ANDV linelist analysis

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Prior sensitivity ​

The joint model ships with a default set of priors: μ_inc, σ_inc on the incubation lognormal; μ_δ, σ_δ on the transmission timing normal; 1/√k on the offspring NB dispersion; and σ_rw on the weekly random walk for log R(t). This page sweeps one prior at a time and overlays the resulting marginal posteriors so the influence of each prior on the headline parameters is visible at a glance.

Each scenario is a single joint_model(...) call with one submodel kwarg overridden, fitted with sample_fit under a shared seed. Posterior summaries are read directly from the fitted chains; no numbers are hard-coded.

julia
using TransmissionLinelist
using AlgebraOfGraphics: data, mapping, visual, draw
using DataFrames: DataFrame, nrow
using DataFramesMeta: @chain, @rtransform, @rsubset, @select, @transform
using Dates: Date, Day
using Distributions: Normal, truncated
using Random
using Statistics: quantile
using CairoMakie
using CairoMakie: Band, Lines
using FlexiChains: FlexiChains
using Turing: DynamicPPL
using Logging: Logging

Random.seed!(20260514)
# Silence the NUTS "Found initial step size" Info logs that would
# otherwise clutter the rendered example output.
Logging.disable_logging(Logging.Info)
LogLevel(1)

Setup ​

A single retrospective line list provides the data. R(t) knots use the package default. All scenarios share the same data and seed; the only difference between them is the prior on one submodel.

julia
ll = load_linelist();
t0_ref = minimum(ll.onset_date) - Day(60)
d_retro = build_data(ll; t0 = t0_ref)
edges_ref = prepare_rt_edges(t0_ref)
seed = 20260514
20260514

Scenarios ​

Five scenarios, all fit to the closed-out outbreak:

  • baseline — package defaults; σ_rw ~ N⁺(0, 0.2), μ_inc ~ N(3.0, 0.5), σ_δ ~ N⁺(0, 1.0), 1/√k ~ N⁺(0, 1.0).

  • wider σ_rw — the old default σ_rw ~ N⁺(0, 0.5), allowing week-on-week R(t) jumps that are 2.5× larger a priori.

  • tighter Inc μ — μ_inc ~ N(3.0, 0.1) instead of (3.0, 0.5); the population log-mean incubation is pinned closer to 3.0.

  • k prior shifted — 1/√k ~ N⁺(0.5, 0.5), pulling the prior on k toward smaller values (more overdispersion).

  • δ wider — σ_δ ~ N⁺(0, 3.0), widening the prior on the per-pair transmission-timing SD.

julia
scenarios = [
    (name = "baseline", model = joint_model(d_retro, edges_ref)),
    (name = "wider σ_rw",
        model = joint_model(d_retro, edges_ref;
            rt = random_walk_rt_model(length(edges_ref);
                sigma_prior = truncated(Normal(0.0, 0.5); lower = 0)))),
    (name = "tighter Inc μ",
        model = joint_model(d_retro, edges_ref;
            incubation = incubation_model(;
                μ_prior = Normal(3.0, 0.1)))),
    (name = "k prior shifted",
        model = joint_model(d_retro, edges_ref;
            dispersion = nb_dispersion_model(
                truncated(Normal(0.5, 0.5); lower = 0)))),
    (name = "δ wider",
        model = joint_model(d_retro, edges_ref;
            transmission = transmission_delta_model(;
                σ_prior = truncated(Normal(0.0, 3.0); lower = 0))))
]
5-element Vector{@NamedTuple{name::String, model::DynamicPPL.Model{typeof(joint_model), (:d, :edges), (:incubation, :transmission, :rt, :dispersion, :cases), (), Tuple{@NamedTuple{t0::Dates.Date, onset_lo_day::Vector{Float64}, onset_hi_day::Vector{Float64}, exp_lo_day::Vector{Union{Missing, Float64}}, exp_hi_day::Vector{Union{Missing, Float64}}, source_idx::Vector{Int64}, Zobs::Vector{Int64}, N::Int64, obs_time::Nothing}, Vector{Float64}}, Tuple{DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}, typeof(case_model)}, DynamicPPL.DefaultContext, false}}}:
 (name = "baseline", model = DynamicPPL.Model{typeof(joint_model), (:d, :edges), (:incubation, :transmission, :rt, :dispersion, :cases), (), Tuple{@NamedTuple{t0::Dates.Date, onset_lo_day::Vector{Float64}, onset_hi_day::Vector{Float64}, exp_lo_day::Vector{Union{Missing, Float64}}, exp_hi_day::Vector{Union{Missing, Float64}}, source_idx::Vector{Int64}, Zobs::Vector{Int64}, N::Int64, obs_time::Nothing}, Vector{Float64}}, Tuple{DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}, typeof(case_model)}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.joint_model, (d = (t0 = Dates.Date("2018-09-04"), onset_lo_day = [60.0, 80.0, 77.0, 84.0, 83.0, 82.0, 94.0, 100.0, 99.0, 97.0  …  124.0, 122.0, 126.0, 125.0, 132.0, 139.0, 142.0, 147.0, 155.0, 155.0], onset_hi_day = [61.0, 81.0, 78.0, 85.0, 84.0, 83.0, 95.0, 101.0, 100.0, 98.0  …  125.0, 123.0, 127.0, 126.0, 133.0, 140.0, 143.0, 148.0, 156.0, 156.0], exp_lo_day = Union{Missing, Float64}[missing, 60.0, 60.0, 60.0, 60.0, 60.0, 80.0, 80.0, 80.0, 77.0  …  111.0, 113.0, 99.0, 102.0, 97.0, 99.0, 119.0, 125.0, 119.0, 119.0], exp_hi_day = Union{Missing, Float64}[missing, 61.0, 61.0, 61.0, 61.0, 61.0, 81.0, 81.0, 81.0, 85.0  …  112.0, 114.0, 100.0, 103.0, 98.0, 100.0, 120.0, 126.0, 120.0, 120.0], source_idx = [0, 1, 1, 1, 1, 1, 2, 2, 2, 2  …  13, 14, 9, 8, 10, 9, 22, 28, 22, 22], Zobs = [5, 6, 0, 0, 1, 0, 0, 1, 10, 3  …  0, 0, 0, 1, 0, 0, 0, 0, 0, 0], N = 34, obs_time = nothing), edges = [69.0, 76.0, 83.0, 90.0, 97.0, 104.0, 111.0, 118.0, 125.0, 132.0, 139.0, 146.0, 153.0]), (incubation = DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.incubation_model, NamedTuple(), (dist_constructor = Distributions.LogNormal, μ_prior = Distributions.Normal{Float64}(μ=3.0, σ=0.5), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0)), DynamicPPL.DefaultContext()), transmission = DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.transmission_delta_model, NamedTuple(), (dist_constructor = Distributions.Normal, μ_prior = Distributions.Normal{Float64}(μ=0.0, σ=5.0), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0)), DynamicPPL.DefaultContext()), rt = DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.random_walk_rt_model, (n_knots = 13,), (init_prior = Distributions.Normal{Float64}(μ=0.4054651081081644, σ=1.0), sigma_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.2); lower=0.0)), DynamicPPL.DefaultContext()), dispersion = DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.nb_dispersion_model, (phi_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0),), NamedTuple(), DynamicPPL.DefaultContext()), cases = TransmissionLinelist.case_model), DynamicPPL.DefaultContext()))
 (name = "wider σ_rw", model = DynamicPPL.Model{typeof(joint_model), (:d, :edges), (:incubation, :transmission, :rt, :dispersion, :cases), (), Tuple{@NamedTuple{t0::Dates.Date, onset_lo_day::Vector{Float64}, onset_hi_day::Vector{Float64}, exp_lo_day::Vector{Union{Missing, Float64}}, exp_hi_day::Vector{Union{Missing, Float64}}, source_idx::Vector{Int64}, Zobs::Vector{Int64}, N::Int64, obs_time::Nothing}, Vector{Float64}}, Tuple{DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}, typeof(case_model)}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.joint_model, (d = (t0 = Dates.Date("2018-09-04"), onset_lo_day = [60.0, 80.0, 77.0, 84.0, 83.0, 82.0, 94.0, 100.0, 99.0, 97.0  …  124.0, 122.0, 126.0, 125.0, 132.0, 139.0, 142.0, 147.0, 155.0, 155.0], onset_hi_day = [61.0, 81.0, 78.0, 85.0, 84.0, 83.0, 95.0, 101.0, 100.0, 98.0  …  125.0, 123.0, 127.0, 126.0, 133.0, 140.0, 143.0, 148.0, 156.0, 156.0], exp_lo_day = Union{Missing, Float64}[missing, 60.0, 60.0, 60.0, 60.0, 60.0, 80.0, 80.0, 80.0, 77.0  …  111.0, 113.0, 99.0, 102.0, 97.0, 99.0, 119.0, 125.0, 119.0, 119.0], exp_hi_day = Union{Missing, Float64}[missing, 61.0, 61.0, 61.0, 61.0, 61.0, 81.0, 81.0, 81.0, 85.0  …  112.0, 114.0, 100.0, 103.0, 98.0, 100.0, 120.0, 126.0, 120.0, 120.0], source_idx = [0, 1, 1, 1, 1, 1, 2, 2, 2, 2  …  13, 14, 9, 8, 10, 9, 22, 28, 22, 22], Zobs = [5, 6, 0, 0, 1, 0, 0, 1, 10, 3  …  0, 0, 0, 1, 0, 0, 0, 0, 0, 0], N = 34, obs_time = nothing), edges = [69.0, 76.0, 83.0, 90.0, 97.0, 104.0, 111.0, 118.0, 125.0, 132.0, 139.0, 146.0, 153.0]), (incubation = DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.incubation_model, NamedTuple(), (dist_constructor = Distributions.LogNormal, μ_prior = Distributions.Normal{Float64}(μ=3.0, σ=0.5), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0)), DynamicPPL.DefaultContext()), transmission = DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.transmission_delta_model, NamedTuple(), (dist_constructor = Distributions.Normal, μ_prior = Distributions.Normal{Float64}(μ=0.0, σ=5.0), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0)), DynamicPPL.DefaultContext()), rt = DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.random_walk_rt_model, (n_knots = 13,), (init_prior = Distributions.Normal{Float64}(μ=0.4054651081081644, σ=1.0), sigma_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0)), DynamicPPL.DefaultContext()), dispersion = DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.nb_dispersion_model, (phi_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0),), NamedTuple(), DynamicPPL.DefaultContext()), cases = TransmissionLinelist.case_model), DynamicPPL.DefaultContext()))
 (name = "tighter Inc μ", model = DynamicPPL.Model{typeof(joint_model), (:d, :edges), (:incubation, :transmission, :rt, :dispersion, :cases), (), Tuple{@NamedTuple{t0::Dates.Date, onset_lo_day::Vector{Float64}, onset_hi_day::Vector{Float64}, exp_lo_day::Vector{Union{Missing, Float64}}, exp_hi_day::Vector{Union{Missing, Float64}}, source_idx::Vector{Int64}, Zobs::Vector{Int64}, N::Int64, obs_time::Nothing}, Vector{Float64}}, Tuple{DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}, typeof(case_model)}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.joint_model, (d = (t0 = Dates.Date("2018-09-04"), onset_lo_day = [60.0, 80.0, 77.0, 84.0, 83.0, 82.0, 94.0, 100.0, 99.0, 97.0  …  124.0, 122.0, 126.0, 125.0, 132.0, 139.0, 142.0, 147.0, 155.0, 155.0], onset_hi_day = [61.0, 81.0, 78.0, 85.0, 84.0, 83.0, 95.0, 101.0, 100.0, 98.0  …  125.0, 123.0, 127.0, 126.0, 133.0, 140.0, 143.0, 148.0, 156.0, 156.0], exp_lo_day = Union{Missing, Float64}[missing, 60.0, 60.0, 60.0, 60.0, 60.0, 80.0, 80.0, 80.0, 77.0  …  111.0, 113.0, 99.0, 102.0, 97.0, 99.0, 119.0, 125.0, 119.0, 119.0], exp_hi_day = Union{Missing, Float64}[missing, 61.0, 61.0, 61.0, 61.0, 61.0, 81.0, 81.0, 81.0, 85.0  …  112.0, 114.0, 100.0, 103.0, 98.0, 100.0, 120.0, 126.0, 120.0, 120.0], source_idx = [0, 1, 1, 1, 1, 1, 2, 2, 2, 2  …  13, 14, 9, 8, 10, 9, 22, 28, 22, 22], Zobs = [5, 6, 0, 0, 1, 0, 0, 1, 10, 3  …  0, 0, 0, 1, 0, 0, 0, 0, 0, 0], N = 34, obs_time = nothing), edges = [69.0, 76.0, 83.0, 90.0, 97.0, 104.0, 111.0, 118.0, 125.0, 132.0, 139.0, 146.0, 153.0]), (incubation = DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.incubation_model, NamedTuple(), (dist_constructor = Distributions.LogNormal, μ_prior = Distributions.Normal{Float64}(μ=3.0, σ=0.1), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0)), DynamicPPL.DefaultContext()), transmission = DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.transmission_delta_model, NamedTuple(), (dist_constructor = Distributions.Normal, μ_prior = Distributions.Normal{Float64}(μ=0.0, σ=5.0), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0)), DynamicPPL.DefaultContext()), rt = DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.random_walk_rt_model, (n_knots = 13,), (init_prior = Distributions.Normal{Float64}(μ=0.4054651081081644, σ=1.0), sigma_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.2); lower=0.0)), DynamicPPL.DefaultContext()), dispersion = DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.nb_dispersion_model, (phi_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0),), NamedTuple(), DynamicPPL.DefaultContext()), cases = TransmissionLinelist.case_model), DynamicPPL.DefaultContext()))
 (name = "k prior shifted", model = DynamicPPL.Model{typeof(joint_model), (:d, :edges), (:incubation, :transmission, :rt, :dispersion, :cases), (), Tuple{@NamedTuple{t0::Dates.Date, onset_lo_day::Vector{Float64}, onset_hi_day::Vector{Float64}, exp_lo_day::Vector{Union{Missing, Float64}}, exp_hi_day::Vector{Union{Missing, Float64}}, source_idx::Vector{Int64}, Zobs::Vector{Int64}, N::Int64, obs_time::Nothing}, Vector{Float64}}, Tuple{DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}, typeof(case_model)}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.joint_model, (d = (t0 = Dates.Date("2018-09-04"), onset_lo_day = [60.0, 80.0, 77.0, 84.0, 83.0, 82.0, 94.0, 100.0, 99.0, 97.0  …  124.0, 122.0, 126.0, 125.0, 132.0, 139.0, 142.0, 147.0, 155.0, 155.0], onset_hi_day = [61.0, 81.0, 78.0, 85.0, 84.0, 83.0, 95.0, 101.0, 100.0, 98.0  …  125.0, 123.0, 127.0, 126.0, 133.0, 140.0, 143.0, 148.0, 156.0, 156.0], exp_lo_day = Union{Missing, Float64}[missing, 60.0, 60.0, 60.0, 60.0, 60.0, 80.0, 80.0, 80.0, 77.0  …  111.0, 113.0, 99.0, 102.0, 97.0, 99.0, 119.0, 125.0, 119.0, 119.0], exp_hi_day = Union{Missing, Float64}[missing, 61.0, 61.0, 61.0, 61.0, 61.0, 81.0, 81.0, 81.0, 85.0  …  112.0, 114.0, 100.0, 103.0, 98.0, 100.0, 120.0, 126.0, 120.0, 120.0], source_idx = [0, 1, 1, 1, 1, 1, 2, 2, 2, 2  …  13, 14, 9, 8, 10, 9, 22, 28, 22, 22], Zobs = [5, 6, 0, 0, 1, 0, 0, 1, 10, 3  …  0, 0, 0, 1, 0, 0, 0, 0, 0, 0], N = 34, obs_time = nothing), edges = [69.0, 76.0, 83.0, 90.0, 97.0, 104.0, 111.0, 118.0, 125.0, 132.0, 139.0, 146.0, 153.0]), (incubation = DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.incubation_model, NamedTuple(), (dist_constructor = Distributions.LogNormal, μ_prior = Distributions.Normal{Float64}(μ=3.0, σ=0.5), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0)), DynamicPPL.DefaultContext()), transmission = DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.transmission_delta_model, NamedTuple(), (dist_constructor = Distributions.Normal, μ_prior = Distributions.Normal{Float64}(μ=0.0, σ=5.0), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0)), DynamicPPL.DefaultContext()), rt = DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.random_walk_rt_model, (n_knots = 13,), (init_prior = Distributions.Normal{Float64}(μ=0.4054651081081644, σ=1.0), sigma_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.2); lower=0.0)), DynamicPPL.DefaultContext()), dispersion = DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.nb_dispersion_model, (phi_prior = Truncated(Distributions.Normal{Float64}(μ=0.5, σ=0.5); lower=0.0),), NamedTuple(), DynamicPPL.DefaultContext()), cases = TransmissionLinelist.case_model), DynamicPPL.DefaultContext()))
 (name = "δ wider", model = DynamicPPL.Model{typeof(joint_model), (:d, :edges), (:incubation, :transmission, :rt, :dispersion, :cases), (), Tuple{@NamedTuple{t0::Dates.Date, onset_lo_day::Vector{Float64}, onset_hi_day::Vector{Float64}, exp_lo_day::Vector{Union{Missing, Float64}}, exp_hi_day::Vector{Union{Missing, Float64}}, source_idx::Vector{Int64}, Zobs::Vector{Int64}, N::Int64, obs_time::Nothing}, Vector{Float64}}, Tuple{DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}, DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}, typeof(case_model)}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.joint_model, (d = (t0 = Dates.Date("2018-09-04"), onset_lo_day = [60.0, 80.0, 77.0, 84.0, 83.0, 82.0, 94.0, 100.0, 99.0, 97.0  …  124.0, 122.0, 126.0, 125.0, 132.0, 139.0, 142.0, 147.0, 155.0, 155.0], onset_hi_day = [61.0, 81.0, 78.0, 85.0, 84.0, 83.0, 95.0, 101.0, 100.0, 98.0  …  125.0, 123.0, 127.0, 126.0, 133.0, 140.0, 143.0, 148.0, 156.0, 156.0], exp_lo_day = Union{Missing, Float64}[missing, 60.0, 60.0, 60.0, 60.0, 60.0, 80.0, 80.0, 80.0, 77.0  …  111.0, 113.0, 99.0, 102.0, 97.0, 99.0, 119.0, 125.0, 119.0, 119.0], exp_hi_day = Union{Missing, Float64}[missing, 61.0, 61.0, 61.0, 61.0, 61.0, 81.0, 81.0, 81.0, 85.0  …  112.0, 114.0, 100.0, 103.0, 98.0, 100.0, 120.0, 126.0, 120.0, 120.0], source_idx = [0, 1, 1, 1, 1, 1, 2, 2, 2, 2  …  13, 14, 9, 8, 10, 9, 22, 28, 22, 22], Zobs = [5, 6, 0, 0, 1, 0, 0, 1, 10, 3  …  0, 0, 0, 1, 0, 0, 0, 0, 0, 0], N = 34, obs_time = nothing), edges = [69.0, 76.0, 83.0, 90.0, 97.0, 104.0, 111.0, 118.0, 125.0, 132.0, 139.0, 146.0, 153.0]), (incubation = DynamicPPL.Model{typeof(incubation_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.incubation_model, NamedTuple(), (dist_constructor = Distributions.LogNormal, μ_prior = Distributions.Normal{Float64}(μ=3.0, σ=0.5), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0)), DynamicPPL.DefaultContext()), transmission = DynamicPPL.Model{typeof(transmission_delta_model), (), (:dist_constructor, :μ_prior, :σ_prior), (), Tuple{}, Tuple{UnionAll, Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.transmission_delta_model, NamedTuple(), (dist_constructor = Distributions.Normal, μ_prior = Distributions.Normal{Float64}(μ=0.0, σ=5.0), σ_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=3.0); lower=0.0)), DynamicPPL.DefaultContext()), rt = DynamicPPL.Model{typeof(random_walk_rt_model), (:n_knots,), (:init_prior, :sigma_prior), (), Tuple{Int64}, Tuple{Distributions.Normal{Float64}, Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.random_walk_rt_model, (n_knots = 13,), (init_prior = Distributions.Normal{Float64}(μ=0.4054651081081644, σ=1.0), sigma_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.2); lower=0.0)), DynamicPPL.DefaultContext()), dispersion = DynamicPPL.Model{typeof(nb_dispersion_model), (:phi_prior,), (), (), Tuple{Distributions.Truncated{Distributions.Normal{Float64}, Distributions.Continuous, Float64, Float64, Nothing}}, Tuple{}, DynamicPPL.DefaultContext, false}(TransmissionLinelist.nb_dispersion_model, (phi_prior = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=1.0); lower=0.0),), NamedTuple(), DynamicPPL.DefaultContext()), cases = TransmissionLinelist.case_model), DynamicPPL.DefaultContext()))

Fitting ​

All scenarios share the same NUTS settings (default warmup, 1000 samples × 4 chains) and the same seed, so any difference in the overlaid marginals is attributable to the prior swap.

julia
fits = map(scenarios) do s
    chn = sample_fit(s.model; seed = seed)
    (; s.name, chn, post = summarise(chn))
end;

Sampler health ​

One row per scenario. R̂ near 1 and zero divergences are the targets; if a wider prior pushes the sampler into a bad geometry it shows up here before the marginals are even plotted.

julia
diag_df = DataFrame(
    map(fits) do f
    merge((; scenario = f.name), first(diagnostics_table(f.chn)))
end)
5×5 DataFrame
Rowscenariorhat_maxess_mindivergencesruntime_seconds
StringFloat64Float64Int64Float64
1baseline1.003981791.84019.5445
2wider σ_rw1.0042019.65016.1744
3tighter Inc μ1.004872505.87016.7595
4k prior shifted1.0071897.96019.1098
5δ wider1.004461767.72017.9231

Marginal posteriors ​

Overlaid marginals of (μ_inc, σ_inc, μ_δ, σ_δ, k) across scenarios. One row per scenario, one column per parameter; the baseline density is repeated as a fixed reference inside every row so each row is a direct overlay of one variant against the baseline.

Long-form DataFrame of scalar parameter draws from one summarise posterior NamedTuple; mirrors the helper used in the real-time page.

julia
function post_long(post, params; scenario, fit)
    rows = mapreduce(vcat, params) do p
        vals = getproperty(post, p)
        DataFrame(scenario = scenario, fit = fit,
            param = String(p), value = collect(vals))
    end
    return rows
end

let
    params = [:μ_inc, :σ_inc, :μ_δ, :σ_δ, :k, :σ_rw]
    baseline = fits[1]
    df = @chain fits[2:end] begin
        map(_) do f
            vcat(
                post_long(f.post, params;
                    scenario = f.name, fit = f.name),
                post_long(baseline.post, params;
                    scenario = f.name, fit = "baseline"))
        end
        reduce(vcat, _)
    end
    # Cap `k` at its 99% quantile (pooled across scenarios) so the
    # long right tail under the shifted-k prior doesn't compress the
    # other panels visually.
    k_cap = @chain df begin
        @rsubset :param == "k" && isfinite(:value)
        quantile(_.value, 0.99)
    end
    df_capped = @rsubset(df, :param != "k" || :value <= k_cap)
    plot_marginal_overlay(df_capped;
        row_col = :scenario, col_col = :param,
        size_kw = (1900, 1500))
end

R(t) under each scenario ​

Posterior medians with 80% CrI ribbons over the weekly knots, one colour per scenario. The widely shared knots make it easy to read off where the random-walk prior matters most: at the start of the outbreak (few cases) and the tail (few completed offspring chains).

julia
let
    band_rows = DataFrame[]
    for f in fits
        tbl = rt_band(f.post)
        tbl.scenario .= f.name
        push!(band_rows, tbl)
    end
    df = reduce(vcat, band_rows)
    band_spec = data(df) *
                mapping(:bin => "Bin index",
                    :lo => "R(t) (80% CrI)", :hi;
                    color = :scenario) *
                visual(Band; alpha = 0.15)
    line_spec = data(df) *
                mapping(:bin => "Bin index",
                    :med => "R(t) (80% CrI)";
                    color = :scenario) *
                visual(Lines; linewidth = 2)
    draw(band_spec + line_spec;
        axis = (; limits = (nothing, (0.0, 4.0))),
        figure = (; size = (1100, 600)))
end

Where the priors actually bite ​

Reading across the panels:

  • σ_rw prior. Widening to N⁺(0, 0.5) mostly inflates the R(t) ribbon at the edges of the knot grid. Headline scalars (μ_inc, μ_δ, k) shift very little, since the random walk prior is essentially independent of the delay submodels in the likelihood.

  • Inc μ prior. A tight N(3.0, 0.1) on μ_inc collapses the μ_inc marginal onto its prior mean; σ_inc then absorbs the slack and shifts upward to keep the fitted lognormal close to the data. δ and k are largely unaffected.

  • k prior shift. Pushing the prior on 1/√k toward larger values pulls posterior k toward smaller values (more dispersion) and widens its right tail. Delay marginals are insensitive.

  • δ wider. Allowing a larger σ_δ mainly fattens the σ_δ posterior itself; μ_δ and the incubation parameters barely move, because the per-pair likelihood already pins δ tightly.

The takeaway is that the joint model's headline incubation and δ summaries are robust to the exact shape of their submodel priors within the ranges swept here. The two priors that matter most for downstream R(t) inference are the random-walk SD (which sets the smoothness of the R(t) curve) and the dispersion k prior (which governs how much pipeline mass each observed source can contribute without observed offspring). See the Real-time monitoring page for how these defaults behave under real-time cut-offs.