Information transfer that characterizes the information feature variation can have a crucial impact on big data analytics and processing. Actually, the measure for information transfer can reflect the system change from the statistics by using the variable distributions, similar to Kullback-Leibler (KL) divergence and Renyi divergence. Furthermore, to some degree, small probability events may carry the most important part of the total message in an information transfer of big data. Therefore, it is significant to propose an information transfer measure with respect to the message importance from the viewpoint of small probability events. In this paper, we present the message importance transfer measure (MITM) and analyze its performance and applications in three aspects. First, we discuss the robustness of MITM by using it to measuring information distance. Then, we present a message importance transfer capacity by resorting to the MITM and give an upper bound for the information transfer process with disturbance. Finally, we apply the MITM to discuss the queue length selection, which is the fundamental problem of caching operation on mobile edge computing.

This paper focuses on the problem of finding a particular data recommendation strategy based on the user preference and a system expected revenue. To this end, we formulate this problem as an optimization by designing the recommendation mechanism as close to the user behavior as possible with a certain revenue constraint. In fact, the optimal recommendation distribution is the one that is the closest to the utility distribution in the sense of relative entropy and satisfies expected revenue. We show that the optimal recommendation distribution follows the same form as the message importance measure (MIM) if the target revenue is reasonable, i.e., neither too small nor too large. Therefore, the optimal recommendation distribution can be regarded as the normalized MIM, where the parameter, called importance coefficient, presents the concern of the system and switches the attention of the system over data sets with different occurring probability. By adjusting the importance coefficient, our MIM based framework of data recommendation can then be applied to systems with various system requirements and data distributions. Therefore, the obtained results illustrate the physical meaning of MIM from the data recommendation perspective and validate the rationality of MIM in one aspect.

Different probabilities of events attract different attention in many scenarios such as anomaly detection and security systems. To characterize the events’ importance from a probabilistic perspective, the message importance measure (MIM) is proposed as a kind of semantics analysis tool. Similar to Shannon entropy, the MIM has its special function in information representation, in which the parameter of MIM plays a vital role. Actually, the parameter dominates the properties of MIM, based on which the MIM has three work regions where this measure can be used flexibly for different goals. When the parameter is positive but not large enough, the MIM not only provides a new viewpoint for information processing but also has some similarities with Shannon entropy in the information compression and transmission. In this regard, this paper first constructs a system model with message importance measure and proposes the message importance loss to enrich the information processing strategies. Moreover, the message importance loss capacity is proposed to measure the information importance harvest in a transmission. Furthermore, the message importance distortion function is discussed to give an upper bound of information compression based on the MIM. Additionally, the bitrate transmission constrained by the message importance loss is investigated to broaden the scope for Shannon information theory.

This paper mainly focuses on the problem of lossy compression storage based on the data value that represents the subjective assessment of users when the storage size is still not enough after the conventional lossless data compression. To this end, we transform this problem to an optimization, which pursues the least importance-weighted reconstruction error in data reconstruction within limited total storage size, where the importance is adopted to characterize the data value from the viewpoint of users. Based on it, this paper puts forward an optimal allocation strategy in the storage of digital data by the exponential distortion measurement, which can make rational use of all the storage space. In fact, the theoretical results show that it is a kind of restrictive water-filling. It also characterizes the trade-off between the relative weighted reconstruction error and the available storage size. Consequently, if a relatively small part of total data value is allowed to lose, this strategy will improve the performance of data compression. Furthermore, this paper also presents that both the users’ preferences and the special characteristics of data distribution can trigger the small-probability event scenarios where only a fraction of data can cover the vast majority of users’ interests. Whether it is for one of the reasons above, the data with highly clustered message importance is beneficial to compression storage. In contrast, from the perspective of optimal storage space allocation based on data value, the data with a uniform information distribution is incompressible, which is consistent with that in the information theory.

Rare events attract more attention and interests in many scenarios of big data such as anomaly detection and security systems. To characterize the rare events importance from probabilistic perspective, the message importance measure (MIM) is proposed as a kind of semantics analysis tool. Similar to Shannon entropy, the MIM has its special functional on information processing, in which the parameter $\varpi$ of MIM plays a vital role. Actually, the parameter $\varpi$ dominates the properties of MIM, based on which the MIM has three work regions where the corresponding parameters satisfy $ 0 \le \varpi \le 2/\max\{p(x_i)\}$, $\varpi > 2/\max\{p(x_i)\}$ and $\varpi < 0$ respectively. Furthermore, in the case $ 0 \le \varpi \le 2/\max\{p(x_i)\}$, there are some similarity between the MIM and Shannon entropy in the information compression and transmission, which provide a new viewpoint for information theory. This paper first constructs a system model with message importance measure and proposes the message importance loss to enrich the information processing strategies. Moreover, we propose the message importance loss capacity to measure the information importance harvest in a transmission. Furthermore, the message importance distortion function is presented to give an upper bound of information compression based on message importance measure. Additionally, the bitrate transmission constrained by the message importance loss is investigated to broaden the scope for Shannon information theory.

Information collection is a fundamental problem in big data, where the size of sampling sets plays a very important role. This work considers the information collection process by taking message importance into account. Similar to differential entropy, we define differential message importance measure (DMIM) as a measure of message importance for continuous random variable. It is proved that the change of DMIM can describe the gap between the distribution of a set of sample values and a theoretical distribution. In fact, the deviation of DMIM is equivalent to Kolmogorov-Smirnov statistic, but it offers a new way to characterize the distribution goodness-of-fit. Numerical results show some basic properties of DMIM and the accuracy of the proposed approximate values. Furthermore, it is also obtained that the empirical distribution approaches the real distribution with decreasing of the DMIM deviation, which contributes to the selection of suitable sampling points in actual system.