Understanding Maximum Likelihood Estimation in STAN

Maximum likelihood estimation (MLE) is a widely used method for estimating the parameters of a statistical model. In STAN, MLE is implemented using the optimizing function, which takes a model, data, and optimization algorithm as input and returns an optimized parameter vector. However, one common question among users is how to obtain standard errors or confidence intervals for these estimates.

The Role of the Hessian Matrix

The Hessian matrix plays a crucial role in calculating standard errors and confidence intervals. In STAN, the Hessian matrix is computed by the hessian argument when calling the optimizing function. This matrix represents the variance-covariance of the estimated parameters. To compute standard errors or confidence intervals, we need to extract the Cholesky decomposition of the negative Hessian matrix and invert it.

Extracting Standard Errors

To extract standard errors from the optimized parameter vector, we can use the following approach:

1. Compute the Cholesky decomposition of the negative Hessian matrix.
2. Invert the resulting matrix.
3. Draw many times from a multivariate normal distribution with mean vector equal to the estimated parameters and variance-covariance represented by the inverted matrix.

The standard deviation of each column in this distribution represents the standard error of the corresponding parameter estimate.

Using the Delta Method

Alternatively, we can use the delta method to approximate the standard errors. This involves computing the partial derivatives of the log-likelihood function with respect to each parameter and evaluating them at the optimized parameters. The resulting variance-covariance matrix is then used to compute the standard errors.

However, this approach requires careful handling of the Jacobian matrix and can be computationally expensive for large models.

Implementing the Solution in R

Here’s an example implementation of the solution in R:

# 1: Compile and save a model (make sure to pass the data here)
model <- stan(file="model.stan", data=c("N","K","X","y"), chains = 0, iter = 0)

# 2: Fit that model
fit <- optimizing(object=get_stanmodel(model), as_vector = FALSE,
                   data=c("N","K","X","y"), hessian = TRUE)

# 3: Extract the vector theta_hat and the Hessian for the unconstrained parameters
theta_hat <- unlist(fit$par)
upars <- unconstrain_pars(linear, relist(theta_hat, fit$par))
Hessian <- fit$hessian

# 4: Extract the Cholesky decomposition of the (negative) Hessian and invert
R <- chol(-Hessian)
V <- chol2inv(R)
rownames(V) <- colnames(V) <- colnames(Hessian)

# 5: Produce a matrix with some specified number of simulation draws from a multinormal
SIMS <- 1000
len <- length(theta_hat)
unconstrained <- upars + t(chol(V)) %*% 
  matrix(rnorm(SIMS * len), nrow = len, ncol = SIMS)

# 6: Produce estimated standard errors for the constrained parameters
se <- apply(unconstrained, 2, sd)

Conclusion

In conclusion, calculating standard errors or confidence intervals in STAN requires the use of the Hessian matrix. By extracting the Cholesky decomposition and inverting it, we can draw many times from a multivariate normal distribution to estimate the standard errors of our model parameters. The delta method provides an alternative approach, but it may require careful handling of the Jacobian matrix. This implementation demonstrates how to extract standard errors using the Hessian matrix in R.

Additional Considerations

There are several additional considerations when working with maximum likelihood estimation and variance-covariance matrices:

• Checking for Numerical Instability: Be cautious when dealing with large or ill-conditioned Hessian matrices, as they can lead to numerical instability.
• Choosing the Right Algorithm: Select an optimization algorithm that is suitable for your model size and complexity. For example, using the lbfgs algorithm may be beneficial for smaller models, while CG may be more efficient for larger models.
• Regularization Techniques: Consider incorporating regularization techniques, such as L1 or L2 regularization, to improve numerical stability and prevent overfitting.

By following these guidelines and implementing the solution in R, you can obtain reliable standard errors and confidence intervals for your STAN model parameters.

Last modified on 2024-09-03