In this paper, we study axiomatic foundations of Bayesian persuasion, where a principal (i.e., sender) delegates the task of choice making after informing a biased agent (i.e., receiver) about the payoff relevant uncertain state (see, e.g., Kamenica and Gentzkow (2011)). Our characterizations involve novel models of Bayesian persuasion, where the principal can steer the agent's bias after acquiring costly information. Importantly, we provide an elicitation method using only observable menu-choice data of the principal, which shows how to construct the principal's subjective costs of acquiring information even when he anticipates managing the agent's bias.