 0.0788 560.0360 663.0874 0.9 ⋯
b_Ω 0.0412 1.8456 0.0825 576.5257 676.4312 1.0 ⋯
b_θ -0.0104 1.8644 0.1042 357.6878 786.3909 1.0 ⋯
b_tp 45115.0905 4212.4893 137.9517 964.6217 820.4943 0.9 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 1.0149 1.1332 1.1998 1.2615 1.3860
plx 49.9605 49.9858 50.0005 50.0129 50.0416
b_a 8.2186 9.3902 10.0639 10.7489 12.0129
b_e 0.0318 0.2420 0.4877 0.7274 0.9724
b_i 0.3651 1.1140 1.5572 2.0372 2.8279
b_ωy -1.0530 -0.6767 0.0887 0.7767 1.0619
b_ωx -1.0718 -0.7316 -0.0617 0.6107 1.0648
b_Ωy -1.0466 -0.7057 -0.0742 0.6609 1.0490
b_Ωx -1.0723 -0.7083 0.0185 0.7256 1.0747
b_θy -1.0582 -0.7526 -0.1055 0.6796 1.0603
b_θx -1.0545 -0.6241 0.0639 0.7006 1.0752
b_ω -2.9887 -1.5715 -0.1183 1.2978 2.9528
b_Ω -2.9127 -1.6390 0.0511 1.6671 2.9742
b_θ -2.9971 -1.7029 0.1093 1.5750 2.9931
b_tp 37733.2034 41242.4624 45477.7792 49461.4940 50400.8239
We now examine the posterior and verify that it matches our KDE prior:
dat = chain[:b_a][:]
@show mean(dat) std(dat)mean(dat) = 10.08791324480641
std(dat) = 0.9853012482722847Observable 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