title: GPflow Predictive Mean/Variance for Poisson Likelihood
body: <p>Context: I trained a model using SVGP with a Poisson Likelihood (and a log/exp link).
  Aside from predicting the counts, I'd like to have an uncertainty measure .</p>
  
  <p>The function <code>m.predict_y</code> provides both the predictive mean and predictive variance. With a <strong>Poisson likelihood</strong>, it's using the default implementation of <code>Likelihood.predict_mean_and_var</code>with Gauss-Hermite quadrature.</p>
  
  <p>Is this calculating the expectations of the counts (i.e. our target/dependent variable), or of the poisson parameter?</p>
  
  <ul>
  <li><p>(i) If it is predictive mean/variance of the poisson parameter, how would one use this variance to get meaningful credible intervals?</p></li>
  <li><p>(ii) If it is the predictive mean/var of the counts, shouldn't the mean be identical to the variance (since we integrate out everything but y and y is poisson distributed)? Or am I misunderstanding the concept of predictive variance? </p></li>
  </ul>
  
  <p>If I understand the code correctly, we get the predictive mean/variance of the poisson parameter (i).</p>
  
  <p>In my current approach I rewrote <code>predict_mean_and_var</code> so that it uses the mcmc method <code>ndiag_mc_perc</code> instead of the hermite-gauss  approximation and then just returning the 10%/90% percentiles of the samples.</p>