Bayesian Statistics: An Introduction, 4th Edition

10.3 Variational Bayesian methods: general case

The variational Bayes approach can also be used in cases where we have more than two parameters or sets of parameters. In cases where we seek an approximation to the density of the form

We need to write

With this notation, a straightforward generalization of the argument in the previous section leads to

and so to

10.3.1 A mixture of multivariate normals

We sometimes encounter data in two or more dimensions which is clearly not adequately modelled by a multivariate normal distribution but which might well be modelled by a mixture of two or more such distributions. An example is the data on waiting time between eruptions and the duration of the eruption for the Old Faithful geyser in Yellowstone National Park, Wyoming, USA taken from Härdle (1991), which is illustrated in Figure 10.3. A more complicated example of variational Bayes techniques ...