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 = 31, risk_level = 3)
(issue_age = 68, risk_level = 1)
(issue_age = 39, risk_level = 2)
(issue_age = 37, risk_level = 3)
(issue_age = 53, risk_level = 2)
(issue_age = 30, risk_level = 2)
(issue_age = 33, risk_level = 2)
(issue_age = 70, risk_level = 1)
(issue_age = 39, risk_level = 1)
(issue_age = 44, risk_level = 2)
⋮
(issue_age = 45, risk_level = 1)
(issue_age = 67, risk_level = 2)
(issue_age = 65, risk_level = 2)
(issue_age = 53, risk_level = 1)
(issue_age = 41, risk_level = 3)
(issue_age = 57, risk_level = 2)
(issue_age = 46, risk_level = 1)
(issue_age = 60, risk_level = 1)
(issue_age = 53, 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 | 75 | 0 | 0.0 |
| 2 | 30 | 30 | 2 | 82 | 0 | 0.0 |
| 3 | 30 | 30 | 3 | 76 | 0 | 0.0 |
| 4 | 30 | 31 | 1 | 75 | 0 | 0.0 |
| 5 | 30 | 31 | 2 | 82 | 0 | 0.0 |
| 6 | 30 | 31 | 3 | 76 | 0 | 0.0 |
| 7 | 30 | 32 | 1 | 75 | 0 | 0.0 |
| 8 | 30 | 32 | 2 | 82 | 0 | 0.0 |
| 9 | 30 | 32 | 3 | 76 | 0 | 0.0 |
| 10 | 30 | 33 | 1 | 75 | 0 | 0.0 |
| 11 | 30 | 33 | 2 | 82 | 0 | 0.0 |
| 12 | 30 | 33 | 3 | 76 | 0 | 0.0 |
| 13 | 30 | 34 | 1 | 75 | 0 | 0.0 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| 604 | 70 | 71 | 1 | 103 | 1 | 0.00970874 |
| 605 | 70 | 71 | 2 | 83 | 5 | 0.060241 |
| 606 | 70 | 71 | 3 | 86 | 3 | 0.0348837 |
| 607 | 70 | 72 | 1 | 102 | 3 | 0.0294118 |
| 608 | 70 | 72 | 2 | 78 | 2 | 0.025641 |
| 609 | 70 | 72 | 3 | 83 | 5 | 0.060241 |
| 610 | 70 | 73 | 1 | 99 | 3 | 0.030303 |
| 611 | 70 | 73 | 2 | 76 | 1 | 0.0131579 |
| 612 | 70 | 73 | 3 | 78 | 4 | 0.0512821 |
| 613 | 70 | 74 | 1 | 96 | 2 | 0.0208333 |
| 614 | 70 | 74 | 2 | 75 | 3 | 0.04 |
| 615 | 70 | 74 | 3 | 74 | 5 | 0.0675676 |
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 30 233 0 0.0
2 │ 30 31 233 0 0.0
3 │ 30 32 233 0 0.0
4 │ 30 33 233 0 0.0
5 │ 30 34 233 0 0.0
6 │ 31 31 218 1 0.00458716
7 │ 31 32 217 0 0.0
8 │ 31 33 217 0 0.0
⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮
199 │ 69 72 247 9 0.0364372
200 │ 69 73 238 10 0.0420168
201 │ 70 70 279 7 0.0250896
202 │ 70 71 272 9 0.0330882
203 │ 70 72 263 10 0.0380228
204 │ 70 73 253 8 0.0316206
205 │ 70 74 245 10 0.0408163
190 rows omitted, deaths = [0, 0, 0, 0, 0, 1, 0, 0, 2, 0 … 4, 8, 6, 9, 10, 7, 9, 10, 8, 10]), 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 = 2.08 seconds
Compute duration = 8.21 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 0.0087 0.0004 0.0000 739.2302 984.6658 1.0036 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
q 0.0079 0.0084 0.0087 0.0090 0.0095
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)0.008680449675753694However, 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}:
0.008619788009461815
0.008501715283944257
0.00863829916911474
0.008534309043623008
0.008548886636930566
0.009015823843725004
0.008631078079973296
0.008959770158964118
0.008581889416265311
0.008166368819955657
⋮
0.008745406235479064
0.008363059920630897
0.008036400737086091
0.008804744442405244
0.008939660609940472
0.008725058484678791
0.008599700311582394
0.008158724288367893
0.00870038603314416531.4 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 = 5.9 seconds
Compute duration = 22.22 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 0.0001 0.0000 0.0000 584.7235 672.9357 1.0042 ⋯
b 0.0874 0.0054 0.0002 581.3264 681.4963 1.0045 ⋯
k 1.9436 0.8760 0.0288 656.5508 499.7290 1.0011 ⋯
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 0.0001 0.0001 0.0001
b 0.0774 0.0837 0.0873 0.0910 0.0986
k 1.0311 1.2973 1.6984 2.3343 4.2571
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 0.0884 0.0056 0.0004 236.3095 352.3040 1.0079 ⋯ ā 0.0002 0.0003 0.0000 306.6128 326.3798 1.0002 ⋯ a[1] 0.0000 0.0000 0.0000 248.0581 352.1054 1.0084 ⋯ a[2] 0.0001 0.0000 0.0000 241.2689 365.3431 1.0090 ⋯ a[3] 0.0001 0.0000 0.0000 250.2072 346.8012 1.0068 ⋯ k 2.0106 1.1428 0.0451 440.6973 254.5145 1.0044 ⋯ 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 = 27.11 seconds
Compute duration = 27.11 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 0.1032 0.0087 0.0007 162.3063 194.7900 1.0339 ⋯
ā 0.0001 0.0001 0.0000 211.9405 286.5972 1.0300 ⋯
a[1] 0.0000 0.0000 0.0000 162.8706 172.3253 1.0326 ⋯
a[2] 0.0000 0.0000 0.0000 150.3623 222.1351 1.0341 ⋯
a[3] 0.0000 0.0000 0.0000 156.1219 135.7489 1.0368 ⋯
c 0.0011 0.0004 0.0000 243.3639 499.4269 1.0118 ⋯
k 2.0483 1.0739 0.0432 364.4628 237.0317 1.0007 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
b 0.0875 0.0968 0.1027 0.1090 0.1211
ā 0.0000 0.0000 0.0000 0.0001 0.0004
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 0.0004 0.0008 0.0010 0.0013 0.0019
k 1.0197 1.3139 1.7338 2.4389 4.6872
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] 0.0860 0.2978 0.0090 1082.3866 1010.5515 1.0004 ⋯
deaths[2] 0.1370 0.3747 0.0122 942.2927 941.8609 0.9991 ⋯
deaths[3] 0.1360 0.3487 0.0112 971.8526 1004.0241 0.9998 ⋯
deaths[4] 0.0960 0.3080 0.0097 1021.0017 682.2254 1.0001 ⋯
deaths[5] 0.1250 0.3457 0.0115 907.9073 892.3067 0.9994 ⋯
deaths[6] 0.1310 0.3660 0.0121 920.3683 933.4012 1.0041 ⋯
deaths[7] 0.0990 0.3087 0.0111 761.7172 790.5959 0.9990 ⋯
deaths[8] 0.1380 0.3675 0.0117 983.3500 952.1043 1.0001 ⋯
deaths[9] 0.1410 0.3785 0.0117 1045.1133 1025.2438 0.9990 ⋯
deaths[10] 0.0960 0.3208 0.0110 841.1203 831.2332 0.9996 ⋯
deaths[11] 0.1370 0.3555 0.0118 909.0094 919.7658 1.0034 ⋯
deaths[12] 0.1190 0.3507 0.0112 987.3264 990.3357 0.9999 ⋯
deaths[13] 0.1170 0.3397 0.0109 946.5897 907.6851 1.0007 ⋯
deaths[14] 0.1340 0.3823 0.0117 1079.0924 835.7793 1.0031 ⋯
deaths[15] 0.1520 0.3963 0.0124 1031.8385 948.5814 0.9998 ⋯
deaths[16] 0.0810 0.2802 0.0083 1137.7476 1008.0890 0.9991 ⋯
deaths[17] 0.1310 0.3577 0.0122 837.0730 849.3045 1.0001 ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
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