# Bayesian estimation for association models with multiplicative predictors

I would like to know if it is possible to fit the Lee-Carter model in a Bayesian setting. This model is used to forecast population mortality dynamics and has the following form:
log (\mu_{xt})=\alpha_x+\beta_x \kappa_t ,
where \beta_x and \kappa_t are factors for age and year, respectively. Due to the multiplication of predictors, this model does not fit in the GLM framework and cannot be fitted with the stan_glm function of the rstanarm package. In the frequentist setting, this model can be fitted with the gnm package but I couldnâ€™t find a Bayesian equivalent.

Do you know if this can easily be implemented in Stan?

Thank you very much for your help!

Here is the start of some code to do a Lee-Carter model:

data {
int<lower = 1> J;                    // number of age categories
int<lower = 1> T;                    // number of years
matrix[J, T] m;                      // mortality matrix
int<lower = 1, upper = T> anomaly;   // when does the pandemic occur
}
transformed data {
vector[J * T] y = to_vector(log(m));
}
parameters {
vector[J] a;
simplex[J] b;                        // strictly positive and sums to 1

real alpha;                          // drift in random walk
real beta;                           // impact of the pandemic
vector[T] epsilon;                   // unscaled errors in the random walk

real<lower = 0> sigma[2];            // error standard deviations
}
transformed parameters {
vector[J] k;
k[1] = alpha + epsilon[1];
for (t in 2:(anomaly - 1)) k[t] = k[t - 1] + alpha + sigma[1] * epsilon[t];
k[anomaly] = k[anomaly - 1] + alpha + beta + sigma[1] * epsilon[anomaly];
for (t in (anomaly + 1):T) k[t] = k[t - 1] + alpha + sigma[1] * epsilon[t];
}
model {
vector[J * T] mu;
int pos = 1;
for (t in 1:T) for (x in 1:J) {
mu[pos] = a[x] + b[x] * k[t];
pos += 1;
}
target += normal_lpdf(y | mu, sigma[2]);
target += std_normal_lpdf(epsilon);
target += exponential_lpdf(sigma | 1);
}


This does not have the restriction that the k sums to zero. Nor does it have a generated quantities block with forecasts yet.

EDITS:

• Corrected the sizes of b and epsilon
2 Likes

Thank you very much for the start of the code! I would have three small comments regarding the code:

• I think we should have : simplex[J] b and vector[T] epsilon since the first one refers to ages and the second to years.
• In the definition of k, shouldnâ€™t we have sigma[2] instead of sigma[1] since there is no defined sigma[1].
• In the last line, shouldnâ€™t we have: target += exponential_lpdf(sigma[2]| 1)

Youâ€™re correct. I just now edited the code to reflect that.

The real<lower = 0> sigma[2] declaration in the parameters block is an array of size two positive numbers. You can think of it as \sigma_1 and \sigma_2. \sigma_1 is the scale on the innovations in the random walk for k and \sigma_2 is the scale of the errors in m.

That just puts an exponential prior with a rate of 1 on both \sigma_1 and \sigma_2, but that is definitely something that could be changed.

1 Like

Thanks for the reply! The code works with the small change vector[T] k in the transformed parametersâ€™ block as it refers to years.

Concerning the sum-to-zero constraint, what is the optimal method to impose that the sum of k is zero? If we impose that the last observation of the time series is the subtraction of the previous elements, I am not sure it is correct (= one less random noise in the sequence).