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=0.1)
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 = 0.91 seconds
Compute duration = 0.91 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 1.1992 0.1006 0.0035 874.4261 566.3778 1.0 ⋯
plx 49.9993 0.0204 0.0007 936.7313 676.4312 1.0 ⋯
b_a 10.0800 1.0919 0.0457 619.8236 316.1480 1.0 ⋯
b_e 0.4890 0.2782 0.0085 990.3864 614.9544 1.0 ⋯
b_i 1.5835 0.6686 0.0200 1078.2827 769.3002 0.9 ⋯
b_ωy -0.0379 0.7052 0.0327 509.2819 741.5125 0.9 ⋯
b_ωx 0.0779 0.7201 0.0379 404.1475 500.7570 0.9 ⋯
b_Ωy 0.0147 0.7247 0.0377 375.6385 789.4449 0.9 ⋯
b_Ωx 0.0625 0.7056 0.0366 377.7605 745.3954 0.9 ⋯
b_θy 0.0367 0.7175 0.0419 285.0305 792.1349 0.9 ⋯
b_θx 0.0255 0.7057 0.0372 383.1695 596.9429 1.0 ⋯
b_ω 0.1797 1.8441 0.0854 574.2227 868.2807 0.9 ⋯
b_Ω 0.1823 1.7964 0.0825 542.9553 661.3184 0.9 ⋯
b_θ 0.0971 1.7674 0.0821 533.2708 715.8530 1.0 ⋯
b_tp 45024.3925 4263.4986 150.8081 818.4966 743.4487 1.0 ⋯
2 columns omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
M 0.9836 1.1371 1.1979 1.2630 1.3923
plx 49.9598 49.9857 49.9992 50.0131 50.0383
b_a 8.0404 9.3628 10.0605 10.7812 12.2996
b_e 0.0282 0.2583 0.4981 0.7306 0.9471
b_i 0.3913 1.0676 1.5752 2.0862 2.8344
b_ωy -1.0874 -0.7156 -0.0904 0.6389 1.0593
b_ωx -1.0773 -0.6261 0.1534 0.7691 1.0667
b_Ωy -1.0676 -0.7054 0.0326 0.7462 1.0808
b_Ωx -1.0779 -0.6232 0.1252 0.7498 1.0662
b_θy -1.0408 -0.6777 0.0947 0.7381 1.0794
b_θx -1.0665 -0.6709 0.0348 0.6894 1.0636
b_ω -3.0046 -1.4495 0.3407 1.7844 2.9989
b_Ω -2.9710 -1.2844 0.2498 1.7679 2.9992
b_θ -2.9518 -1.3261 0.0486 1.6200 2.9055
b_tp 37258.9828 41277.7187 45210.5155 49420.5183 50399.6036
We now examine the posterior and verify that it matches our KDE prior:
dat = chain[:b_a][:]
@show mean(dat) std(dat)mean(dat) = 10.080030370224607
std(dat) = 1.0919213483850636Observable 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