IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 5, SEPTEMBER 1997 1549 The Quadratic Gaussian CEO Problem Harish Viswanathan, Student Member, IEEE, and Toby Berger, Fellow, IEEE Abstract— A firm’s CEO employs a team of agents who observe independently corrupted versions of a data sequence Let be the total data rate at which the agents may communicate information about their observations to the CEO. The agents are not allowed to convene. Berger, Zhang, and Viswanathan determinined the asymptotic behavior of the minimal error frequency in the limit as and tend to infinity for the case in which the source and observations are discrete and memoryless. We consider the same multiterminal source coding problem when is independent and identically distributed (i.i.d.) Gaussian random variable corrupted by inde- pendent Gaussian noise. We study, under quadratic distortion, the rate–distortion tradeoff in the limit as and tend to infinity. As in the discrete case, there is a significant loss between the cases when the agents are allowed to convene and when they are not. As , if the agents may pool their data before communicating with the CEO, the distortion decays exponentially with the total rate ; this corresponds to the distortion-rate function for an i.i.d. Gaussian source. However, for the case in which they are not permitted to convene, we establish that the distortion decays asymptotically only as . Index Terms— Decentralized estimation, Fisher information, Gaussian source, mean-squared error, multiterminal source cod- ing. I. INTRODUCTION A N IMPORTANT class of problems involves observing events in space–time and making estimates or predictions based on the phenomena observed. Often the sensor–estimator system is distributed in the sense that data are collected at several spatially separated sites and have to be transmitted to a central estimator/decision maker over communication channels of limited capacity. The following problem in mul- titerminal source coding was introduced in [6]. A firm’s CEO is interested in a data sequence which cannot be observed directly. The CEO employs a team of agents who observe independently corrupted versions of Let be the total data rate at which the agents may communicate information about their observations to the CEO. The agents are not allowed to convene. In [6] and [7], Berger, Zhang, and Viswanathan determine the asymptotic behavior of the minimal error frequency in the limit as and tend to infinity. Their result is for the case in which the source and observations are discrete and memoryless. One would also like to solve the CEO problem for the cases in which the source and observations take values in arbitrary alphabets. In this paper we consider a special case of Manuscript received July 3, 1996; revised February 2, 1997. The material in this paper was presented at the 1995 IEEE International Symposium on Information Theory, Whistler, BC, Canada, September 1995. The authors are with the School of Electrical Enginnering, Cornell Univer- sity, Ithaca, NY 14853 USA. Publisher Item Identifier S 0018-9448(97)05012-8. the continuous source and observations problem. We assume that the source is an independent and identically distributed (i.i.d.) sequence of zero-mean Gaussian random variables and the observations are corrupted by identical independent memoryless Gaussian noise The CEO is interested in reconstructing the source with minimum mean-squared error. We study the asymptotic behavior of the minimum achievable distortion in the limit as first and then tends to infinity. That is, we study the behavior of In most information theory problems the solution for a continuous alphabet does not differ significantly from that for a discrete alphabet. This is not so in the case of the CEO problem. In the discrete alphabet version the average distortion decays exponentially in the limit of large total rate, but in the Gaussian case the distortion exhibits a dependence on the total rate. This difference may be explained as follows. Whereas most rate–distortion problems are not solved via hypothesis testing or parameter estimation, the CEO problem is, especially asymptotically in the limit of a large number of agents. Hence, the inherent difference between parameter estimation and hypothesis testing manifests itself in a different rate–distortion tradeoff in the CEO problem for continuous as opposed to discrete sources. Our main result is that the distortion decays at best as , where is a constant. We also derive the upper bound These results should be contrasted with the fact that, if the agents were allowed to convene before communicating to the CEO, they could smooth out their noisy observations and achieve a rate–distortion performance corresponding to that of the Gaussian source , i.e., the distortion would decay as Thus there is a significant performance degradation in the isolated agents case. The quadratic Gaussian CEO problem also provides strong connections between information theory and statistics. Among these is that we use the Cramer–Rao bound for random parameter estimation when lower-bounding the achievable distortion. Such interesting connections have also appeared in the investigation of the multiterminal estimation problem in- troduced in [3] and studied in [17], [10], [1], and [2]. The CEO problem is also related to the multisensor data fusion problem. Spatial and physical limitations often mean that only partial information can be provided by a single sensor. Multisensor systems aim to overcome such shortcomings of single-sensor systems through redundancy, diversity, and complimentarity in the information sensed by employing multiple sensors. The interested reader is referred to [12] and the references cited therein. 0018–9448/97$10.00  1997 IEEE