# Bayesian Mortality Modeling

    library(StanMoMo)

## Introduction: Bayesian Lee-Carter with Stan

First, we explain how to estimate the Lee-Carter model using the `lc_stan` function in the `StanMoMo` package.

For illustration, the package already includes the object `FRMaleData` containing deaths (`FRMaleData$Dxt`) and central exposures (`FRMaleData$Ext`) of French males for the period 1816-2017 and for ages 0-110 extracted from the Human Mortality Database. In this example, we focus on ages 60-90 and the period 1980-2010. This can be obtained via:

    ages.fit<-60:90
    years.fit<-1980:2010
    deathFR<-FRMaleData$Dxt[formatC(ages.fit),formatC(years.fit)]
    exposureFR<-FRMaleData$Ext[formatC(ages.fit),formatC(years.fit)]

As a reminder, the Lee-Carter model assumes that the log death rates are given by \[ \log \mu_{xt}=\alpha_x+\beta_x \kappa_t \] To ensure identifiability of the model, we assume that \[ \sum_x \beta_x=1,\kappa_1=0 \]

Moreover, we assume that the period parameter follows a random walk with drift: \[ \kappa_t \sim \mathcal{N}(c+ \kappa_{t-1},\sigma) \] The choice of priors for each parameter can be found in the documentation of each function.

## Bayesian Estimation

All the parameters can be estimated either under a Poisson model with argument `family = "poisson"` or under a Negative-Binomial model which includes an additional overdispersion parameter \(\phi\) with argument `family="nb"`: \[ D_{xt} \sim Poisson (e_{xt}\mu_{xt}) \text{ or } D_{xt} \sim NB (e_{xt}\mu_{xt},\phi) \] Given the matrix of deaths `deathFR` and the matrix of central exposures `exposureFR`, we can infer the posterior distribution of all the parameters and obtain death rates forecasts for the next 10 years under a Poisson model by a simple call to the `lc_stan` function:

    fitLC=lc_stan(death = deathFR,exposure=exposureFR, forecast = 10, family = "poisson",chains=1,iter=1000,cores=1)

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Internally, the sampling is performed through the `rstan::sampling` function. By default, Stan samples four Markov chains of 2000 iterations. For each chain, there are 1000 warmup iterations (hence, 4000 post warm-up draws in total). Moreover, by default, Stan uses 1 core but we recommend using as many processors as the hardware and RAM allow (up to the number of chains). All this can be set using the following additional arguments : for instance, if one wants to sample 2 chains of 3000 iterations with 1000 warm-up samples, one can add `chains=2,iter=3000,warmup=1000`.

For additional arguments, the reader may refer to the `rstan::sampling` documentation.

The output is an object of class `stanfit` (cf. the rstan package) which contains, among others, posterior draws, posterior summary statistics and convergence diagnostics.

The easiest way to extract the posterior draws is to call the `rstan::extract` function which returns a list with named components corresponding to the model parameters.

    #Extract model parameters from the fitted model
    params<-rstan::extract(fitLC)
    print(names(params))

    ##  [1] "a"       "b"       "c"       "ks"      "sigma"   "k"       "phi"
    ##  [8] "k_p"     "mufor"   "log_lik" "pos"     "pos2"    "pos3"    "lp__"

Among these parameters, we find

- `a` : \(\alpha_x\).
- `b` : \(\beta_x\).
- `k` : \(\kappa_t\).
- `k_p` : forecasts of \(\kappa_t\).
- `mufor` : forecasts of death rates \(\mu_{xt}\).
- `log_lik` : log-likelihoods.
- `phi` : overdispersion parameter for the NB model.
- `c` and `sigma` : drift and standard deviation of the random walk with drift.

The user can have access to an interface for interactive MCMC diagnostics, plots and tables helpful for analyzing posterior samples through the `shinystan` package (fore more details, you can check the shinystan web page)

    library(shinystan)
    launch_shinystan(fitLC)

Note: you need to close the shiny app before continuing in R.

Moreover, the package also includes functions to represent boxplots of the posterior distribution based on the `ggplot2` package.

For instance, boxplots for the parameters from the Lee-Carter model can be obtained via:

    boxplot_post_dist(fitLC, "a", ages.fit, years.fit)

    boxplot_post_dist(fitLC, "b", ages.fit, years.fit)

    boxplot_post_dist(fitLC, "k", ages.fit, years.fit)

## Forecasting

All forecast death rates \(\mu_{xt}\) are stored in the output `mufor`. We point out that the output `mufor` is a matrix of dimension \(N\) \(\times\) \((A . M)\) where the number of rows \(N\) is the posterior sample size and the number of columns \(A.M\) is the product of \(A\) the age dimension and \(M\) the number of forecast years.

## Other models

In a similar fashion, the models RH, APC, CBD and M6 can be estimated with the following functions:

Model | Predictor
--- | ---
LC | \(\log \mu_{xt} = \alpha_x + \beta_x\kappa_t\)
RH | \(\log \mu_{xt} = \alpha_x + \beta_x\kappa_t+\gamma_{t-x}\)
APC | \(\log \mu_{xt} = \alpha_x + \kappa_t +\gamma_{t-x}\)
CBD | \(\log \mu_{xt} = \kappa_t^{(1)} + (x-\bar{x})\kappa_t^{(2)}\)
M6 | \(\log \mu_{xt} = \kappa_t^{(1)} + (x-\bar{x})\kappa_t^{(2)}+\gamma_{t-x}\)

    fitRH=rh_stan(death = deathFR,exposure=exposureFR, forecast = 10, family = "poisson",cores=4)
    fitAPC=apc_stan(death = deathFR,exposure=exposureFR, forecast = 10, family = "poisson",cores=4)
    fitCBD=cbd_stan(death = deathFR,exposure=exposureFR, age=ages.fit, forecast=10,family = "poisson",cores=4)
    fitM6=m6_stan(death = deathFR,exposure=exposureFR, age=ages.fit,forecast = 10, family = "poisson",cores=4)

For information, the estimated parameters \(\kappa_t^{(1)}\), \(\kappa_t^{(2)}\) and \(\gamma_{t-x}\) are respectively stored in the variables `k`,`k2` and `g`. The forecast parameters are respectively stored in the variables `k_p`,`k2_p` and `g_p`.

You can check the next vignette to know more about model averaging techniques based on leave-future-out validation.