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In this study, we propose a head-to-head type (H2H-type) inter-personal multimodal Dirichlet mixture (Inter-MDM) by modifying the original Inter-MDM, which is a probabilistic generative model that represents the symbol emergence between two agents as multiagent multimodal categorization. A Metropolis--Hastings method-based naming game based on the Inter-MDM enables two agents to collaboratively perform multimodal categorization and share signs with a solid mathematical foundation of convergence. However, the conventional Inter-MDM presumes a tail-to-tail connection across a latent word variable, causing inflexibility of the further extension of Inter-MDM for modeling a more complex symbol emergence. Therefore, we propose herein a head-to-head type (H2H-type) Inter-MDM that treats a latent word variable as a child node of an internal variable of each agent in the same way as many prior studies of multimodal categorization. On the basis of the H2H-type Inter-MDM, we propose a naming game in the same way as the conventional Inter-MDM. The experimental results show that the H2H-type Inter-MDM yields almost the same performance as the conventional Inter-MDM from the viewpoint of multimodal categorization and sign sharing.