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Consider a data set \((X_{vi})\) with responses from persons \(v\) to items \(i\)

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The Rasch model is defined by the probabilities \[P(X_{vi}=x|\theta_v=\theta)=\frac{\exp(x(\theta-\beta_i))}{1+\exp(\theta-\beta_i)}\] for dichotomous responses, while the Rasch model for polytomous responses is defined by the probabilities \[P(X_{vi}=h|\theta_v=\theta)=\frac{\exp(h\theta-\beta_{ih}))}{\sum_{k=0}^{m_1}\exp(k\theta-\beta_{ik})}\] with the restriction \(\beta_{i0}=0\)

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