Discussion

1. The Bayes Density Estimator provides a kernel density estimate, without the need for bandwidth choice by the user. This has the distinct advantage that, when applied to a series of projections, the bandwidth need not be assumed to be the same for all projections. The corollary to this is that the data need not be distorted in trying to ensure uniform smoothness for all projections.
2. The model found by the BKDE should be the best available from the family of models provided (Berk, 1966), which, in the case of KDE models, gives a wide range to choose from. This leads to a high degree of confidence in the density estimate obtained.
3. The bandwidth found is not a point estimate but a density with location and shape parameters. This allows for examination of the chosen bandwidth for suitability and might lead, in further work, to a system of kernel suitability evaluation.
4. The choice of kernel affects the final model, some kernels being more suitable for particular types of data, for example, survival data is not well served by any kernel that allows negative values. The BKDE allows for rapid selection of kernel and might lead to suitability analysis of the kernel.
5. The prior allows for the modelling of belief in the smoothness of the underlying density. The strength of that belief can also be represented and allows for a wide variation in the balance between that belief and the information from the data.
6. With the large samples obtained from MCMC simulation, the prior is dominated by the information from the data in the likelihood, however it is possible to force a ``wrong'' prior on the system. A very strongly held belief in a prior, for example $N(0.1,0.01)$ in section 2 needs significantly more data to modify than that of $N(0,1)$. used in graph 1. of Figure 2.

The examples from Berlinet and Devroye (1994) are designed to be difficult and the results in that paper are all averages of 20 different samples of size 100. The overwhelming conclusion to be drawn from it is that no one KDE will do well at all densities and some experimentation with method is needed. However, the BKDE in one of its forms produced acceptable estimates of a large number of the densities without the need for human intervention. As a method KDE compares well to several others and can be considered at least the equal of most.