My question is primarily a math question about the Multinomial to Poisson transformation,

referring to a post about how to use the multinominal distribution in brms.

See: https://github.com/paul-buerkner/brms/issues/338#issuecomment-369760462

The math based on:

https://stats.stackexchange.com/questions/105282/logistic-multinomial-regression-as-two-multiple-poisson-regressions

\log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j

is transformed to:

\log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^*

and the brms fit by Paul is:

`fit2 <- brm(y ~ key + key:x + (1 | obs), df_long, family = poisson())`

Since \alpha_j = \alpha^*_j − \alpha^*_J

it follows: \alpha_J = 0 (the reference group).

\eta is the intercept in fit2, and `(1 | obs)`

refering to \theta_i.

Is this model identifiable? Why do we need \theta_i?

The brms fit:

```
Family: poisson
Links: mu = log
Formula: y ~ key + key:x + (1 | obs)
Data: df_long (Number of observations: 300)
Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup samples = 4000
ICs: LOO = NA; WAIC = NA; R2 = NA
Group-Level Effects:
~obs (Number of levels: 100)
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sd(Intercept) 0.02 0.02 0.00 0.06 2184 1.00
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept 1.57 0.06 1.44 1.69 2509 1.00
keyy2 1.18 0.07 1.04 1.33 2655 1.00
keyy3 1.66 0.07 1.52 1.79 2653 1.00
keyy1:x 0.77 0.08 0.62 0.93 2886 1.00
keyy2:x -0.36 0.06 -0.46 -0.25 4000 1.00
keyy3:x -0.85 0.05 -0.95 -0.75 4000 1.00
```

Considering glm:

fit3 <- glm(y ~ key + key:x, data=df_long, family = poisson)

summary(fit3)

```
Call:
glm(formula = y ~ key + key:x, family = poisson, data = df_long)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.11253 -0.65366 0.01812 0.60466 2.06867
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.56862 0.06455 24.302 < 2e-16 ***
keyy2 1.17994 0.07380 15.988 < 2e-16 ***
keyy3 1.65664 0.07044 23.519 < 2e-16 ***
keyy1:x 0.77511 0.07801 9.936 < 2e-16 ***
keyy2:x -0.35613 0.05575 -6.387 1.69e-10 ***
keyy3:x -0.85128 0.05155 -16.515 < 2e-16 ***
```

and the multinom:

```
multinom(formula = Y ~ x, data = df)
Coefficients:
(Intercept) x
2 1.179938 -1.131244
3 1.656643 -1.626395
Std. Errors:
(Intercept) x
2 0.07380438 0.09588863
3 0.07043854 0.09350489
```

These fit matches, the slope math:

```
> -0.35613 - 0.77511
[1] -1.13124
> -0.85128 - 0.77511
[1] -1.62639
```

There is no need for \theta_i, so why brms needs the `(1 | obs)`

term in the formula?

This is I don’t understand. Is it a Stan thing?