A BIVARIATE DENSITY ESTIMATION PROBLEM As a second illustration of the mixture model software, I generated bivariate real data from a mixture of two component distributions, with probabilities 0.3 and 0.7. These two distributions were not exactly Gaussian, and the two real variables were not exactly independent within one of these components. Accordingly, modeling this data well with a mixture of Gaussian distributions will require more than two components in the mixture. For the exact distribution used, see the source of the generation program, in xrgen.c. I generated 1000 cases with this program, stored in 'xrdata', of which the first 500 are used for training (the others are not used for anything at the moment). A two-component mixture model for the density estimation problem. We can first see what happens when we model this data with a mixture of two Gaussians - even though we know the data cannot be perfectly modeled in this way. We specify this two-component model using the 'mix-spec' and 'model-spec' commands, as follows: > mix-spec xrlog.2 0 2 2 / 1 0.05:0.5:0.2 10 > model-spec xrlog.2 real 0.05:0.5:0.5:1 The 'mix-spec' command creates the log file "xrlog.2". The arguments following the log file name are the number of input attributes in a case (always 0 at present), the number of target attributes (2 for this bivariate problem), and the number of mixture components to use (2 for this model). The Dirichlet concentration parameter follows the "/" - in this model, its value is 1 (unscaled, since there's no 'x'), which produces a uniform prior for the single number determining the probabilities of the two components. The "offset" parameters of the two components represent the Gaussian means when modeling real data. Hyperparameters determine the prior means and standard deviations of these offsets (separately for the two target attributes); priors for these hyperparameters are specified in 'mix-spec'. In the above command, the prior for the mean of an offset is Gaussian with standard deviation 10 (the last argument). The standard deviations for the offsets are given a hierarchical prior, with a higher-level hyperparameter common to both the lower-level standard deviations. The top-level precision (standard deviation to the power -2) is given a Gamma prior with mean 0.05 and shape parameter 0.5; the precisions for the lower-level hyperparameters have Gamma priors with mean given by the higher-level precision, and shape parameter 0.2. This is all specified by the second-to-last argument of 'mix-spec'. A similar hierarchical scheme is used for the "noise" standard deviations (the standard deviations of the Gaussian distributions in the mixture), except that this scheme has three levels - a top-level hyperparameter, a hyperparameter for each target attribute, and hyperparameters for each target for each component. The 'model-spec' command gives the top-level mean, and the shape parameters for the Gamma priors going down the hierarchy. We next specify where the data comes from, with 'data-spec': > data-spec xrlog.2 0 2 / xrdata@1:500 . Here again, we specify that there are 0 input attributes and 2 target attributes. For this finite model, we can specify that all the Markov chain updates should be done with Gibbs sampling, as follows: > mc-spec xrlog.2 repeat 20 gibbs-indicators gibbs-params gibbs-hypers The "repeat 20" just repeats these operations in a single interation, to reduce the volume of data stored in the log file. Finally, we run the Markov chain simulation for 100 iterations: > mix-mc xrlog.2 100 Once this is done (after less than a minute on our SGI machine), we can look at the hyperparameters and component parameters at various iterations. The last iteration should be something like the following: > mix-display xrlog.2 MIXTURE MODEL IN FILE "xrlog.2" WITH INDEX 100 HYPERPARAMETERS Standard deviations for component offsets: 0.585: 4.108 40.500 Means for component offsets: -0.549 -2.258 Standard deviations for Gaussian target distributions: 1.464: 0.959 3.693 PARAMETERS AND FREQUENCIES FOR COMPONENTS OF THE MIXTURE 1: 0.712 -2.046 +11.371 1.068 5.440 2: 0.288 +1.757 +20.956 0.949 7.198 We see above that the two components are associated with fractions of approximately 0.7 and 0.3 of the training cases, as expected from the way the data was generated. For each component, the two offset parameters, giving the Gaussian means, are shown on the first line, and the standard deviation parameters on the following line. These component parameters are approximately what we would expect, but of course the two Gaussian components cannot perfectly model the actual distribution. An infinite mixture model for the density estimation problem. To more closely approximate the true distribution, we can use a mixture model with a countably infinite number of Gaussian components. An infinite mixture is used if we simply omit the argument giving the number of components in the 'mix-spec' command. We must also change the specification for the Dirichlet concentration parameter, preceding it with an 'x' to indicate that it should be scaled so as to produce a sensible infinite limit. In the specification below, the moderate value of 5 is chosen for this specification in order to indicate that we believe that a fairly large number of components will have substantial probability (the other prior specifications are the same as before): > mix-spec xrlog.inf 0 2 / x5 0.05:0.5:0.2 10 The 'model-spec' and 'data-spec' commands are the same as before: > model-spec xrlog.inf real 0.05:0.5:0.5:1 > data-spec xrlog.inf 0 2 / xrdata@1:500 . The 'mc-spec' command must be altered, however, since it is not possible to do Gibbs sampling for component indicators when there are an infinite number of components. The met-indicators operation is used instead, with 10 changes being proposed to every indicator: > mc-spec xrlog.inf repeat 20 met-indicators 10 gibbs-params gibbs-hypers We can now run the simulation for 100 iteratons (which takes about a minute and a half on our SGI machine): > mix-mc xrlog.inf 100 If we now examine the state with 'mix-display', we will find that quite a few (eg, 20) mixture components are associated with training cases - though fewer components account for the bulk of the cases. We can see how well the model has captured the true distribution by generating a sample of cases from a distribution drawn form the posterior, as represented by the state at a particular interation. We do this as follows: > mix-cases xrlog.inf 100 new 1000 This command generates 1000 new cases based on iteration 100, and stores them (one per line) in the file "new". We can now use a plot program to view a scatterplot of the data in "new", and compare it with a scatterplot of data from the actual distribution. Note that the data in "new" is taken jointly from one example of a distribution from the posterior distribution. If 'mix-cases' is called for another iteration, it will produce data from a different distribution from the posterior, which in general could be quite different. This variation represents the uncertainly regarding the true distribution that remains when only a finite amount of training data is available. A representation of the predictive distribution for a single new data point, which is the average of distributions drawn from the posterior, could be obtained by combining the output of 'mix-cases' for a number of iterations.