Simple Dirichlet model with partial pooling

Questions
1

I am trying to implement a simple Dirichlet model with partial pooling to estimate changes to preferences between two samples of count data (think vote intentions between two polls as an example). I can estimate the preferences for candidates A,B and C in two polls using the simple models describe below. I try to implement a hierarchical model and I was expecting that it would show the means for a candidate in each poll would shrink towards the mean for the candidate in both polls… However in my case, there is no change to the means. I am assuming I have incorrectly setup the model.

I was following similar ideas as to what is described here: https://stats.stackexchange.com/questions/44144/multinomial-dirichlet-model-with-hyperprior-distribution-on-the-concentration-pa

A reproducible example is:

#data, row = "poll", candidate = column
counts=np.array([[317,120,563],[308,85,607]])

shape=counts.shape
totals=np.sum(counts,axis=1)

#Non-pooled model
with pm.Model() as simple_model:
    
    theta = pm.Dirichlet("theta", a=np.ones(shape),shape=shape)
    results = pm.Multinomial("results", n=totals, p=theta, observed=counts)
    
with simple_model:
    simple_model = pm.sample(draws=1000)

#check means:
#pm.summary(simple_model)

#Hierarchical model
with pm.Model() as hierarchical_model:
 
    beta = pm.Dirichlet("beta", a=np.ones(shape[1]),shape=shape[1])
    lambd = pm.Exponential('lambd',np.ones(shape[1]),shape=shape[1])
    
    theta = pm.Dirichlet("theta", beta*lambd,shape=shape)
    results = pm.Multinomial("results", n=totals, p=theta, observed=counts)
    
with hierarchical_model:
    hierarchical_model_trace = pm.sample(draws=10000)
    
#check means:
#pm.summary(hierarchical_model_trace)
1 Like
2

The estimation of theta in both model should be nearly identical based on the way you model it (essentially, you have one free parameter for one observation). However, if you look at beta and lambd, you should see some partial pooling property like shrinkage (wont be too obvious as well as there is not a lot of data).