# Incrementally compute the conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase or not; and I want to update the current conditional entropy value each time we have a new element c.

Basically, I want to know if it is possible to have an incremental expression for this conditional entropy, like it is the case for example for the mean which can be computed by: $\bar X_n = n^{-1}[X_n + (n-1)\bar X_{n-1}]$

• Conditional entropy formula 1: $$H(C|K) = - \sum_{k=1}^{|K|} \sum_{c=1}^{|C|} \frac{a_{ck}}{N} log \frac{a_{ck}}{\sum_{c=1}^{|C|} a_{ck}}$$ $$H(C) = - \sum_{c=1}^{|C|} \frac{\sum_{k=1}^{|K|} a_{ck}}{N} log \frac{\sum_{k=1}^{|K|} a_{ck}}{N}$$

• Conditional entropy formula 2: $$H(K|C) = - \sum_{c=1}^{|C|} \sum_{k=1}^{|K|} \frac{a_{ck}}{N} log \frac{a_{ck}}{\sum_{k=1}^{|K|} a_{ck}}$$ $$H(K) = - \sum_{k=1}^{|K|} \frac{\sum_{c=1}^{|C|} a_{ck}}{N} log \frac{\sum_{c=1}^{|C|} a_{ck}}{N}$$

Note: $a_{ck}$ may refer to something like a distance between elements c and k, or the number of elements of type c that are in k, or something like that ... and $N$ is $\sum_{k=1}^{|K|} \sum_{c=1}^{|C|} a_{ck}$

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