add poisson_log_lcdf and poisson_log_lccdf functions

Summary:

[From @bob-carpenter: I edited the title to highlight the root cause. You can skip all the discussion; just missing the underlying functions at this point.]

I'm fitting a model with the poisson_log function (i.e. a Poisson distribution parameterized by log(lambda) instead of lambda).

I'm running into two problems:

• Large number of divergent transitions with a simple model (well over 50% of the post-warmup iterations)
• If I add truncation, I start getting errors about negative lambda values and my log(rate) parameter is squashed up against 0.

My best guess is that truncation flips a switch somewhere that leads to stan::math::poisson_log_log when it should go to to stan::math::poisson_log. I can't explain the divergent transitions, but I think it also might be related. Maybe an incorrect nonlinear transform?

Reproducible Steps:

compiled = stan_model(model_code =
"data {
  int<lower=1> N;
  int<lower=1> y[N];
}
parameters {
  real log_rate;
}
model {
  log_rate ~ normal(0.0, 1.0);
  for(i in 1:N){
    y[i] ~ poisson_log(log_rate) T[1, ];
  }
}
"
)

y = rpois(1000, 1)
y = y[y>0]
data = list(y = y, N = length(y))

m = sampling(compiled, data = data)

As noted above, the non-truncated version (i.e. the above code with T[1, ] removed on line 12) also has problems.

Current Output:

Sampling without truncation:

• There were 2768 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help.
• samples of log_rate approximately Gaussian and can span 0, depending on the distribution of y in data.

Sampling with truncation:

• If the model initializes, I get many iterations of Exception thrown at line 12: poisson_log: Rate parameter is -2.24632e-13, but must be >= 0! 1
• With an earlier (more complicated) version of the model that didn't initialize, I instead got [596] "Exception thrown at line 12: stan::math::poisson_log: Rate parameter is -0.989166, but must be >= 0!" Note the reference to stan::math::poisson_log instead of stan::math::poisson_log_log
• There were 1065 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help

Expected Output:

MCMC samples

Current Version:

rstan 2.11.1