Candidate density matched curvature normal

In general, this matched curvature normal will not be a suitable candidate density, because we determined a global property of the density (the variance) from a local property (the curvature of the density at a single point). The spread may be much too small. However, it will provide us the basis for finding a suitable density. 

Use Student s t candidate density that is similar to the matched curvature normai. The way forward is to find a candidate density that is similar shape to the matched curvature normal, but with heavier tails than the target. The Student s t with low degrees of freedom will work very well. The Student s t with 1 degree of freedom will dominate all target distributions. 

Figure 7 The traceplot and the histogram (with the taiget density) for the first 1000 draws from the chain using the matched curvature normal candidate distribution.

A heavy-tailed matched curvature candidate density can be found by matching the mode and curvature at the target with a normal, then replacing the normal with a Student s t with degrees of freedom equal to 1. 

This matched curvature multivariate normal candidate density does not really have any relationship with the spread of the target. If we used it is likely that it does not dominate the target in the tails. So, instead, we use the multivariate Student s r[m, V] with low degrees of freedom. First we find the lower triangular matrix L that satisfies V = LL by using the Cholesky decomposition. Then we draw a random sample of Student s t random variables with k degrees of freedom. 

We suggest that when the posterior has multiple nodes a similar strategy be used. At each mode, we find the multivariate normal distribution that matches the curvature of the target at that mode. The candidate density should be a mixture of the multivariate Student s t distributions with low degrees of fieedom where each component of the mixture corresponds to the multivariate normal found for that node. Using this mixture density as the independent candidate density will give fast convergence to the posterior. 

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