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## 3.2 Partially known parameters

In some situations, such as when a multivariate probability function has partially observed outcomes or parameters, it will be necessary to create a vector mixing known (data) and unknown (parameter) values. This can be done in Stan by creating a vector or array in the transformed parameters block and assigning to it.

The following example involves a bivariate covariance matrix in which the variances are known, but the covariance is not.

data {
int<lower=0> N;
vector y[N];
real<lower=0> var1;
real<lower=0> var2;
}
transformed data {
real<lower=0> max_cov = sqrt(var1 * var2);
real<upper=0> min_cov = -max_cov;
}
parameters {
vector mu;
real<lower=min_cov, upper=max_cov> cov;
}
transformed parameters {
matrix[2, 2] Sigma;
Sigma[1, 1] = var1;     Sigma[1, 2] = cov;
Sigma[2, 1] = cov;      Sigma[2, 2] = var2;
}
model {
y ~ multi_normal(mu, Sigma);
}

The variances are defined as data in variables var1 and var2, whereas the covariance is defined as a parameter in variable cov. The $$2 \times 2$$ covariance matrix Sigma is defined as a transformed parameter, with the variances assigned to the two diagonal elements and the covariance to the two off-diagonal elements.

The constraint on the covariance declaration ensures that the resulting covariance matrix sigma is positive definite. The bound, plus or minus the square root of the product of the variances, is defined as transformed data so that it is only calculated once.

The vectorization of the multivariate normal is critical for efficiency here. The transformed parameter Sigma could be defined as a local variable within the model block if it does not need to be included in the sampler’s output.