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In the studies on symbol emergence and emergent communication in a populationof agents, a computational model was employed in which agents participate invarious language games. Among these, the Metropolis-Hastings naming game (MHNG)possesses a notable mathematical property: symbol emergence through MHNG isproven to be a decentralized Bayesian inference of representations shared bythe agents. However, the previously proposed MHNG is limited to a two-agentscenario. This paper extends MHNG to an N-agent scenario. The maincontributions of this paper are twofold: (1) we propose the recursiveMetropolis-Hastings naming game (RMHNG) as an N-agent version of MHNG anddemonstrate that RMHNG is an approximate Bayesian inference method for theposterior distribution over a latent variable shared by agents, similar toMHNG; and (2) we empirically evaluate the performance of RMHNG on synthetic andreal image data, enabling multiple agents to develop and share a symbol system.Furthermore, we introduce two types of approximations -- one-sample andlimited-length -- to reduce computational complexity while maintaining theability to explain communication in a population of agents. The experimentalfindings showcased the efficacy of RMHNG as a decentralized Bayesian inferencefor approximating the posterior distribution concerning latent variables, whichare jointly shared among agents, akin to MHNG. Moreover, the utilization ofRMHNG elucidated the agents' capacity to exchange symbols. Furthermore, thestudy discovered that even the computationally simplified version of RMHNGcould enable symbols to emerge among the agents.

Recursive Metropolis-Hastings Naming Game: Symbol Emergence in a Multi-agent System based on Probabilistic Generative Models