Proof that the EQI for cn is the sum of EQI for the individual ck

In order to accomodate the more general result that does not presuppose the independence conditions we must first generalize the definitions of QI and of EQI. In what follows we will treat the case where only condition-independence is assumed. If result-independence holds as well, all occurrences of ‘(ck−1·ek−1)’ may be dropped, which gives the theorem stated in the text. If neither independence condition holds, all occurrences of ‘ck·(ck−1·ek−1)’ are replaced by ‘cn·ek−1’ and occurrences of ‘b·ck−1’ are replaced by ‘b·cn’.

Definition: QI — the Quality of the Information.
For each experiment or observation ck, define the quality of the information provided by oku for distinguishing hj from hi (given b·ck, relative to the past evidence (ck−1·ek−1)), as follows:
QI[oku | hi/hj | b·ck·(ck−1·ek−1)]  =  log[P[oku | hi·b·ck·(ck−1·ek−1)] / P[oku | hj·b·ck·(ck−1·ek−1)]].

Similarly, define:

QI[en | hi/hj | b·cn]  = log[P[en | hi·b·cn] / P[en | hj·b·cn]].

Definition: EQI — the Expected Quality of the Information.
Let's call hj outcome-compatible with hi on evidence stream ck just when for each possible outcome sequence ek of ck, if P[ek | hi·b·ck] > 0, then P[ek | hj·b·ck] > 0.

We also adopt the following convention:

If P[oku | hj·b·ck·(ck−1·ek−1)] = 0, then the term QI[oku | hi/hj | b·ck·(ck−1·ek−1)] · P[oku | hi·b·ck·(ck−1·e k−1)] = 0, since the outcome oku has 0 probability of occurring given hi·b·ck·(ck−1·ek−1).

Now, for hj outcome-compatible with hi on ck, we define

EQI[ck | hi/hj | b·(ck−1·ek−1)]   =  u  QI[oku | hi/hj | b·ck·(ck−1·ek−1)]· P[oku | hi·b·ck·(ck−1·ek−1)];

and define

EQI[ck | hi/hj | b·ck−1]   =   {ek−1} EQI[ck | hi/hj | b·(ck−1·ek−1)] · P[ek−1 | hi·b·ck−1].

Also define

EQI[cn | hi/hj | b] {en} QI[en | hi/hj | b] · P[en | hi·b·cn].

Then we have the following generalization of Equation (16) in the main text:

Theorem: The EQI Decomposition Theorem:
 EQI[cn | hi/hj | b] = n∑k=1 EQI[ck |hi/hj | b·ck−1].

Proof:

EQI[cn | hi/hj | b]

=  {en} QI[en | hi/hj | b·cn] · P[en | hi·b·cn]
=  {en} log[P[en | hi·b·cn]/P[en | hj·b·cn]] · P[en | hi·b·cn]
=  {en−1}{en} (log[P[en | hi·b·cn·(cn−1·en−1)]/P[en | hj·b·cn·(cn−1·en−1)]] + log[P[en−1 | hi·b·cn−1]/P[en−1 | hj·b·cn−1]]) ·
P[en | hi·b·cn·(cn−1·en−1)] · P[en−1 | hi·b·cn−1]
=  (∑{en−1}{en} log[P[en | hi·b·cn·(cn−1·en−1)]/P[en | hj·b·cn·(cn−1·en−1)]] · P[en | hi·b·cn·(cn−1·en−1)] · P[en−1 | hi·b·cn−1]) + (∑{en−1} log[P[en−1 | hi·b·cn−1]/P[en−1 | hj·b·cn−1]] ·  P[en−1 | hi·b·cn−1] · {en} P[en | hi·b·cn·(cn−1·en−1)])
=  EQI[cn | hi/hj | b·cn−1] + EQI[cn−1 | hi/hj | b]
=

=
 n∑k=1 EQI[ck | hi/hj | b·ck−1].

The average expected quality of information then becomes:

Definition: The Average Expected Quality of Information
EQI[c | hi/hj | b] = EQI[cn | hi/hj | b] ÷ n

=
 (1/n) · n∑k=1 EQI[ck | hi/hj | b·ck−1].