Context-specific independencies in hierarchical multinomial marginal models

Abstract

Marginal or conditional independencies are well known relationships among variables involved in a contingency table. In this paper we handle with categorical (ordinal) variables and we focus on the (in)dependence relationships under this marginal and conditional perspective in addition to context-specific point of view. The last statement concerns independencies holding only in a subspace of the outcome space. We take advantage from the Hierarchical Multinomial Marginal models environment and we provide several original results about the representation of context-specific independencies through these models. An application about the innovation degree of the Italian enterprises is provided.

Context-specific independencies in stratified chain regression graphical models

“…The following lemma and corollary are needed to simplify the proofs of Theorem 4.1 and Theorem 4.4. Nicolussi, F. and Cazzaro, M. (2019) proved these results, whatever is the type of logits chosen.…”

“…Furthermore, this simplification allows us to see that the constraints are not linear dependent easily. See also Nicolussi, F. and Cazzaro, M. (2019) for further details. However, some constraints may imply more reliable conditions, when the addition of constraints for a CSI implies new conditional independence.…”

“…The constraints (iii ) and (iv ) come from the linear constraints on HMM parameters that satisfy the CSIs (see La Rocca, L. and Roverato, A. (2017); Nicolussi, F. and Cazzaro, M. (2019)). Indeed, given a CSI as < A, B|C; K l × I C\l >, if the variables in the vector X C are based on the baseline logit, then the constraints on the HMM parameters are:…”

“…This new formulation leads to a new constraint on the regression parameters. In particular, by following Nicolussi, F. and Cazzaro, M. (2019), we have that whatever the chosen aggregation criterion, the CSIs in formula ( 14) are represented by the following constraints on the HMM parameters in formula (3):…”

“…First, given an independence like < A, B|C >, the probability distribution of X ABC obeys the independence if, and only if, the HMM parameters η M abc = 0, where a ⊆ A, b ⊆ B, c ⊆ C, a, b = ∅ and M is any subset of V , see Bergsma, W. P. and Rudas, T. (2002). Second, given a generic parameter η M L (i L ), the choice of the unspecified category of the variable X j with j ∈ M\L is arbitrary and we set equal to the first category without loss of generality, see Nicolussi, F. and Cazzaro, M. (2019). Finally, in force of the consideration in the proof of Theorem 4.2, all the parameters η M vc are null if v ⊆ T h and c ⊆ pre(T h )\(pa T (T h )).…”