Changes

The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.

The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.

:* The uncertainty reduction of <math>(\log 5 - \log 3)</math> gets a weight of 3.

:* The uncertainty reduction of <math>(\log 5 - \log 3)\!</math> gets a weight of 3.

:* The uncertainty reduction of <math>(\log 5 - \log 2)</math> gets a weight of 2.

:* The uncertainty reduction of <math>(\log 5 - \log 2)\!</math> gets a weight of 2.

Finally, the weighted average of these two reductions is:

Finally, the weighted average of these two reductions is:

: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))</math>

: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))\!</math>

Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as ''n''&nbsp;=&nbsp;''k''₁&nbsp;+&nbsp;''k''₂.

Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as <math>n = k_1 + k_2.\!</math>

{| cellspacing="6"  

{| cellspacing="6"  

| Capacity

| Capacity

| of a channel {"A", "B"} that bears the odds of 60 "A" to 40 "B"

| of a channel {&ldquo;A&rdquo;, &ldquo;B&rdquo;} that bears the odds of 60 &ldquo;A&rdquo; to 40 &ldquo;B&rdquo;

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))</math>

| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))\!</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)</math>

| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)\!</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)</math>

| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)\!</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})</math>

| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})\!</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)</math>

| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)\!</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)</math>

| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)\!</math>

|-

|-

| &nbsp;

| &nbsp;

| <math>=\quad 0.971</math>

| <math>=\quad 0.971\!</math>

|}

|}

* [[Charles Sanders Peirce (Bibliography)]]

* [[Charles Sanders Peirce (Bibliography)]]

* Peirce, C.S. (1867), "Upon Logical Comprehension and Extension", [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].

* Peirce, C.S. (1867), &ldquo;Upon Logical Comprehension and Extension&rdquo;, [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].

==See also==

==See also==