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Assume that you're fitting a multinomial regression model with J categories where the contrast between j and the last category J is modeled as $$ \log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j $$ where $X_i$ is a vector of covariates associated with the $i$th case. This is rather a hard optimisation problem because the parameters are coupled by the multinomial's conditioning on $N_i$, the $i$-th case's marginal total.

You can instead fit J separate Poisson regressions with linear predictor of the form $$ \log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^* $$ where each case gets one nuisance parameter $\theta$.

Subtracting the expression for $\log\, \mu_{ij}$ from $\log\, \mu_{iJ}$ to construct $\log (\mu_{ij}/\mu_{iJ})$ shows that the parameters are related as $\alpha_j = \alpha^*_j − \alpha^*_J$ and $\beta_j = \beta^*_j - \beta^*_J$ (the nuisance parameters and $\eta$ cancel). Thus you get what you came for: fitting multiple Poisson regression models gives you the same result as fitting a single more complicated multinomial regression model.

This exposition is essentially the one in Gérman Rodríguez's always excellent notes, in the section on the 'equivalent log-linear model'.

In general this manoeuvre is called the multinomial-Poisson transformation and was written about by Baker in 1994.

Notice also that you don't need to adjust $X$ at all to use it.

Assume that you're fitting a multinomial regression model with J categories where the contrast between j and the last category J is modeled as $$ \log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j $$ where $X_i$ is a vector of covariates associated with the $i$th case. This is rather a hard optimisation problem because the parameters are coupled by the multinomial's conditioning on $N_i$, the $i$-th case's marginal total.

You can instead fit J separate Poisson regressions with linear predictor of the form $$ \log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^* $$ where each case gets one nuisance parameter $\theta$.

Subtracting the expression for $\log\, \mu_{ij}$ from $\log\, \mu_{iJ}$ to construct $\log (\mu_{ij}/\mu_{iJ})$ shows that the parameters are related as $\alpha_j = \alpha^*_j − \alpha^*_J$ and $\beta_j = \beta^*_j - \beta^*_J$ (the nuisance parameters and $\eta$ cancel). Thus you get what you came for: fitting multiple Poisson regression models gives you the same result as fitting a single more complicated multinomial regression model.

This exposition is essentially the one in Gérman Rodríguez's always excellent notes, in the section on the 'equivalent log-linear model'.

In general this manoeuvre is called the multinomial-Poisson transformation and was written about by Baker in 1994.

Notice also that you don't need to adjust $X$ at all to use it.

Assume that you're fitting a multinomial regression model with J categories where the contrast between j and the last category J is modeled as $$ \log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j $$ where $X_i$ is a vector of covariates associated with the $i$th case. This is rather a hard optimisation problem because the parameters are coupled by the multinomial's conditioning on $N_i$, the $i$-th case's marginal total.

You can instead fit J separate Poisson regressions with linear predictor of the form $$ \log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^* $$ where each ase get's one nuisance parameter $\theta$.

Subtracting the expression for $\log\, \mu_{ij}$ from $\log\, \mu_{iJ}$ to construct $\log (\mu_{ij}/\mu_{iJ})$ shows that the parameters are related as $\alpha_j = \alpha^*_j − \alpha^*_J$ and $\beta_j = \beta^*_j - \beta^*_J$ (the nuisance parameters and $\eta$ cancel). Thus you get what you came for: fitting multiple Poisson regression models gives you the same result as fitting a single more complicated multinomial regression model.

This exposition is essentially the one in Gérman Rodríguez's always excellent notes, in the section on the 'equivalent log-linear model'.

In general this manoeuvre is called the multinomial-Poisson transformation and was written about by Baker in 1994.

Notice also that you don't need to adjust $X$ at all to use it.

Assume that you're fitting a multinomial regression model with J categories where the contrast between j and the last category J is modeled as $$ \log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j $$ where $X_i$ is a vector of covariates associated with the $i$th case. This is rather a hard optimisation problem because the parameters are coupled by the multinomial's conditioning on $N_i$, the $i$-th case's marginal total.

You can instead fit J separate Poisson regressions with linear predictor of the form $$ \log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^* $$ where each case gets one nuisance parameter $\theta$.

Subtracting the expression for $\log\, \mu_{ij}$ from $\log\, \mu_{iJ}$ to construct $\log (\mu_{ij}/\mu_{iJ})$ shows that the parameters are related as $\alpha_j = \alpha^*_j − \alpha^*_J$ and $\beta_j = \beta^*_j - \beta^*_J$ (the nuisance parameters and $\eta$ cancel). Thus you get what you came for: fitting multiple Poisson regression models gives you the same result as fitting a single more complicated multinomial regression model.

This exposition is essentially the one in Gérman Rodríguez's always excellent notes, in the section on the 'equivalent log-linear model'.

In general this manoeuvre is called the multinomial-Poisson transformation and was written about by Baker in 1994.

Notice also that you don't need to adjust $X$ at all to use it.

Assume that you're fitting a multinomial regression model with J categories where the contrast between j and the last category J is modeled as $$ log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j $$ where $X_i$ is a vector of covariates associated with the $i$th case. This is rather a hard optimisation problem because the parameters are coupled by the multinomial's conditioning on $N_i$, the $i$-th case's marginal total.

You can instead fit J separate Poisson regressions with linear predictor of the form $$ log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^* $$ where each ase get's one nuisance parameter $\theta$.

Subtracting the expression for $log\, \mu_{ij}$ from $log\, \mu_{iJ}$ to construct $log (\mu_{ij}/\mu_{iJ})$ shows that the parameters are related as $\alpha_j = \alpha^*_j − \alpha^*_J$ and $\beta_j = \beta^*_j - \beta^*_J$ (the nuisance parameters and $\eta$ cancel). Thus you get what you came for: fitting multiple Poisson regression models gives you the same result as fitting a single more complicated multinomial regression model.

This exposition is essentially the one in Gérman Rodríguez's always excellent notes, in the section on the 'equivalent log-linear model'.

In general this manoeuvre is called the multinomial-Poisson transformation and was written about by Baker in 1994.

Notice also that you don't need to adjust X at all to use it.

Assume that you're fitting a multinomial regression model with J categories where the contrast between j and the last category J is modeled as $$ \log \frac{\mu_{ij}}{\mu_{iJ}} = \alpha_j + X_i \beta_j $$ where $X_i$ is a vector of covariates associated with the $i$th case. This is rather a hard optimisation problem because the parameters are coupled by the multinomial's conditioning on $N_i$, the $i$-th case's marginal total.

You can instead fit J separate Poisson regressions with linear predictor of the form $$ \log\, \mu_{ij} = \eta + \theta_i + \alpha_j^* + X_i\beta_j^* $$ where each ase get's one nuisance parameter $\theta$.

Subtracting the expression for $\log\, \mu_{ij}$ from $\log\, \mu_{iJ}$ to construct $\log (\mu_{ij}/\mu_{iJ})$ shows that the parameters are related as $\alpha_j = \alpha^*_j − \alpha^*_J$ and $\beta_j = \beta^*_j - \beta^*_J$ (the nuisance parameters and $\eta$ cancel). Thus you get what you came for: fitting multiple Poisson regression models gives you the same result as fitting a single more complicated multinomial regression model.

This exposition is essentially the one in Gérman Rodríguez's always excellent notes, in the section on the 'equivalent log-linear model'.

In general this manoeuvre is called the multinomial-Poisson transformation and was written about by Baker in 1994.

Notice also that you don't need to adjust $X$ at all to use it.