How close is the expected length of Huffman coding and entropy?
$begingroup$
If I want to use entropy as an approximation for the expected length of Huffman coding, how good is the approximation? I know the following identity:
$H(X)leq L(X)< H(X)+1 $
where expected length is:
$L(X)= sum_i p_il_i$
and entropy is:
$H(X)= sum_i p_ilog_2(1/p_i)$
But it's not tight enough, or in other words, helpful enough in some cases. Is there a way to approximate the residual using $p_i$, like what we do in Taylor series?
information-theory entropy
edited Jan 6 at 17:21
leonbloy
41.7k647108
asked Jan 6 at 3:27
John AoJohn Ao
12
$endgroup$
add a comment |
$begingroup$
If I want to use entropy as an approximation for the expected length of Huffman coding, how good is the approximation? I know the following identity:
$H(X)leq L(X)< H(X)+1 $
where expected length is:
$L(X)= sum_i p_il_i$
and entropy is:
$H(X)= sum_i p_ilog_2(1/p_i)$
But it's not tight enough, or in other words, helpful enough in some cases. Is there a way to approximate the residual using $p_i$, like what we do in Taylor series?
information-theory entropy
edited Jan 6 at 17:21
leonbloy
41.7k647108
asked Jan 6 at 3:27
John AoJohn Ao
12
$endgroup$
1
$begingroup$
"But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p approx 0$...
$endgroup$
– leonbloy
Jan 6 at 14:09
add a comment |
$begingroup$
If I want to use entropy as an approximation for the expected length of Huffman coding, how good is the approximation? I know the following identity:
$H(X)leq L(X)< H(X)+1 $
where expected length is:
$L(X)= sum_i p_il_i$
and entropy is:
$H(X)= sum_i p_ilog_2(1/p_i)$
But it's not tight enough, or in other words, helpful enough in some cases. Is there a way to approximate the residual using $p_i$, like what we do in Taylor series?
information-theory entropy
edited Jan 6 at 17:21
leonbloy
41.7k647108
asked Jan 6 at 3:27
John AoJohn Ao
12
$endgroup$
If I want to use entropy as an approximation for the expected length of Huffman coding, how good is the approximation? I know the following identity:
$H(X)leq L(X)< H(X)+1 $
where expected length is:
$L(X)= sum_i p_il_i$
and entropy is:
$H(X)= sum_i p_ilog_2(1/p_i)$
But it's not tight enough, or in other words, helpful enough in some cases. Is there a way to approximate the residual using $p_i$, like what we do in Taylor series?
information-theory entropy
information-theory entropy
edited Jan 6 at 17:21
leonbloy
41.7k647108
asked Jan 6 at 3:27
John AoJohn Ao
12
edited Jan 6 at 17:21
leonbloy
41.7k647108
asked Jan 6 at 3:27
John AoJohn Ao
12
edited Jan 6 at 17:21
leonbloy
41.7k647108
edited Jan 6 at 17:21
leonbloy
41.7k647108
edited Jan 6 at 17:21
leonbloy
41.7k647108
41.7k647108
asked Jan 6 at 3:27
John AoJohn Ao
12
asked Jan 6 at 3:27
John AoJohn Ao
12
asked Jan 6 at 3:27
John AoJohn Ao
12
12
1
$begingroup$
"But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p approx 0$...
$endgroup$
– leonbloy
Jan 6 at 14:09
add a comment |
1
$begingroup$
"But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p approx 0$...
$endgroup$
– leonbloy
Jan 6 at 14:09
1
1
$begingroup$
"But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p approx 0$...
$endgroup$
– leonbloy
Jan 6 at 14:09
$begingroup$
"But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p approx 0$...
$endgroup$
– leonbloy
Jan 6 at 14:09
add a comment |
0
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$begingroup$
"But it's not tight enough". Well, in some sense it is. Take for example a Bernoulli variable with $p approx 0$...
$endgroup$
– leonbloy
Jan 6 at 14:09