  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.
