We started discussing the paper "Nested sampling  for general Bayesian computation" by Skilling. Nested sampling estimates directly how the likelihood function relates to previous value of the discrete random variable. It can be used to compare two data models for the same set by comparing the conditional probability that is assigned after the relevant evidence is taken into account.
This method computes the marginal likelihood directly by integration. Moreover, we can get samples from the unobserved as conditionals on the observed optionally. The sampling proceeds based on the nested contours of the likelihood function and not on their values. This technique allows the method to overcome the limitations that creep into annealing methods.
We looked at the direct summation performed by this method by taking the Bayes in the form
Likelihood x Prior = Evidence x Posterior. The evidence can then be written as a summation over the prior mass elements.
The calculation of the evidence allows different model assumptions to be compared through the ratios of evidence values known as Bayes factors. This works well even for future models because evidence does not have to be recalculated for the current one. In fact the evidence and the posterior are valuable information for anybody who wants to compare models. The evidence gives a certain information on the strength of the model and the posterior gives how the observed data modulates the prior beliefs. Both of them together gives good information on how the likelihood is constructed but also how it is used.
This is an improvement over Markov Chain Monte Carlo methods discussed in earlier posts because they yielded a set of samples representing the normalized posterior. Here we get evidence as well which was not always easy because it was found as intermediary distributions that bridge prior and posterior. Annealing methods took direct likelihood values as intermediary steps. It was expensive to compute, it was a by-product and it had secondary importance with regard to posterior. This approach reverses the importance and directly calculates the evidence first. Although here too we sort the parameters by their likelihood values, they are summed up over the function to give the evidence. Since there are usually far too many points to do this continuously, sampling is nested to do it in steps.
#codingexercise
Find the lexicographic minimum in a circular array, e.g. for the array BCABDADAB, the lexicographic minimum is ABBCABDAD.
String GetMinRotation(string input)
{
String arr[input.Length];
String combined = input + input;
string min = input;
for (int I = 0; I < input.Length; i++)
{
    String candidate = input.substring(I, input.Length);
    if (candidate < min) 
       min = candidate;
}

return min;

}

Get Count rotation to desired output

int GetCount(string input, string output)

{

String arr[input.Length];
String combined = input + input;
int count = 0;
for (int I = 0; I < input.Length; i++)
{
    String candidate = input.substring(I, input.Length);

    count++;

    if (candidate == output) 
       return count;
}

return -1;

}