After optimization Cholesky of Hessian can fail

Summary:

After optimization Cholesky of Hessian can fail. Since this likely due to a numerical inaccuracy, instead of error and fail, it would be better to add some jitter, give a warning and provide a result. When using the coming PSIS diagnostic and correction it doesn't matter if jitter causes extra variation in draws.

Reproducible Steps:

Data in http://www.stat.columbia.edu/~gelman/regression/

earnings_all <- read.csv("Earnings/data","earnings.csv")
earnings_all$positive <- earnings_all$earn > 0
# only non-zero earnings
earnings <- earnings_all[earnings_all$positive, ]
M_1 <- stan_glm(earn ~ height + male, data = earnings, algorithm="optimizing")

Current Output:

Optimization terminated normally: 
  Convergence detected: relative gradient magnitude is below tolerance
Error in chol.default(-H) : 
  the leading minor of order 4 is not positive definite
Error in out$theta_tilde[, mark] %*% t(R_inv) : 
  requires numeric/complex matrix/vector arguments

Expected Output:

Optimization terminated normally: 
  Convergence detected: relative gradient magnitude is below tolerance
Warning: Hessian is close to a singular  (up to to the numerical accuracy). Addded 1-e10 jitter.

RStan Version:

The version of RStan you are running (e.g., from packageVersion("rstan"))
[1] ‘2.17.3’

R Version:

The version of R you are running (e.g., from R.version.string)
[1] "R version 3.4.4 (2018-03-15)"

Start by reproducing the stan_glm(..., algorithm="optimizing") example with the supplied earnings data and inspect the chol.default(-H) failure. Done means the Hessian case produces a warning about jitter and returns a result instead of failing with the matrix error.