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This paper presents a Bayesian inferential framework for estimating joint, conditional, and marginal probabilities in directed acyclic graphs (DAGs) applied to the study of the progression of hospitalized patients with severe influenza. Using data from the PIDIRAC retrospective cohort study in Catalonia, we model patient pathways from admission through different stages of care until discharge, death, or transfer to a long-term care facility. Direct transition probabilities are estimated through a Bayesian approach combining conjugate Dirichlet-multinomial inferential processes, while posterior distributions associated to absorbing state or inverse probabilities are assessed via simulation techniques. Bayesian methodology quantifies uncertainty through posterior distributions, providing insights into disease progression and improving hospital resource planning during seasonal influenza peaks. These results support more effective patient management and decision making in healthcare systems. Keywords: Confirmed influenza hospitalization; Directed acyclic graphs (DAGs); Dirichlet-multinomial Bayesian inferential process; Healthcare decision-making; Transition probabilities.