Balancing Suspense and Surprise: Timely Decision Making with Endogenous Information Acquisition
We develop a Bayesian model for decision-making under time pressure with endogenous information acquisition. In our model, the decision-maker decides when to observe (costly) information by sampling an underlying continuoustime stochastic process (time series) that conveys information about the potential occurrence/non-occurrence of an adverse event which will terminate the decisionmaking process. In her attempt to predict the occurrence of the adverse event, the decision-maker follows a policy that determines when to acquire information from the time series (continuation), and when to stop acquiring information and make a final prediction (stopping). We show that the optimal policy has a "rendezvous" structure, i.e. a structure in which whenever a new information sample is gathered from the time series, the optimal "date" for acquiring the next sample becomes computable. The optimal interval between two information samples balances a trade-off between the decision maker's "surprise", i.e. the drift in her posterior belief after observing new information, and "suspense", i.e. the probability that the adverse event occurs in the time interval between two information samples. Moreover, we characterize the continuation and stopping regions in the decisionmaker's state-space, and show that they depend not only on the decision-maker's beliefs, but also on the "context", i.e. the current realization of the time series.
This paper proposes an interesting model of decision making in a situation where the sensory observation depends on the occurrence of the adverse event, i.e., the deadline. The subject needs to infer the deadline, i.e., occurrence of adverse event, based on the history of sensory information.  The model demonstrates how to compute the optimal “date” for gathering the next information by balancing the Bayesian surprise and suspense. The optimal policy, the rendezvous policy, can be obtained using Bellman optimality condition. The papers also shows that the continuation and stopping regions depend not only on subject’s belief, but also the context of the sensory information.  This paper is interesting and  technically sound, though a practical application of this model is not clear.  I expect that this paper should be able to draw certain attention from a small group of researchers in this community. However, I personally think this paper is more suitable for publication in a journal like SICON.
Technical quality: This paper proposes an interesting model and derives three main theorems. The paper demonstrates when the decision maker acquires new information (continuation) and when makes a final prediction (stop) in a scenario where the deadline depends on the sensory information. It also shows that the subject's belief state is a supermartingle, and that the optimal policy has a "rendezvous" structure. The continuation and stopping region in this model depend both on the subject's belief state and "context" of the sensory observations. The paper is interesting and technically sound.   Novelty: Most decision making models for 2AFC and stop signal task assume that the  deadline is independent of the sensory information. The  main reason is because  those cognitive tasks are designed in a particular way such that the sensory information does not convey the information about the occurrence of the deadline. For example, in 2AFC, the subject needs to decide the moving direction of the stimulus within a fixed or random deadline, which does not include any information about the stimulus. This paper proposes an interesting model of decision making in a situation where the sensory observation depends on the occurrence of the adverse event, i.e., the deadline. The model is interesting and has certain novelty.  Potential impact : Although the model is conceptually novel, the structure of cost function is similar to previous Bayesian decision models in 2AFC. The dependence between observation and deadline can also be included in previous Bayesian decision models. From computational point of view, the impact of this paper is limited. It is not clear how we can build a practical algorithm and run simulations based on the result of this paper. Again, I think this paper is more suitable for publication in a theoretical journal because the proof of the theorem is critical and should at least be sketched in the paper.   Clarity and presentation: This paper is well written and easy to follow. The author makes effort in explaining each theorem beside the equations. I would recommend the following improvement: 1. Provide the expected cost function to be minimized to give readers a clearer idea about the dependency between the deadline and observation context. 2. Use different notations for the sigma-algebra generated by the process' survival  and by the stopping event. 3. Check a few typos existing in this paper.  ----------------------- I have read the rebuttal. Like I said in my previous review, I like the main idea of this paper which tries to solve an important problem in decision making. My main concern is still the same. It is not clear how we can build a practical algorithm and run simulations based on the result of this paper. The whole paper is written in the language of probability measure theory and makes it very hard to be understood by the majority of the audience in the NIPS community. I definitely appreciate the theoretical contribution of this paper. However, I highly recommend the author submitting this paper to a more theoretically oriented journal in control theory or probability theory.