Turing.jl: fix some parameters to prior distribution

[MFairley]

@trappmartin

Every ~ indicates that we perform inference for that parameter and that we account for it in the joint probability. But you can simply draw beta using the rand function and add its log probability to the joint.

Does that help?
May I also ask why you don’t want to update the prior?

Great, so something like this:

@model bayes_sir(y) = begin
    # Calculate number of timepoints
    l = length(y)
    i₀  ~ Uniform(0.0,1.0)
    β = rand(Uniform(0.0,1.0))
    I = i₀*1000.0
    u0=[1000.0-I,I,0.0,0.0]
    p=[β,10.0,0.25]
    tspan = (0.0,float(l))
    prob = ODEProblem(sir_ode!,
            u0,
            tspan,
            p)
    sol = solve(prob,
                Tsit5(),
                saveat = 1.0)
    sol_C = Array(sol)[4,:] # Cumulative cases
    sol_X = sol_C[2:end] - sol_C[1:(end-1)]
    l = length(y)
    for i in 1:l
      y[i] ~ Poisson(sol_X[i])
    end
end;

and then use the trick from Is there a turing alternative to pm.Deterministic from pymc3? - #3 by torfjelde to keep track of the samples from beta?

There are a couple of applications I had in mind:

1. For computational efficiency, when I know that a parameter has no influence on the likelihood.

2. When the prior was estimated using some external source that I think is more reliable than my own data. For example, say I observe deaths in patients from treatments A and B. A clinical trial reports a hazard rate ratio (hr) for the effectiveness of A vs. B. I assume the likelihood is: deaths_a ~ Poisson(lambda * hr) and deaths_b ~ Poisson(lambda). I want to run inference on lambda but I want to leave hr unchanged from its prior.