Bayesian Minitab Macros in Bayesian Computation Using Minitab By Jim Albert

CHAPTER 2 - Simulating Games of Chance

PROGRAM craps -- Plays game of craps.

To run: Type exec 'craps'. The program will let you play the game a number of times. At the end of the games, some summary statistics are displayed.
SAMPLE MINITAB OUTPUT FROM 'craps'

PROGRAM yahtzee -- Plays simplified version of Yahtzee.

To run: Type exec 'yahtzee'. Five dice are rolled three times. After each roll, you decide which dice to keep (if you wish to keep the first, second and fourth dice, you type 1 1 0 1 0). After each roll, the result is printed (two of a kind, small straight, etc.)

SAMPLE MINITAB OUTPUT FROM 'yahtzee'

PROGRAM yahtz_au -- Computer plays Yahtzee using a particular strategy.

To run: Type exec 'yahtz_au'. The computer will decide which dice to keep.

PROGRAM yahtz_re -- Play the computer version of Yahtzee repeatedly.

To run: Type exec 'yahtz_re' K, where K is the number of times you wish to repeat the game. (The program 'yahtz_au' must be run first.) The results for each game in coded form are displayed. The results of the first three rolls are placed in the columns 'ROLL_1', 'ROLL_2', 'ROLL_3'. The column 'F_ROLL' contains the results of the final roll.

PROGRAM bball -- Simulates a baseball season using a Bradley-Terry model.

To run: Type exec 'bball'. One inputs the strength values for the teams. A season is played where each team plays every other team 8 games. The results of all of the games are displayed in a 2-way contingency table. The number of games won by all of the teams are displayed.
SAMPLE MINITAB OUTPUT FROM 'bball'

PROGRAM bball_re -- Repeats `bball'. Simulates a number of baseball seasons.

To run: Type exec 'bball_re' K, where K is the number of times you wish to repeat the simulation. (The program 'bball' must be run first.) The number of games won for all teams in each season are displayed. The number of wins for all teams are stored in the columns 'NWIN_1', 'NWIN_2', etc. ('NWIN_1' contains the number of games won by the team with the first strength value, etc.) The place finishes for all teams are stored in the columns 'PLACE_1', 'PLACE_2', etc.

CHAPTER 3 - Introduction to Inference Using Bayes' Rule

PROGRAM bayes_se -- Sets up models, prior probabilities, and likelihoods for Bayes' rule.

To run: Type exec 'bayes_se'. You input the number of models, the prior probabilities, the number of observations and names, and the likelihoods.

PROGRAM bayes -- Implements Bayes' rule sequentially for independent sequence of outcomes.

To run: Type exec 'bayes'. (The program 'bayes_se' must be run first.) You input the sequence of independent observations and Bayes rule is implemented sequentially.
SAMPLE MINITAB OUTPUT FROM 'bayes_se' AND 'bayes'

CHAPTER 4 - Learning About a Proportion

PROGRAM p_disc -- Computes the posterior distribution for p when there is a discrete set of models.

To run: Name two Minitab columns 'P' and 'PRIOR'. Place the value of p in 'P' and the prior probabilities in 'PRIOR'. Then type exec 'p_disc'. The posterior probabilities are stored in the column 'POST'.
SAMPLE MINITAB OUTPUT FROM 'p_disc'

PROGRAM p_disc_p -- Computes predictive distribution for number of successes in future binomial experiment.

To run: The values of p are contained in a column 'P' and the probabilities are contained either in the columns 'PRIOR' or 'POST'. Type exec 'p_disc_p'. You will input the number of trials in the future experiment, if you want to use prior or posterior probabilities for p, and the range of number of successes that you are interested in. The predictive distribution is stored in the columns 'SUCC' and 'PRED'.
SAMPLE MINITAB OUTPUT FROM 'p_disc_p'

PROGRAM beta_sel -- Finds parameters of beta distribution which matches two predictive probabilities.

To run: Type exec 'beta_sel'. You input the probability of a future success, and the probability of a second success conditional on a first success. The program gives the matching values of the beta parameters a and b.
SAMPLE MINITAB OUTPUT FROM 'beta_sel'

PROGRAM p_beta -- Summarizes beta distribution for p.

To run: Type exec 'p_beta'. After inputting the beta parameters a and b, you can see a graph of the density, compute cumulative probabilities for a list of values of interest, and compute percentiles of the distribution.
SAMPLE MINITAB OUTPUT FROM 'p_beta'

PROGRAM p_beta_p -- For beta prior or posterior, computes predictive distribution for number of successes in future experiment.

To run: Type exec 'p_beta_p'. You input the beta parameters, the number of trials, and the range of numbers of successes that you are interested in. The predictive distribution is stored in the columns 'SUCC' and 'PRED'.
SAMPLE MINITAB OUTPUT FROM 'p_beta_p'

PROGRAM p_beta_t -- Tests the hypothesis that p is equal to a specific value.

To run: Type exec 'p_beta_t'. You input the value of p that you wish to test, the prior probability of this value, the parameters of the beta distribution under the alternative hypothesis, and the number of successes and failures in the experiment. The output is the Bayes factor and the posterior probability that the proportion is equal to the specific value.
SAMPLE MINITAB OUTPUT FROM 'p_beta_t'

PROGRAM p_hist_p -- For a histogram prior for p, finds posterior probabilities of intervals using simulation.

To run: Type exec 'p_hist_p'. Suppose that you can divide the interval of plausible p values into equally spaced subintervals and assign a probability to each subinterval. You input the list of midpoints of the subintervals, the corresponding prior probabilities, and the data (number of successes and failures). The posterior probabilities of the intervals are displayed in the output.
SAMPLE MINITAB OUTPUT FROM 'p_hist_p'

CHAPTER 5 - Comparing Two Proportions

PROGRAM pp_disc -- Finds posterior distribution for (p1, p2) when a uniform prior on a grid is used.

To run: Type exec 'pp_disc'. You input the low and high values of each proportion on the grid, the number of models, and the number of successes and failures for each sample. The table of posterior probabilities of (p1, p2) is displayed and graphed. The posterior probabilities for the difference d = p2 - p1 is shown. The values of p1 and p2 are stored in the columns 'P1' and 'P2'; the values of the prior and posterior distribution are stored in the columns 'PRIOR' and 'POST'. The probability distribution for d is stored in the columns 'DIFF' and 'P_DIFF'.
SAMPLE MINITAB OUTPUT FROM 'pp_disc'

PROGRAM pp_disct -- Using a prior which gives a specific probabilitity to p1=p2, summarizes posterior.

To run: Type exec 'pp_disct'. For each proportion, you input the low and high values and the number of models. Also, you input the prior probability that p1=p2 and the binomial data. The output is the same as the program 'pp_disc'.

PROGRAM pp_discm -- Finds posterior distribution when informative prior is given in table form.

To run: Place the values of p1 in column C1 (say), the values of p2 in column C2, and the table of prior probabilities in columns C3-CN (each row corresponds to the corresponding value of p1 and all values of p2). Type exec 'pp_discm'. You tell the program where the prior is located and input the number of successes and failures for each sample. The output is the same as the program 'pp_disc'.

PROGRAM pp_beta -- Using simulation, summarizes distribution of p2-p1 when proportions have independent beta distributions.

To run: -- Type exec 'pp_beta'. Input the parameters of the beta distribution for each proportion and the number of values to simulate. Program gives dotplot of distribution of p2-p1 and computes probabilities that p2-p1 exceeds each of a list of values that is inputted. The simulated samples of p1, p2, and p2-p1 are stored in the columns 'P1', 'P2', and 'P2-P1'.
SAMPLE MINITAB OUTPUT FROM 'pp_beta'

PROGRAM pp_bet_t -- Constructs test that p1=p2 using beta priors.

To run: Type exec 'pp_bet_t'. Input the prior probability of equality, the beta distribution for the common value p1=p2, the independent beta distributions for p1 and p2 when p1 is not equal to p2, and the data. The output is the Bayes factor and the posterior probability of equality.
SAMPLE MINITAB OUTPUT FROM 'pp_bet_t'

PROGRAM pp_exch -- Using an exchangeable prior on the logits, summarizes the posterior distribution of p1, p2.

To run: Type exec 'pp_exch'. You input the data, the prior standard deviation of the normal prior distribution on the logits, and the number of values to simulate. The simulated samples from the prior and posterior distributions are stored in the columns 'PRIOR_P1','PRIOR_P2', 'POST_P1', and 'POST_P2'.
SAMPLE MINITAB OUTPUT FROM 'pp_exch'

CHAPTER 6 - Learning About a Normal Mean

PROGRAM m_disc -- Computes the posterior distribution for M when there is a discrete set of models (sampling variance known).

To run: Name two Minitab columns 'M' and 'PRIOR'. Place the value of M in 'M' and the prior probabilities in 'PRIOR'. If data is in raw form, then place it into a particular column of the worksheet . Then type exec 'm_disc'. You input the value of the known standard deviation and the column number of the data. (if data is in summary form, input the sample size, sample mean, and sample standard deviation). The posterior probabilities are stored in the column 'POST'.
SAMPLE MINITAB OUTPUT FROM 'm_disc'

PROGRAM normal_s -- Finds parameters of normal distribution which matches two prior percentiles.

To run: Type exec 'normal_s'. One inputs two percentiles (probability to the left and the corresponding percentile) and the program gives the mean and standard deviation of the matching normal curve.
SAMPLE MINITAB OUTPUT FROM 'normal_s'

PROGRAM m_cont -- Summarizes posterior distribution for normal mean M with a normal prior (approximate method).

To run: If data is in raw form, then place it into a particular column of the worksheet. (Summary data may also be input.) Type exec 'm_cont'. You input the mean and standard deviation of the normal prior distribution and the column number of the data (if data is in summary form, input the sample size, sample mean, and sample standard deviation). The output is the mean and standard deviation of the approximate normal marginal posterior density for M. Also, the normal parameters of the approximate predictive density for one future observation are given.
SAMPLE MINITAB OUTPUT FROM 'm_cont'

PROGRAM normal -- Plots and performs normal density calculations.

To run: Type exec 'normal'. You input the mean and standard deviation of the normal density. The density is graphed and it computes cumulative probabilities and percentiles of the distribution.
SAMPLE MINITAB OUTPUT FROM 'normal'

PROGRAM m_norm_t -- Constructs test that M is equal to specific value using normal priors (sampling variance known).

To run: Place data into a particular column. (Summary data may also be input.) Type exec 'm_norm_t'. You input the value to be tested, the prior probability of this value, a list of plausible values for the prior standard deviation of M under the alternative hypothesis and the population standard deviation. In addition, you input the number of the column where the data is located. The output is the Bayes factor and the posterior probability of the hypothesis for each value of the prior standard deviation.
SAMPLE MINITAB OUTPUT FROM 'm_norm_t'

PROGRAM m_nchi -- Gives exact analysis for M and S using vague priors.

To run: Place data into a particular column. (Summary data may also be input.) Type exec 'm_nchi'. Plots and gives summary statistics from the marginal posterior densities of M and the standard deviation S.
SAMPLE MINITAB OUTPUT FROM 'm_nchi'

CHAPTER 7 - Learning About Two Normal Means

PROGRAM mm_cont -- Gives mean and standard deviation for the normal distribution for the difference M1-M2 when M1 and M2 have independent normal distributions.

To run: Type exec 'mm_cont'. You input the mean and standard deviation for each of the two means and the program gives the mean and standard deviation for the difference in means.
SAMPLE MINITAB OUTPUT FROM 'mm_cont'

PROGRAM mm_tt -- Uses simulation to summarize the difference M2-M1 (difference of two t distributions).

To run: Place the two datasets into two columns. (Summary data may also be used.) Type exec 'mm_tt'. The output is dotplots of the marginal posterior densities for M1 and M2 and a dotplot and summary statistics for the difference in means.
SAMPLE MINITAB OUTPUT FROM 'mm_tt'

CHAPTER 8 - Learning About Relationships

PROGRAM lin_reg -- Gives inference about slope and predictive response for simple regression model.

To run: Place the x and y data in two columns. Type exec 'lin_reg'. You input the numbers of the columns of the data. The output is the mean and standard deviation of the regression slope. In addition, for each value of x that are inputted, the mean and standard deviation of the predicted response are given.
SAMPLE MINITAB OUTPUT FROM 'lin_reg'

PROGRAM c_table -- Computes a Bayes factors for a two-way contingency table using uniform priors.

To run: Place the contingency table in consecutive columns of the worksheet. Type exec 'c_table'. Input the numbers of the columns which contain the table. The output is the usual chi-squared test and a Bayes factor against the hypothesis of independence.
SAMPLE MINITAB OUTPUT FROM 'c_table'

CHAPTER 9 - Learning About Discrete Models

PROGRAM mod_disc -- Computes a posterior distribution for discrete models.

To run: Place model values in the column 'MODEL' and the prior probabilities in 'PRIOR'. Type exec 'mod_disc'. You input the number of the likelihood (binomial, normal, Poisson, etc.) and the data. The posterior probabilities are stored in the column 'POST'.
SAMPLE MINITAB OUTPUT FROM 'mod_disc' (learning about the number of successes in a finite population)

PROGRAM disc_sum -- Summarizes a discrete probability distribution.

To run: Type exec 'disc_sum'. Input the numbers of the columns which contain the values of the variable and the probabilities. The distribution is graphed and summaries (mode, mean and standard deviation) are given. Cumulative probabilities and probability intervals are computed for sets of values of interest.
SAMPLE MINITAB OUTPUT FROM 'disc_sum'

PROGRAM mod_crit -- Compares two discrete prior distributions.

To run: Place model values in the column 'MODEL' and the two sets of prior probabilities in the columns 'PRIOR1' and 'PRIOR2'. Type exec 'mod_crit'. You input the number of the likelihood (binomial, normal, Poisson, etc.) and the data. The two sets of corresponding posterior probabilities are stored in the columns 'POST1' and 'POST2'. The Bayes factor in support of the first prior is given; this value is stored in the column 'BAYES_F'.
SAMPLE MINITAB OUTPUT FROM 'mod_crit'

CHAPTER 10 - Learning About Continuous Models

PROGRAM mod_cont -- Computes the posterior distribution for continuous models.

To run Simulate a large number of values from the prior distribution of the parameter. Place these values in the column 'PRIOR_S'. Type exec 'mod_cont'. You input the number of the likelihood (binomial, normal, Poisson, etc.) and the data. A simulated sample from the posterior density is stored in the column 'POST_S'.
SAMPLE MINITAB OUTPUT FROM 'mod_cont' (learning about the size of a population by capture-recapture sampling)

CHAPTER 11 - Summarizing Posterior Distributions

SETUP FOR ONE PARAMETER

For one real-valued parameter, place the definition of the logarithm of the posterior density in the macro 'logpost1' -- the input column is 'X' and the output column is 'F'.

PROGRAM laplace1 -- Summarizes a one-parameter posterior density defined in macro 'logpost1' using the Laplace method.

To run: Type exec 'laplace1'. Input a guess at the mode and the number of iterations. The output is an estimate at the posterior mode and standard deviation and an estimate at the logarithm of the normalizing constant.
SAMPLE MINITAB OUTPUT FROM 'laplace1'

To run: Type exec 'ad_quad1'. Input guesses at the mean and standard deviation of the parameter and the number of iterations. Estimates at the posterior mean and standard deviation and the logarithm of the normalizing constant are output. The grid, density values, and weights are stored in the columns 'X', 'F', 'WT'.

PROGRAM metrop -- Summarizes a one-parameter posterior density defined in macro 'logpost1' using simulation.

To run: Type exec 'metrop'. Input a starting location and the number of values to simulate. The simulated values are stored in the column 'POST_S'.
SAMPLE MINITAB OUTPUT FROM 'metrop'

SETUP FOR TWO PARAMETERS

For two real-valued parameters, place the definition of the logarithm of the posterior density in the macro 'logpost2' -- the input columns are 'X' and 'Y' and the output column is 'F'.

PROGRAM laplace2 -- Summarizes a two-parameter posterior density defined in macro 'logpost2' using the Laplace method.

To run: Type exec 'laplace2'. Input a guess at the mode and the number of iterations. The output is an estimate at the mode and posterior standard deviations and covariance and an estimate at the logarithm of the normalizing constant.
SAMPLE MINITAB OUTPUT FROM 'laplace2'