I am trying to model a data with 6 repeats (number of rows) and 40 categories (number of columns) using multinomial distribution (see below for the data and code). I tried to model this data using two different shape arguments to the likelihood function which I thought would implicitly be broadcasted to the same thing but the results turned out be completely different for low count observations:

```
import pymc3 as pm
import arviz as az
import numpy as np
default_sample_parameters = {
'draws':1000,
'tune':1000,
'chains':6,
'cores':6,
'return_inferencedata':True,
'progressbar':True,
'target_accept':0.95
}
def fit_multinomial(observed, sample_parameters=None, hdi_prob=0.95,
lshape=None):
if sample_parameters is None:
sample_parameters = {}
sample_parameters = dict(default_sample_parameters, **sample_parameters)
nrepeats, ncategories = observed.shape
if lshape is None:
lshape = (nrepeats)
with pm.Model() as model:
p0 = pm.Dirichlet("p0", a=np.ones(ncategories))[None,:]
n0 = np.sum(observed, axis=1)[:,None]
pm.Multinomial(f"initial_sample", n=n0, p=p0, shape=lshape,
observed=observed)
trace = pm.sample(**sample_parameters)
with model:
log_p0 = pm.Deterministic("log_p0", pm.math.log(p0))
ppt = pm.sample_posterior_predictive(trace, var_names=["log_p0"],
keep_size=True)
summary = az.summary(ppt, var_names=["log_p0"],
hdi_prob=hdi_prob, skipna=True)
return summary
observed = np.array(
[[388156, 24806, 35, 6540, 4952, 186, 2381, 5482,
27, 112, 172, 196, 10050, 9847, 284, 14606,
322, 758, 330, 221, 12, 1057, 12, 5,
112, 298, 563, 78, 480, 17, 782, 139,
20, 4552, 940, 4689, 7462, 596, 804, 7919],
[375116, 25604, 7, 8479, 3496, 116, 2054, 4539,
338, 124, 171, 511, 12027, 12498, 275, 8833,
816, 781, 647, 347, 5, 633, 13, 11,
119, 216, 882, 149, 611, 10, 795, 126,
3, 4091, 1315, 4456, 8873, 531, 942, 19440],
[378762, 17261, 37, 8731, 4920, 273, 2228, 4673,
258, 93, 268, 820, 13556, 13737, 116, 14917,
698, 625, 451, 328, 13, 974, 14, 10,
193, 119, 789, 109, 719, 12, 1187, 86,
24, 5416, 980, 5262, 7077, 544, 610, 13110],
[378996, 17881, 53, 8014, 4439, 218, 2744, 4601,
0, 120, 264, 411, 11867, 10576, 547, 11806,
1073, 1190, 788, 430, 19, 766, 24, 15,
118, 202, 610, 184, 591, 16, 922, 95,
38, 4625, 1064, 5943, 9623, 632, 844, 17651],
[372024, 21671, 29, 6377, 3952, 253, 3054, 4753,
156, 170, 148, 843, 5277, 13510, 469, 13860,
528, 822, 510, 375, 15, 1051, 9, 8,
196, 234, 1270, 259, 559, 26, 1008, 93,
7, 6050, 1202, 5209, 11087, 396, 1085, 21455],
[373764, 19591, 34, 9503, 4824, 209, 2843, 7107,
114, 153, 254, 596, 9898, 10053, 489, 14993,
797, 1021, 710, 303, 11, 781, 28, 2,
64, 235, 1200, 135, 770, 16, 996, 208,
28, 4613, 1151, 5190, 10907, 623, 832, 14954]])
nrepeats, ncategories = observed.shape
lshape = (nrepeats)
observed_proportions = observed/np.sum(observed, axis=1)[:, None]
summary=\
fit_multinomial(observed, lshape=lshape)
```

The first shape argument I supplied was lshape = (nrepeats, ncategories) and the second one I tried was (nrepeats). Initially I expected them to be the same with implicit broadcasting but they came out quite different for low count observations. The data I am supplying above does not fit this model exactly, I know that. So I was not expecting the model to work very well. But somehow when lshape=(nrepeats) it works quite well (when I scatter plot fitted log_p0 vs np.log(np.nanmean(observed_proportions,axis=0)) whenever well defined). So this made me realize I may not be understanding how the shape parameter is handled and I dont want to take lshape=nrepeats without understanding why it comes out better. I still suspect the first version is the more correct one because I would expect some discrepancy for lower count observations. So my question is how is each shape argument handled in the likelihood? I have also seen that with stupd shape arguments like (nrepeats, nrepeats) the model runs eventhough with quite bad results for r values and such. So this makes me think that I donβt understand at all how the shape argument is handled internally. Thanks

ps: In the following page, in the first model, they also use something that is anologous to (nrepeats, ncategories):

dirichlet mixtures of multinomials