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 = 0.97 seconds
Compute duration = 0.97 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 1.1993 0.1005 0.0029 1191.8838 712.4322 1.0 ⋯
plx 50.0006 0.0202 0.0006 1071.7069 436.7351 1.0 ⋯
b_a 10.0293 1.0246 0.0325 996.0405 618.5840 1.0 ⋯
b_e 0.5064 0.2824 0.0081 1129.3961 414.1687 0.9 ⋯
b_i 1.5309 0.6452 0.0192 1083.3080 738.0395 1.0 ⋯
b_ωy -0.0590 0.7174 0.0381 387.3104 906.1256 1.0 ⋯
b_ωx -0.0170 0.7152 0.0421 294.3155 443.6906 1.0 ⋯
b_Ωy 0.0353 0.7127 0.0460 288.6507 588.2839 1.0 ⋯
b_Ωx 0.0122 0.7053 0.0354 459.4530 768.4756 1.0 ⋯
b_θy 0.0529 0.7186 0.0397 349.3876 806.2388 0.9 ⋯
b_θx -0.0208 0.7024 0.0330 488.0166 826.4583 1.0 ⋯
b_ω -0.0817 1.8805 0.0958 480.5521 790.9754 1.0 ⋯
b_Ω 0.0333 1.7800 0.0800 567.1196 708.1083 1.0 ⋯
b_θ -0.0239 1.7691 0.0726 666.9666 708.3337 1.0 ⋯
b_tp 45236.5702 4253.6124 144.0297 885.8951 703.6541 1.0 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 0.9989 1.1284 1.1991 1.2727 1.3906
plx 49.9599 49.9880 50.0006 50.0141 50.0396
b_a 8.0764 9.3236 10.0427 10.7698 12.1475
b_e 0.0358 0.2637 0.5088 0.7556 0.9625
b_i 0.3379 1.0453 1.5330 1.9928 2.7529
b_ωy -1.1146 -0.7350 -0.1227 0.6294 1.0654
b_ωx -1.0963 -0.7109 -0.0293 0.6881 1.0537
b_Ωy -1.0579 -0.6814 0.0802 0.7363 1.0611
b_Ωx -1.0759 -0.6896 0.0696 0.7026 1.0353
b_θy -1.0644 -0.6478 0.0961 0.7510 1.0739
b_θx -1.0581 -0.7197 -0.0445 0.6510 1.0579
b_ω -2.9942 -1.8206 -0.0866 1.5631 2.9986
b_Ω -2.9814 -1.4681 0.0984 1.5135 3.0118
b_θ -2.9984 -1.5140 -0.0856 1.4162 2.9978
b_tp 37982.0214 41310.1726 45825.1263 49525.0370 50400.1569
We now examine the posterior and verify that it matches our KDE prior:
dat = chain[:b_a][:]
@show mean(dat) std(dat)mean(dat) = 10.029276290199228
std(dat) = 1.0245746640740667Observable 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