log_mix on the logit scale

From @bob-carpenter on February 3, 2015 18:31

Now that we have log_mix defined on the natural probability scale:

log_mix(theta, lp1, lp2) 
  = log(theta * exp(lp1) + (1 - theta) * exp(lp2))
  = log_sum_exp(log(theta * exp(lp1)), log((1 - theta) * exp(lp2))
  = log_sum_exp(log(theta) + lp1, log(1-theta) + lp2)

it'd be nice to roll in the link function (inv_logit) and just work on the unconstrained log-odds scale with

log_mix_logit(alpha, lp1, lp2) 
  = log_mix(inv_logit(alpha), lp1, lp2)
  = log_sum_exp(log(inv_logit(alpha) + lp1, log(1 - inv_logit(theta)) + lp2)

Special functions log_inv_logit and log1m_inv_logit already exist.

Copied from original issue: stan-dev/stan#1260