using MortalityTables
using Turing
using DataFramesMeta
using MCMCChains
using LinearAlgebra
using CairoMakie
using StatsBase31 Bayesian Mortality Modeling
“After a year of intense mental struggle, however, [Arthur Bailey] realized to his consternation that actuarial sledgehammering worked. He even preferred [the Bayesian underpinnings of credibility theory] to the elegance of frequentism. He positively liked formulae that described ‘actual data. . . . I realized that the hard-shelled underwriters were recognizing certain facts of life neglected by the statistical theorists.’ He wanted to give more weight to a large volume of data than to the frequentists’ small sample; doing so felt surprisingly ‘logical and reasonable.’ He concluded that only a ‘suicidal’ actuary would use Fisher’s method of maximum likelihood, which assigned a zero probability to nonevents.” - Sharon Bertsch McGrayne, Excerpt From The Theory That Would Not Die
31.1 Chapter Overview
An example of using a Bayesian MCMC approach to fitting a mortality curve to sample data, with multi-level models and censored data.
31.2 Generating fake data
The problem of interest is to look at mortality rates, which are given in terms of exposures (whether or not a life experienced a death in a given year).
We’ll grab some example rates from an insurance table, which has a “selection” component: When someone enters observation, say at age 50, their mortality is path dependent (so for someone who started being observed at 50 will have a different risk/mortality rate at age 55 than someone who started being observed at 45).
Addtionally, there may be additional groups of interest, such as:
- high/medium/low risk classification
- sex
- group (e.g. company, data source, etc.)
- type of insurance product offered
The example data will start with only the risk classification above.
n = 10_000
inforce = map(1:n) do i
(
issue_age=rand(30:70),
risk_level=rand(1:3),
)
end10000-element Vector{@NamedTuple{issue_age::Int64, risk_level::Int64}}:
(issue_age = 55, risk_level = 2)
(issue_age = 34, risk_level = 2)
(issue_age = 36, risk_level = 1)
(issue_age = 46, risk_level = 3)
(issue_age = 42, risk_level = 2)
(issue_age = 57, risk_level = 2)
(issue_age = 37, risk_level = 2)
(issue_age = 30, risk_level = 1)
(issue_age = 56, risk_level = 1)
(issue_age = 69, risk_level = 3)
⋮
(issue_age = 65, risk_level = 1)
(issue_age = 67, risk_level = 3)
(issue_age = 69, risk_level = 2)
(issue_age = 61, risk_level = 3)
(issue_age = 31, risk_level = 3)
(issue_age = 35, risk_level = 1)
(issue_age = 47, risk_level = 2)
(issue_age = 46, risk_level = 3)
(issue_age = 30, risk_level = 1)base_table = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB")
function tabular_mortality(params, issue_age, att_age, risk_level)
q = params.ultimate[att_age]
if risk_level == 1
q *= 0.7
elseif risk_level == 2
q = q
else
q *= 1.5
end
endtabular_mortality (generic function with 1 method)function model_outcomes(inforce, assumption, assumption_params; n_years=5)
outcomes = map(inforce) do pol
alive = 1
sim = map(1:n_years) do t
att_age = pol.issue_age + t - 1
q = assumption(
assumption_params,
pol.issue_age,
att_age,
pol.risk_level
)
if rand() < q
out = (att_age=att_age, exposures=alive, death=1)
alive = 0
out
else
(att_age=att_age, exposures=alive, death=0)
end
end
filter!(x -> x.exposures == 1, sim)
end
df = DataFrame(inforce)
df.outcomes = outcomes
df = flatten(df, :outcomes)
df.att_age = [x.att_age for x in df.outcomes]
df.death = [x.death for x in df.outcomes]
df.exposures = [x.exposures for x in df.outcomes]
select!(df, Not(:outcomes))
end
exposures = model_outcomes(inforce, tabular_mortality, base_table)
data = combine(groupby(exposures, [:issue_age, :att_age])) do subdf
(exposures=nrow(subdf),
deaths=sum(subdf.death),
fraction=sum(subdf.death) / nrow(subdf))
end
data2 = combine(groupby(exposures, [:issue_age, :att_age, :risk_level])) do subdf
(exposures=nrow(subdf),
deaths=sum(subdf.death),
fraction=sum(subdf.death) / nrow(subdf))
end615×6 DataFrame
590 rows omitted
| Row | issue_age | att_age | risk_level | exposures | deaths | fraction |
|---|---|---|---|---|---|---|
| Int64 | Int64 | Int64 | Int64 | Int64 | Float64 | |
| 1 | 30 | 30 | 1 | 83 | 0 | 0.0 |
| 2 | 30 | 30 | 2 | 71 | 1 | 0.0140845 |
| 3 | 30 | 30 | 3 | 65 | 0 | 0.0 |
| 4 | 30 | 31 | 1 | 83 | 0 | 0.0 |
| 5 | 30 | 31 | 2 | 70 | 0 | 0.0 |
| 6 | 30 | 31 | 3 | 65 | 0 | 0.0 |
| 7 | 30 | 32 | 1 | 83 | 0 | 0.0 |
| 8 | 30 | 32 | 2 | 70 | 0 | 0.0 |
| 9 | 30 | 32 | 3 | 65 | 0 | 0.0 |
| 10 | 30 | 33 | 1 | 83 | 0 | 0.0 |
| 11 | 30 | 33 | 2 | 70 | 1 | 0.0142857 |
| 12 | 30 | 33 | 3 | 65 | 0 | 0.0 |
| 13 | 30 | 34 | 1 | 83 | 1 | 0.0120482 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 604 | 70 | 71 | 1 | 80 | 0 | 0.0 |
| 605 | 70 | 71 | 2 | 74 | 5 | 0.0675676 |
| 606 | 70 | 71 | 3 | 60 | 1 | 0.0166667 |
| 607 | 70 | 72 | 1 | 80 | 3 | 0.0375 |
| 608 | 70 | 72 | 2 | 69 | 2 | 0.0289855 |
| 609 | 70 | 72 | 3 | 59 | 1 | 0.0169492 |
| 610 | 70 | 73 | 1 | 77 | 3 | 0.038961 |
| 611 | 70 | 73 | 2 | 67 | 3 | 0.0447761 |
| 612 | 70 | 73 | 3 | 58 | 6 | 0.103448 |
| 613 | 70 | 74 | 1 | 74 | 2 | 0.027027 |
| 614 | 70 | 74 | 2 | 64 | 3 | 0.046875 |
| 615 | 70 | 74 | 3 | 52 | 2 | 0.0384615 |
31.3 1: A single binomial parameter model
Estimate \(q\), the average mortality rate, not accounting for any variation within the population/sample. Our model is defines as:
\[ q ~ Beta(1,1) p(\text{death}) ~ \text{Binomial}(q) \]
@model function mortality(data, deaths)
q ~ Beta(1, 1)
for i = 1:nrow(data)
deaths[i] ~ Binomial(data.exposures[i], q)
end
end
m1 = mortality(data, data.deaths)DynamicPPL.Model{typeof(mortality), (:data, :deaths), (), (), Tuple{DataFrame, Vector{Int64}}, Tuple{}, DynamicPPL.DefaultContext}(mortality, (data = 205×5 DataFrame
Row │ issue_age att_age exposures deaths fraction
│ Int64 Int64 Int64 Int64 Float64
─────┼───────────────────────────────────────────────────
1 │ 30 30 219 1 0.00456621
2 │ 30 31 218 0 0.0
3 │ 30 32 218 0 0.0
4 │ 30 33 218 1 0.00458716
5 │ 30 34 217 1 0.00460829
6 │ 31 31 268 1 0.00373134
7 │ 31 32 267 0 0.0
8 │ 31 33 267 0 0.0
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮
199 │ 69 72 236 9 0.0381356
200 │ 69 73 227 6 0.0264317
201 │ 70 70 222 8 0.036036
202 │ 70 71 214 6 0.0280374
203 │ 70 72 208 6 0.0288462
204 │ 70 73 202 12 0.0594059
205 │ 70 74 190 7 0.0368421
190 rows omitted, deaths = [1, 0, 0, 1, 1, 1, 0, 0, 1, 1 … 3, 9, 3, 9, 6, 8, 6, 6, 12, 7]), NamedTuple(), DynamicPPL.DefaultContext())
31.3.1 Sampling from the posterior
We use a No-U-Turn-Sampler (NUTS) technique to sample multiple chains at once:
num_chains = 4
chain = sample(m1, NUTS(), MCMCThreads(), 400, num_chains)Chains MCMC chain (400×13×4 Array{Float64, 3}):
Iterations = 201:1:600
Number of chains = 4
Samples per chain = 400
Wall duration = 2.07 seconds
Compute duration = 8.23 seconds
parameters = q
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat e ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
q 0.0086 0.0004 0.0000 703.8364 984.7212 1.0007 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
q 0.0077 0.0083 0.0085 0.0088 0.0094
Here, we have asked for the outcomes to be modeled via a single parameter for the population. We see that the posterior distribution of \(q\) is very close to the overall population mortality rate:
# Posterior mean of q should be close to the pooled fraction
sum(data.deaths) / sum(data.exposures)0.008533636751528944However, We can see that the sampling of possible posterior parameters doesn’t really fit the data very well since our model was so simplified. The lines represent the posterior binomial probability.
This is saying that for the observed data, if there really is just a single probability p that governs the true process that came up with the data, there’s a pretty narrow range of values it could possibly be:
let
data_weight = sqrt.(data.exposures) / 2
f = Figure(title="Parametric Bayesian Mortality"
)
ax = Axis(f[1, 1],
xlabel="age",
ylabel="mortality rate",
limits=(nothing, nothing, -0.01, 0.10),
)
scatter!(ax,
data.att_age,
data.fraction,
markersize=data_weight,
color=(:blue, 0.5),
label="Experience data point (size indicates relative exposure quantity)",)
# show n samples from the posterior plotted on the graph
n = 300
ages = sort!(unique(data.att_age))
q_posterior = sample(chain, n)[:q]
for i in 1:n
hlines!(ax, [q_posterior[i]], color=(:grey, 0.1))
end
sim05 = Float64[]
sim95 = Float64[]
for r in eachrow(data)
outcomes = map(1:n) do i
rand(Binomial(r.exposures, q_posterior[i]), 500)
end
push!(sim05, quantile(Iterators.flatten(outcomes), 0.05) / r.exposures)
push!(sim95, quantile(Iterators.flatten(outcomes), 0.95) / r.exposures)
end
f
end
let
n = 300
q_posterior = sample(chain, n)[:q]
end2-dimensional AxisArray{Float64,2,...} with axes:
:iter, 1:300
:chain, 1:1
And data, a 300×1 Matrix{Float64}:
0.008635668654579745
0.009236679663994952
0.008716594908961631
0.007848755439702043
0.007490891781211297
0.009035257334031129
0.009024333614318777
0.008055042342873787
0.008537078772980598
0.008356924902844105
⋮
0.008517370324275316
0.008608878593954138
0.008199228707438794
0.008474701957349355
0.009068822919529871
0.008583671659268535
0.008039259664180927
0.009452172148041978
0.00898235589133696331.4 2. Parametric model
In this example, we utilize a MakehamBeard parameterization because it’s already very similar in form to a logistic function. This is important because our desired output is a probability (ie the probability of a death at a given age), so the value must be constrained to be in the interval between zero and one.
The prior values for a,b,c, and k are chosen to constrain the hazard (mortality) rate to be between zero and one.
This isn’t an ideal parameterization (e.g. we aren’t including information about the select underwriting period), but is an example of utilizing Bayesian techniques on life experience data. ”
@model function mortality2(data, deaths)
a ~ Exponential(0.1)
b ~ Exponential(0.1)
c = 0.0
k ~ truncated(Exponential(1), 1, Inf)
# use the variables to create a parametric mortality model
m = MortalityTables.MakehamBeard(; a, b, c, k)
# loop through the rows of the dataframe to let Turing observe the data
# and how consistent the parameters are with the data
for i = 1:nrow(data)
age = data.att_age[i]
q = MortalityTables.hazard(m, age)
deaths[i] ~ Binomial(data.exposures[i], q)
end
endmortality2 (generic function with 2 methods)We combine the model with the data and sample from the posterior using a similar call as before:
m2 = mortality2(data, data.deaths)
chain2 = sample(m2, NUTS(), MCMCThreads(), 400, num_chains)Chains MCMC chain (400×15×4 Array{Float64, 3}):
Iterations = 201:1:600
Number of chains = 4
Samples per chain = 400
Wall duration = 5.67 seconds
Compute duration = 20.99 seconds
parameters = a, b, k
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat e ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
a 0.0000 0.0000 0.0000 494.0051 519.4787 1.0045 ⋯
b 0.0912 0.0055 0.0003 485.2947 505.3110 1.0050 ⋯
k 1.8985 0.9014 0.0326 571.9751 466.0100 1.0036 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
a 0.0000 0.0000 0.0000 0.0001 0.0001
b 0.0805 0.0875 0.0912 0.0950 0.1020
k 1.0244 1.2390 1.6170 2.2538 4.2105
31.4.1 Plotting samples from the posterior
We can see that the sampling of possible posterior parameters fits the data well:
let
data_weight = sqrt.(data.exposures) / 2
p = scatter(
data.att_age,
data.fraction,
markersize=data_weight,
alpha=0.5,
label="Experience data point (size indicates relative exposure quantity)",
axis=(
xlabel="age",
limits=(nothing, nothing, -0.01, 0.10),
ylabel="mortality rate",
title="Parametric Bayesian Mortality"
)
)
# show n samples from the posterior plotted on the graph
n = 300
ages = sort!(unique(data.att_age))
for i in 1:n
s = sample(chain2, 1)
a = only(s[:a])
b = only(s[:b])
k = only(s[:k])
c = 0
m = MortalityTables.MakehamBeard(; a, b, c, k)
lines!(ages, age -> MortalityTables.hazard(m, age), alpha=0.1, label="")
end
p
end
let
data_weight = sqrt.(data.exposures) / 2
f = Figure(title="Parametric Bayesian Mortality"
)
ax = Axis(f[1, 1],
xlabel="age",
ylabel="mortality rate",
limits=(nothing, nothing, -0.01, 0.10),
)
scatter!(ax,
data.att_age,
data.fraction,
markersize=data_weight,
color=(:blue, 0.5),
label="Experience data point (size indicates relative exposure quantity)",)
# show n samples from the posterior plotted on the graph
n = 300
ages = sort!(unique(data.att_age))
for i in 1:n
s = sample(chain2, 1)
a = only(s[:a])
b = only(s[:b])
k = only(s[:k])
c = 0
m = MortalityTables.MakehamBeard(; a, b, c, k)
qs = MortalityTables.hazard.(m, ages)
lines!(ax, ages, qs, color=(:grey, 0.1))
end
f
end
Recall that the lines are not plotting the possible outcomes of the claims rates, but the mean claims rate for the given age.
31.5 3. Multi-level model
This model extends the prior to create a multi-level model. Each risk class (risk_level) gets its own \(a\) paramater in the MakhamBeard model. The prior for \(a_i\) is determined by the hyper-parameter \(\bar{a}\).
@model function mortality3(data, deaths)
risk_levels = length(levels(data.risk_level))
b ~ Exponential(0.1)
ā ~ Exponential(0.1)
a ~ filldist(Exponential(ā), risk_levels)
c = 0
k ~ truncated(Exponential(1), 1, Inf)
# use the variables to create a parametric mortality model
# loop through the rows of the dataframe to let Turing observe the data
# and how consistent the parameters are with the data
for i = 1:nrow(data)
risk = data.risk_level[i]
m = MortalityTables.MakehamBeard(; a=a[risk], b, c, k)
age = data.att_age[i]
q = MortalityTables.hazard(m, age)
deaths[i] ~ Binomial(data.exposures[i], q)
end
end
m3 = mortality3(data2, data2.deaths)
chain3 = sample(m3, NUTS(), 1000)
summarize(chain3)parameters mean std mcse ess_bulk ess_tail rhat e ⋯ Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯ b 0.0914 0.0058 0.0004 247.9515 284.7842 1.0040 ⋯ ā 0.0002 0.0005 0.0000 246.3024 227.9634 1.0060 ⋯ a[1] 0.0000 0.0000 0.0000 244.7297 366.9222 1.0036 ⋯ a[2] 0.0000 0.0000 0.0000 259.0605 257.3560 1.0051 ⋯ a[3] 0.0001 0.0000 0.0000 255.9665 273.8641 1.0046 ⋯ k 2.0254 0.9568 0.0330 718.4169 621.7398 0.9995 ⋯ 1 column omitted
let data = data2
data_weight = sqrt.(data.exposures)
color_i = data.risk_level
cm = CairoMakie.Makie.wong_colors()
p, ax, _ = scatter(
data.att_age,
data.fraction,
markersize=data_weight,
alpha=0.5,
color=[(CairoMakie.Makie.wong_colors()[c], 0.7) for c in color_i],
colormap=CairoMakie.Makie.wong_colors(),
label="Experience data point (size indicates relative exposure quantity)",
axis=(
xlabel="age",
limits=(nothing, nothing, -0.01, 0.10),
ylabel="mortality rate",
title="Parametric Bayesian Mortality"
)
)
# show n samples from the posterior plotted on the graph
n = 100
ages = sort!(unique(data.att_age))
for r in 1:3
for i in 1:n
s = sample(chain3, 1)
a = only(s[Symbol("a[$r]")])
b = only(s[:b])
k = only(s[:k])
c = 0
m = MortalityTables.MakehamBeard(; a, b, c, k)
lines!(ages, age -> MortalityTables.hazard(m, age), label="risk level $r", alpha=0.2, color=(CairoMakie.Makie.wong_colors()[r], 0.2))
end
end
axislegend(ax, merge=true)
p
end
Again, the lines are not plotting the possible outcomes of the claims rates, but the mean claims rate for the given age and risk class.
31.6 Handling non-unit exposures
The key is to use the Poisson distribution, which is a continuous approximation to the Binomial distribution:
@model function mortality4(data, deaths)
risk_levels = length(levels(data.risk_level))
b ~ Exponential(0.1)
ā ~ Exponential(0.1)
a ~ filldist(Exponential(ā), risk_levels)
c ~ Beta(4, 18)
k ~ truncated(Exponential(1), 1, Inf)
# use the variables to create a parametric mortality model
# loop through the rows of the dataframe to let Turing observe the data
# and how consistent the parameters are with the data
for i = 1:nrow(data)
risk = data.risk_level[i]
m = MortalityTables.MakehamBeard(; a=a[risk], b, c, k)
age = data.att_age[i]
q = MortalityTables.hazard(m, age)
deaths[i] ~ Poisson(data.exposures[i] * q)
end
end
m4 = mortality4(data2, data2.deaths)
chain4 = sample(m4, NUTS(), 1000)Chains MCMC chain (1000×19×1 Array{Float64, 3}):
Iterations = 501:1:1500
Number of chains = 1
Samples per chain = 1000
Wall duration = 26.16 seconds
Compute duration = 26.16 seconds
parameters = b, ā, a[1], a[2], a[3], c, k
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat e ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
b 0.1108 0.0097 0.0008 148.2068 291.8095 1.0127 ⋯
ā 0.0000 0.0001 0.0000 236.9165 337.4538 1.0070 ⋯
a[1] 0.0000 0.0000 0.0000 155.5239 298.8641 1.0131 ⋯
a[2] 0.0000 0.0000 0.0000 151.6236 330.0986 1.0126 ⋯
a[3] 0.0000 0.0000 0.0000 148.6310 279.3477 1.0138 ⋯
c 0.0011 0.0004 0.0000 285.6626 376.7046 1.0054 ⋯
k 2.1275 1.1868 0.0491 338.1275 247.8097 1.0043 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
b 0.0932 0.1040 0.1098 0.1170 0.1316
ā 0.0000 0.0000 0.0000 0.0000 0.0002
a[1] 0.0000 0.0000 0.0000 0.0000 0.0000
a[2] 0.0000 0.0000 0.0000 0.0000 0.0000
a[3] 0.0000 0.0000 0.0000 0.0000 0.0001
c 0.0005 0.0008 0.0011 0.0013 0.0019
k 1.0188 1.3479 1.7929 2.4963 5.3336
risk_factors4 = [mean(chain4[Symbol("a[$f]")]) for f in 1:3]
risk_factors4 ./ risk_factors4[2]
let data = data2
data_weight = sqrt.(data.exposures) / 2
color_i = data.risk_level
p, ax, _ = scatter(
data.att_age,
data.fraction,
markersize=data_weight,
alpha=0.5,
color=color_i,
label="Experience data point (size indicates relative exposure quantity)",
axis=(xlabel="age",
limits=(nothing, nothing, -0.01, 0.10),
ylabel="mortality rate",
title="Parametric Bayesian Mortality"
)
)
# show n samples from the posterior plotted on the graph
n = 100
ages = sort!(unique(data.att_age))
for r in 1:3
for i in 1:n
s = sample(chain4, 1)
a = only(s[Symbol("a[$r]")])
b = only(s[:b])
k = only(s[:k])
c = 0
m = MortalityTables.MakehamBeard(; a, b, c, k)
lines!(ages, age -> MortalityTables.hazard(m, age), label="risk level $r", alpha=0.2, color=(CairoMakie.Makie.wong_colors()[r], 0.2))
end
end
axislegend(ax, merge=true)
p
end
31.7 Model Predictions
We can generate predictive estimates by passing a vector of missing in place of the outcome variables and then calling predict.
We get a table of values where each row is the the prediction implied by the corresponding chain sample, and the columns are the predicted value for each of the outcomes in our original dataset.
preds = predict(mortality4(data2, fill(missing, length(data2.deaths))), chain4)Chains MCMC chain (1000×615×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
parameters = deaths[1], deaths[2], deaths[3], deaths[4], deaths[5], deaths[6], deaths[7], deaths[8], deaths[9], deaths[10], deaths[11], deaths[12], deaths[13], deaths[14], deaths[15], deaths[16], deaths[17], deaths[18], deaths[19], deaths[20], deaths[21], deaths[22], deaths[23], deaths[24], deaths[25], deaths[26], deaths[27], deaths[28], deaths[29], deaths[30], deaths[31], deaths[32], deaths[33], deaths[34], deaths[35], deaths[36], deaths[37], deaths[38], deaths[39], deaths[40], deaths[41], deaths[42], deaths[43], deaths[44], deaths[45], deaths[46], deaths[47], deaths[48], deaths[49], deaths[50], deaths[51], deaths[52], deaths[53], deaths[54], deaths[55], deaths[56], deaths[57], deaths[58], deaths[59], deaths[60], deaths[61], deaths[62], deaths[63], deaths[64], deaths[65], deaths[66], deaths[67], deaths[68], deaths[69], deaths[70], deaths[71], deaths[72], deaths[73], deaths[74], deaths[75], deaths[76], deaths[77], deaths[78], deaths[79], deaths[80], deaths[81], deaths[82], deaths[83], deaths[84], deaths[85], deaths[86], deaths[87], deaths[88], deaths[89], deaths[90], deaths[91], deaths[92], deaths[93], deaths[94], deaths[95], deaths[96], deaths[97], deaths[98], deaths[99], deaths[100], deaths[101], deaths[102], deaths[103], deaths[104], deaths[105], deaths[106], deaths[107], deaths[108], deaths[109], deaths[110], deaths[111], deaths[112], deaths[113], deaths[114], deaths[115], deaths[116], deaths[117], deaths[118], deaths[119], deaths[120], deaths[121], deaths[122], deaths[123], deaths[124], deaths[125], deaths[126], deaths[127], deaths[128], deaths[129], deaths[130], deaths[131], deaths[132], deaths[133], deaths[134], deaths[135], deaths[136], deaths[137], deaths[138], deaths[139], deaths[140], deaths[141], deaths[142], deaths[143], deaths[144], deaths[145], deaths[146], deaths[147], deaths[148], deaths[149], deaths[150], deaths[151], deaths[152], deaths[153], deaths[154], deaths[155], deaths[156], deaths[157], deaths[158], deaths[159], deaths[160], deaths[161], deaths[162], deaths[163], deaths[164], deaths[165], deaths[166], deaths[167], deaths[168], deaths[169], deaths[170], deaths[171], deaths[172], deaths[173], deaths[174], deaths[175], deaths[176], deaths[177], deaths[178], deaths[179], deaths[180], deaths[181], deaths[182], deaths[183], deaths[184], deaths[185], deaths[186], deaths[187], deaths[188], deaths[189], deaths[190], deaths[191], deaths[192], deaths[193], deaths[194], deaths[195], deaths[196], deaths[197], deaths[198], deaths[199], deaths[200], deaths[201], deaths[202], deaths[203], deaths[204], deaths[205], deaths[206], deaths[207], deaths[208], deaths[209], deaths[210], deaths[211], deaths[212], deaths[213], deaths[214], deaths[215], deaths[216], deaths[217], deaths[218], deaths[219], deaths[220], deaths[221], deaths[222], deaths[223], deaths[224], deaths[225], deaths[226], deaths[227], deaths[228], deaths[229], deaths[230], deaths[231], deaths[232], deaths[233], deaths[234], deaths[235], deaths[236], deaths[237], deaths[238], deaths[239], deaths[240], deaths[241], deaths[242], deaths[243], deaths[244], deaths[245], deaths[246], deaths[247], deaths[248], deaths[249], deaths[250], deaths[251], deaths[252], deaths[253], deaths[254], deaths[255], deaths[256], deaths[257], deaths[258], deaths[259], deaths[260], deaths[261], deaths[262], deaths[263], deaths[264], deaths[265], deaths[266], deaths[267], deaths[268], deaths[269], deaths[270], deaths[271], deaths[272], deaths[273], deaths[274], deaths[275], deaths[276], deaths[277], deaths[278], deaths[279], deaths[280], deaths[281], deaths[282], deaths[283], deaths[284], deaths[285], deaths[286], deaths[287], deaths[288], deaths[289], deaths[290], deaths[291], deaths[292], deaths[293], deaths[294], deaths[295], deaths[296], deaths[297], deaths[298], deaths[299], deaths[300], deaths[301], deaths[302], deaths[303], deaths[304], deaths[305], deaths[306], deaths[307], deaths[308], deaths[309], deaths[310], deaths[311], deaths[312], deaths[313], deaths[314], deaths[315], deaths[316], deaths[317], deaths[318], deaths[319], deaths[320], deaths[321], deaths[322], deaths[323], deaths[324], deaths[325], deaths[326], deaths[327], deaths[328], deaths[329], deaths[330], deaths[331], deaths[332], deaths[333], deaths[334], deaths[335], deaths[336], deaths[337], deaths[338], deaths[339], deaths[340], deaths[341], deaths[342], deaths[343], deaths[344], deaths[345], deaths[346], deaths[347], deaths[348], deaths[349], deaths[350], deaths[351], deaths[352], deaths[353], deaths[354], deaths[355], deaths[356], deaths[357], deaths[358], deaths[359], deaths[360], deaths[361], deaths[362], deaths[363], deaths[364], deaths[365], deaths[366], deaths[367], deaths[368], deaths[369], deaths[370], deaths[371], deaths[372], deaths[373], deaths[374], deaths[375], deaths[376], deaths[377], deaths[378], deaths[379], deaths[380], deaths[381], deaths[382], deaths[383], deaths[384], deaths[385], deaths[386], deaths[387], deaths[388], deaths[389], deaths[390], deaths[391], deaths[392], deaths[393], deaths[394], deaths[395], deaths[396], deaths[397], deaths[398], deaths[399], deaths[400], deaths[401], deaths[402], deaths[403], deaths[404], deaths[405], deaths[406], deaths[407], deaths[408], deaths[409], deaths[410], deaths[411], deaths[412], deaths[413], deaths[414], deaths[415], deaths[416], deaths[417], deaths[418], deaths[419], deaths[420], deaths[421], deaths[422], deaths[423], deaths[424], deaths[425], deaths[426], deaths[427], deaths[428], deaths[429], deaths[430], deaths[431], deaths[432], deaths[433], deaths[434], deaths[435], deaths[436], deaths[437], deaths[438], deaths[439], deaths[440], deaths[441], deaths[442], deaths[443], deaths[444], deaths[445], deaths[446], deaths[447], deaths[448], deaths[449], deaths[450], deaths[451], deaths[452], deaths[453], deaths[454], deaths[455], deaths[456], deaths[457], deaths[458], deaths[459], deaths[460], deaths[461], deaths[462], deaths[463], deaths[464], deaths[465], deaths[466], deaths[467], deaths[468], deaths[469], deaths[470], deaths[471], deaths[472], deaths[473], deaths[474], deaths[475], deaths[476], deaths[477], deaths[478], deaths[479], deaths[480], deaths[481], deaths[482], deaths[483], deaths[484], deaths[485], deaths[486], deaths[487], deaths[488], deaths[489], deaths[490], deaths[491], deaths[492], deaths[493], deaths[494], deaths[495], deaths[496], deaths[497], deaths[498], deaths[499], deaths[500], deaths[501], deaths[502], deaths[503], deaths[504], deaths[505], deaths[506], deaths[507], deaths[508], deaths[509], deaths[510], deaths[511], deaths[512], deaths[513], deaths[514], deaths[515], deaths[516], deaths[517], deaths[518], deaths[519], deaths[520], deaths[521], deaths[522], deaths[523], deaths[524], deaths[525], deaths[526], deaths[527], deaths[528], deaths[529], deaths[530], deaths[531], deaths[532], deaths[533], deaths[534], deaths[535], deaths[536], deaths[537], deaths[538], deaths[539], deaths[540], deaths[541], deaths[542], deaths[543], deaths[544], deaths[545], deaths[546], deaths[547], deaths[548], deaths[549], deaths[550], deaths[551], deaths[552], deaths[553], deaths[554], deaths[555], deaths[556], deaths[557], deaths[558], deaths[559], deaths[560], deaths[561], deaths[562], deaths[563], deaths[564], deaths[565], deaths[566], deaths[567], deaths[568], deaths[569], deaths[570], deaths[571], deaths[572], deaths[573], deaths[574], deaths[575], deaths[576], deaths[577], deaths[578], deaths[579], deaths[580], deaths[581], deaths[582], deaths[583], deaths[584], deaths[585], deaths[586], deaths[587], deaths[588], deaths[589], deaths[590], deaths[591], deaths[592], deaths[593], deaths[594], deaths[595], deaths[596], deaths[597], deaths[598], deaths[599], deaths[600], deaths[601], deaths[602], deaths[603], deaths[604], deaths[605], deaths[606], deaths[607], deaths[608], deaths[609], deaths[610], deaths[611], deaths[612], deaths[613], deaths[614], deaths[615]
internals =
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
deaths[1] 0.1090 0.3244 0.0107 913.3787 909.9676 0.9992 ⋯
deaths[2] 0.1110 0.3268 0.0104 990.4136 987.3694 0.9998 ⋯
deaths[3] 0.1100 0.3286 0.0104 1001.5529 1010.5515 0.9990 ⋯
deaths[4] 0.1210 0.3442 0.0116 881.3675 758.0117 1.0029 ⋯
deaths[5] 0.0950 0.3132 0.0097 1029.7060 1014.3181 0.9992 ⋯
deaths[6] 0.0950 0.3132 0.0104 904.7898 913.0043 0.9994 ⋯
deaths[7] 0.1110 0.3329 0.0109 939.9057 960.0101 0.9997 ⋯
deaths[8] 0.1220 0.3482 0.0120 837.3789 822.8354 0.9994 ⋯
deaths[9] 0.1110 0.3175 0.0103 948.2237 941.1233 0.9990 ⋯
deaths[10] 0.1240 0.3532 0.0111 1009.1972 1011.2902 0.9990 ⋯
deaths[11] 0.0960 0.3145 0.0102 947.1295 953.3921 1.0023 ⋯
deaths[12] 0.1040 0.3183 0.0099 996.1686 917.0357 1.0003 ⋯
deaths[13] 0.1130 0.3498 0.0112 958.1668 943.4942 1.0008 ⋯
deaths[14] 0.1170 0.3368 0.0112 922.3999 985.5505 1.0004 ⋯
deaths[15] 0.1100 0.3194 0.0102 980.4853 971.7818 0.9994 ⋯
deaths[16] 0.1040 0.3183 0.0092 1190.2268 1010.1399 0.9993 ⋯
deaths[17] 0.1570 0.3981 0.0134 877.2075 869.1933 1.0018 ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
1 column and 598 rows omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
deaths[1] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[2] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[3] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[4] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[5] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[6] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[7] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[8] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[9] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[10] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[11] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[12] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[13] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[14] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[15] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[16] 0.0000 0.0000 0.0000 0.0000 1.0000
deaths[17] 0.0000 0.0000 0.0000 0.0000 1.0000
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
598 rows omitted