In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy. Named after Claude Shannon, the source coding theorem shows that (in the limit, as the length of a stream of independent and identically-distributed random variable (i.i.d.) data tends to infinity) it is impossible to compress the data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss. The source coding theorem for symbol codes places an upper and a lower bound on the minimal possible expected length of codewords as a function of the entropy of the input word (which is viewed as a random variable) and of the size of the target alphabet. Statements Source coding is a mapping from (a sequence of) symbols from an information source to a sequence of alphabet symbols (usually bits) such that the source symbols can be exactly recovered from the binary bits (lossless source coding) or recovered within some distortion (lossy source coding). This is the concept behind data compression. Source coding theorem In information theory, the source coding theorem (Shannon 1948) informally states that (MacKay 2003, pg. 81, Cover 2006, Chapter 5): i.i.d. random variables each with entropy can be compressed into more than bits with negligible risk of information loss, as ; but conversely, if they are compressed into fewer than bits it is virtually certain that information will be lost.The coded sequence represents the compressed message in a biunivocal way, under the assumption that the decoder knows the source. From a practical point of view, this hypothesis is not always true. Consequently, when the entropy encoding is applied the transmitted message is . Usually, the information that characterizes the source is inserted at the beginning of the transmitted message. Source coding theorem for symbol codes Let denote two finite alphabets and let and denote the set of all finite words from those alphabets (respectively). Suppose that is a random variable taking values in and let be a uniquely decodable code from to where . Let denote the random variable given by the length of codeword . If is optimal in the sense that