Ch2 Understanding difference between sampling random() and mcmc.sample()

[rriv]

Hello,

first thanks a lot to Cameron to bring us such a nice book on MCMC.

Can anyone please tell me if I'm understanding correctly about Parent and Child relationships and distribution sampling.

Let me do the following (from Ch2 example) :

parameter = pm.Exponential("poisson_param", 1)
data_generator = pm.Poisson("data_generator", parameter)

if I do now :

samples1 = [data_generator.random() for i in range(2000)]

I'm taking 2000 random values from data_generator Poisson distribution, but with a constant parameter (initially random chosen from Exponential distribution, but the same for all 2000 sample). I'm telling this following this sentence from PyMC Documentation § 3.4.1 :

Whenever a variable is used as a parent for a child variable, PyMC replaces it with its value attribute when the child’s value or log-probability is computed.

After all, samples1 follows a Poisson distribution. With some parameter, but a Poisson distribution.

If I do now :

model = pm.Model([data_generator,parameter]);
mcmc = pm.MCMC(model)
mcmc.sample(2000);
samples2 = mcmc.trace('data_generator')[:];

Here parameter is drawn each time data_generator is sampled. Now samples2 has no more anything to do with a Poisson distribution. We really mixed the parent random parameter with the child distribution sampling.

So I'd be glad to hear if I'm right or wrong... Thanks in advance,

Robert

[CamDavidsonPilon]

You are correct: the former is just Poisson with a fixed parameter, the latter is a compound distribution of an Exponential and a Poisson.

Perhaps the best way to see this is to plot the samples (using your code, and calling hist on them)

We can see the differences in the distributions quite clearly (ignoring the width, but focusing on the height of each bar).