Update model when new observations are available

Modeling
1

Hi,

I have the following Stan-code


data {
  int<lower=1> N;       // number of observations
  int<lower=1> K;       // dimension of observations
  vector[K] y[N];       // observations: 
  vector[K] mu_x;
  vector[K] mu_y;
}

parameters {
  vector<lower=0>[K] sigma_x;
  vector<lower=0>[K] sigma_y;
  vector<lower=-1, upper=1>[K] rho;
}

model {
  for k in (1:K) {
    vector[2] mu = [mu_x[k], mu_y[k]]';
    // cholesky decomposition of covariance matrix
    matrix[2, 2] L_Sigma = [[sigma_x[k], 0], [rho[k] * sigma_y[k], sigma_y[k] * sqrt(1 - rho[k] * rho[k])]];
    sigma_x[k] ~ normal(0, 20);
    sigma_y[k] ~ normal(0, 20);

    y[k] ~ multi_normal(mu[k], L_Sigma);
}

After running this, I get the posteriors of sigma_x, sigma_y and rho. When new data arrives I want to update the model. I believe the model is suited fort this because of the presence of the multivariate normal distribution. My question is how does the new update-Stan code looks? Do I read the three variables one-by-one as vectors in the data-section? Is the below code the correct way to go?

data {
  int<lower=1> N;       // number of observations
  int<lower=1> K;       // dimension of observations
  vector[K] y[N];       // observations: a list of K vectors
  vector[K] mu_x;
  vector[K] mu_y;
  vector[K] sigma_x;
  vector[K] sigma_y;
  vector[K] rho;
}

parameters {
  vector<lower=0>[K] sigma_x;
  vector<lower=0>[K] sigma_y;
  vector<lower=-1, upper=1>[K] rho;
}

model {
  for k in (1:K) {
    vector[2] mu = [mu_x[k], mu_y[k]]';
    // cholesky decomposition of covariance matrix
    matrix[2, 2] L_Sigma = [[sigma_x[k], 0], [rho[k] * sigma_y[k], sigma_y[k] * sqrt(1 - rho[k] * rho[k])]];
    sigma_x[k] ~ normal(0, 20);
    sigma_y[k] ~ normal(0, 20);

    y[k] ~ multi_normal(mu, L_Sigma);
}

Thanks in advance

2

Sequential inference of this kind is not directly possible in Stan. There are ad hoc ways to attempt it regardless, but see here for a thread on why this can be folly.

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3

Also see a work in progress for nonparametric updating Non-parametric Bayesian updating by @robertgrant

3 Likes
4

See also Using posteriors as new priors.

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