In this section we assume that we know the tree model of the actual source but not its parameters. Can we find a good coding distribution for this case?
In principle this problem is quite simple. Since we know the model we
can for each source symbol determine its suffix. All symbols that
correspond to the same suffix 
form a memoryless subsequence
whose statistic is determined by an unknown parameter 
.
For this subsequence we simply use the KT-estimator.
The estimated probability of the entire source sequence is the product
of the KT-estimates for the subsequences and hence by
(8), we obtain
Example:
Figure 3: A sequence, its subsequences, and the past.
Let 
. The estimated probability of the
sequence 0100110 given the past 
(see figure
3) is 
, where 
,
, and 
are the probabilities of the
subsequences 11, 00, and 010, corresponding to the leaves 00,
10, and 1 respectively.
Again, what is the redundancy of this estimated distribution?
Consider a sequence 
with subsequence corresponding to leaf s
having 
zeroes and 
ones.
Then, using (14) we obtain
where we again used lemma 1 in [7] for bounding the
parameter redundancies for all the subsequences.
Convexity is used to obtain a bound on the sum of the logarithms in terms
of the 
function which is defined as
Note that (15) holds for all and all 
for .
Example:Let and T=1024, then the codeword is never
longer than the ideal codeword length plus
bit.
