Example:  Using a nonconjugate prior distribution in estimating
a binomial probability (see page 41 of Gelman et al)

prior is triangular form
         p(t) = .5,           t < .4 or t > .6
              = .5+55(t-.4), .4 < t < .5
              = .5+55(.6-t), .5 < t < .6
observe 437 successes and 543 failures
want to find posterior median and central 95% credible interval

first we name the Minitab columns
MTB > name c1 't' c2 'prior' c3 'loglike' c4 'logpost' c5 'post' c6 'post_s'

put a grid of values .001,.002, ..., .999 into 't'
MTB > set 't'
DATA> .001:.999/.001
DATA> end

compute the prior on this grid
MTB > let 'prior'=.5+55*('t'-.4)*('t'>.4)*('t'<.5)+55*(.6-'t')*('t'>.5)*('t'<.6)

compute the logarithm of the likelihood function
MTB > let 'loglike'=437*log('t')+543*log(1-'t')

compute the posterior -- important to scale the log values before you exp!
then you normalize the values so that they sum to 1
MTB > let 'logpost'=log('prior')+'loglike'
MTB > let 'post'=exp('logpost'-max('logpost'))
MTB > let 'post'='post'/sum('post')

simulate 1000 values from this discrete posterior distribution
MTB > Random 1000 'post_s';
SUBC>   Discrete 't' 'post'.
MTB > dotplot 'post_s'

Each dot represents 4 points
                                      .
                                .    .: :
                                : .:::: :
                           .:.:::.:::::::..:: .
                           ::::::::::::::::::::.
                         ..:::::::::::::::::::::
                    .  .::::::::::::::::::::::::: ..
               . ..::.::::::::::::::::::::::::::::::::.... .. .
          -+---------+---------+---------+---------+---------+-----post_s  
       0.400     0.420     0.440     0.460     0.480     0.500

the posterior median is approximately the median of this simulated value
MTB > median 'post_s'

Column Median
   Median of post_s  =     0.45000

the 95% interval is the 25th and 975th order statistics of this simulated sample
MTB > sort 'post_s' 'post_s'
MTB > let k1='post_s'(25)
MTB > let k2='post_s'(975)
MTB > prin k1 k2
K1       0.420000
K2       0.480000