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 50-90 and the period 1970-2017. This can be obtained via:<-50:90<-1970:2017

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 estimate 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",cores=4)

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). The argument cores=4 ensures that the four chains are sampled in parallel for time efficiency. 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
##  [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)

As an illustration, we produce fan charts depicting the parameter uncertainty from the Lee-Carter model with the fanplot package.

plot(, colMeans(params$a), ylim=range(params$a),ylab=TeX("$\\alpha_x$"), xlab="Age: x")
fan(data=params$a,[1],type = "interval", ln=NULL,probs = c(0.5,0.8,0.95,0.995),fan.col = colorRampPalette(c("Orange", "white")))

plot(, colMeans(params$b), ylim=range(params$b),ylab=TeX("$\\beta_x$"), xlab="Age: x")
fan(data=params$b,[1],type = "interval", ln = NULL,probs = c(0.5,0.8,0.95,0.995),  fan.col = colorRampPalette(c("red", "white")))

plot(, colMeans(params$k), ylim=range(params$k),ylab=TeX("$\\kappa_t$"), xlab="Year: t")
fan(data=params$k,[1],type = "interval",ln = NULL, probs = c(0.5,0.8,0.95,0.995),fan.col = colorRampPalette(c("blue", "white")))


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.

Predictions of death rates for the next 10 years with prediction intervals can be obtained as follows:

# Resize the forecast deaths rates as an array "Number of draws X Ages X Years to predict"
pred<-array(params$mufor,dim=list(samplesize,length(,length(years.predict)),dimnames = list(c(1:samplesize),formatC(,formatC(years.predict)))
#Fan plots for ages 65,75,85
probs = c(0.5,0.8,0.95,0.995)
qxt <- deathFR / exposureFR
matplot(, t(qxt[c("65", "75", "85"), ]),
        xlim = c(1970, 2027), ylim = c(0.0025, 0.2), pch = 20, col = "black",
        log = "y", xlab = "year", ylab = "death rate (log scale)")
fan(pred[,"65" , ], start = 2018, probs = probs, = 5,
    fan.col =  colorRampPalette(c("green", "white")), ln = NULL)
fan(pred[,"75" , ], start = 2018, probs = probs, = 5,
    fan.col =  colorRampPalette(c("red", "white")), ln = NULL)
fan(pred[,"85" , ], start = 2018, probs = probs, = 5,
    fan.col =  colorRampPalette(c("blue", "white")), ln = NULL)
text(1980, qxt[c("65", "75", "85"), "2000"],
     labels = c("x = 65", "x = 75", "x = 85"))

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,, forecast=10,family = "poisson",cores=4)
fitM6=m6_stan(death = deathFR,exposure=exposureFR,,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.