by the parameters. For example, the values and of a beta distribution can be thought of as corresponding to successes and failures if the posterior mode is used to choose an optimal parameter setting, or successes and failures if the posterior mean is used to choose an optimal parameter setting. In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of pseudo-observations. This can help provide intuition behind the often messy update equations and help choose reasonable hyperparameters for a prior. Dynamical system One can think of conditioning on conjugate priors as defining a kind of (discrete time) dynamical system: from a given set of hyperparameters, incoming data updates these hyperparameters, so one can see the change in hyperparameters as a kind of "time evolution" of the system, corresponding to "learning". Starting at different points yields different flows over time. This is again analogous with the dynamical system defined by a linear operator, but note that since different samples lead to different inferences, this is not simply dependent on time but rather on data over time. For related approaches, see Recursive Bayesian estimation and Data assimilation. Practical example Suppose a rental car service operates in your city. Drivers can drop off and pick up cars anywhere inside the city limits. You can find and rent cars using an app. Suppose you wish to find the probability that you can find a rental car within a short distance of your home address at any time of day. Over three days you look at the app and find the following number of cars within a short distance of your home address: Suppose we assume the data comes from a Poisson distribution. In that case, we can compute the maximum likelihood estimate of the parameters of the model, which is Using this maximum likelihood estimate, we can compute the probability that there will be at least one car available on a given day: This is the Poisson distribution that is the most likely to have generated the observed data . But the data could also have come from another Poisson distribution, e.g., one with , or , etc. In fact, there is an infinite number of Poisson distributions that could have generated the observed data. With relatively few data points, we should be quite uncertain about which exact Poisson distribution generated this data. Intuitively we should instead take