a weighted average of the probability of for each of those Poisson distributions, weighted by how likely they each are, given the data we've observed . Generally, this quantity is known as the posterior predictive distribution where is a new data point, is the observed data and are the parameters of the model. Using Bayes' theorem we can expand therefore Generally, this integral is hard to compute. However, if you choose a conjugate prior distribution , a closed-form expression can be derived. This is the posterior predictive column in the tables below. Returning to our example, if we pick the Gamma distribution as our prior distribution over the rate of the Poisson distributions, then the posterior predictive is the negative binomial distribution, as can be seen from the table below. The Gamma distribution is parameterized by two hyperparameters , which we have to choose. By looking at plots of the gamma distribution, we pick , which seems to be a reasonable prior for the average number of cars. The choice of prior hyperparameters is inherently subjective and based on prior knowledge. Given the prior hyperparameters and we can compute the posterior hyperparameters and Given the posterior hyperparameters, we can finally compute the posterior predictive of This much more conservative estimate reflects the uncertainty in the model parameters, which the posterior predictive takes into account. Table of conjugate distributions Let n denote the number of observations. In all cases below, the data is assumed to consist of n points (which will be random vectors in the multivariate cases). If the likelihood function belongs to the exponential family, then a conjugate prior exists, often also in the exponential family; see Exponential family: Conjugate distributions. When the likelihood function is a discrete distribution When likelihood function is a continuous distribution See also Beta-binomial distribution Notes References Bayesian statistics