... quadrature¹

Numerical quadrature or integration is the process of approximating an integral of the form $\int_a^b f(x) dx$ by a summation of the form $\sum_{k=0}^m w_k f(x_k)$. It can be shown that this can be achieved with any desired level of accuracy by careful choice of the nodes $x_k$ and the weights $w_k$ (for example, see Davis and Rabinowitz, 1984; Shaw, 1986). As the number of nodes needed increases as the dimensionality of the function, numerical quadrature becomes more difficult to apply as the dimensionality of the problem increases.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... present²

Present for the article was taken to be 1983.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... result³

Extracting a sub-sample of the data in this way removes correlation from the result. A sample size of 500 is amenable to computation and allows demonstration of the influence of a strong prior. If 10000 points are used then the information in almost any prior is significantly less than that in the data.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...fig:histoldf2⁴

In these figures, 17 bins refers to the command used to generate the image, in this case hist(oldf2$lengthm, main=" ",xlab=" ", breaks = 17), this is a much higher bin count than would normally be used with this data but is an informed choice.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...dberdev(seq(-n,m,0.01),dnum=1)⁵

Where the value of dnum varies from 1 to 28 giving the appropriate line for each density. The values n and m provide a limit to the range of dberdev so the graph is not swamped.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.