# how to determine a priori probability distribution of sigama2 in montecarlo simulation?

1、the monte carlo simulation code in SAS： Example1: https://support.sas.com/rnd/app/stat/examples/BayesStd/new_example/index.html

proc mcmc data=class outpost=classout nmc=10000 thin=2 seed=246810;
parms beta0 0 beta1 0;
parms sigma2 1;
prior beta0 beta1 ~ normal(mean = 0, var = 1e6);
prior sigma2 ~ igamma(shape = 3/10, scale = 10/3);
mu = beta0 + beta1*height;
model weight ~ n(mu, var = sigma2);
run;


What does the "parm sigma2 1"mean? Does it mean the sigma2's parameter begins from 1 in the simulation and why it begins from 1 but not 0.1、0.011、0.0001 or others?

Just like Example2 below: https://www.phusewiki.org/docs/Conference%202017%20AS%20Papers/AS03.pdf

the variable "a_Sex" begins from 0.1 and the sigma2 begins from 0.0011. So how to determine the variable's initial value?

proc mcmc data=Orthodont_tr outpost=outpost nbi=5000 nmc=25000 thin=15 plots=ALL;
array mean[4];
array a_age[3]      a_age10 a_age12 a_age14;
array a_sex_age[3]  a_sex_age10  a_sex_age12  a_sex_age14;
array dist[4]       dist1 dist2 dist3 dist4;
parms intercept 0.1
a_Sex 0.1
a_age: 0
a_sex_age: 0;
parms S2 0.0011;
parms Rho;

prior intercept   ~ normal(0,var=1e6);
prior a_Sex       ~ normal(0,var=1e6);
prior a_age:      ~ normal(0,var=1e6);
prior  a_sex_age: ~ normal(0,var=1e6);
prior S2          ~ igamma(shape=0.01, scale=0.01);
prior Rho         ~ uniform(-1,1);

mean[1] = intercept + a_Sex * (Sex=1)                            ;/*time = 8*/
mean[2] = intercept + a_Sex * (Sex=1)+ a_age10 +  a_sex_age10 * (Sex=1);/*time = 10*/
mean[3] = intercept + a_Sex * (Sex=1)+ a_age12 +  a_sex_age12 * (Sex=1);/*time = 12*/
mean[4] = intercept + a_Sex * (Sex=1)+ a_age14 +  a_sex_age14 * (Sex=1);/*time = 14*/

model dist ~ mvnar(mean, var=S2, rho=Rho);

run;


2、Why the sigma2's prior distribution in the 1st example is like this:

prior sigma2 ~ igamma(shape = 3/10, scale = 10/3)


But in the 2rd example:

prior S2 ~ igamma(shape=0.01, scale=0.01)


Why are they different? How to determine the sigma2's priori distribution and why the sigma2 has an igamma distribution?

@merv Do you mean I need to use this formula below to calculate the igamma distribution's "shape(α)" and "scale(1/β)"?

But obviously it is impossible to get the "α=0.01" in Example2 through the "α+n/2"……

– Stuart.Sklinar
Commented Dec 27, 2019 at 15:21
• @Stuart.Sklinar I have pasted all of the code in the Example1 and two key codes about the question in the Example 2.
– tumidou
Commented Dec 27, 2019 at 15:30
• PhUse 2017 are coded images - equally, you may be better off at math.stackexchange.com
– Stuart.Sklinar
Commented Dec 27, 2019 at 15:31
• @Stuart.Sklinar Ok,thanks. I have updated it.
– tumidou
Commented Dec 27, 2019 at 15:50
• Have you tried changing the parms values? In MCMC sampling for most models, it shouldn't matter where you initialize - as long as it eventually samples from the typical set, it should be good. If it is sensitive to these init values, then that could indicate a problem, or at least a need to discard a larger burn-in set (though the current nbi=5000 is likely sufficient).
– merv
Commented Dec 27, 2019 at 15:50