#### Supplement to Inductive Logic

## Proof that the EQI for c^{n} is the sum of EQI for the individual c_{k}

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
‘(c^{k−1}·e^{k−1})’
may be dropped, which gives the theorem stated in the text. If neither
*independence condition* holds, all occurrences of
‘c_{k}·(c^{k−1}·e^{k−1})’
are replaced by
‘c^{n}·e^{k−1}’
and occurrences of
‘b·c^{k−1}’ are
replaced by
‘b·c^{n}’.

Definition:QI — the Quality of the Information.

For each experiment or observation c_{k}, definethe quality of the informationprovided by o_{ku}for distinguishing h_{j}from h_{i}(given b·c_{k}, relative to the past evidence (c^{k−1}·e^{k−1})), as follows:QI[o_{ku}| h_{i}/h_{j}| b·c_{k}·(c^{k−1}·e^{k−1})] = log[P[o_{ku}| h_{i}·b·c_{k}·(c^{k−1}·e^{k−1})] / P[o_{ku}| h_{j}·b·c_{k}·(c^{k−1}·e^{k−1})]].Similarly, define:

QI[e^{n}| h_{i}/h_{j}| b·c^{n}] = log[P[e^{n}| h_{i}·b·c^{n}] / P[e^{n}| h_{j}·b·c^{n}]].

Definition:EQI — the Expected Quality of the Information.

Let's call h_{j}outcome-compatiblewith h_{i}on evidence stream c^{k}just whenfor each possible outcome sequence e^{k}of c^{k}, if P[e^{k}| h_{i}·b·c^{k}] > 0, then P[e^{k}| h_{j}·b·c^{k}] > 0.

We also adopt the following convention:

If P[o_{ku}| h_{j}·b·c_{k}·(c^{k−1}·e^{k−1})] = 0, then the term QI[o_{ku}| h_{i}/h_{j}| b·c_{k}·(c^{k−1}·e^{k−1})] · P[o_{ku}| h_{i}·b·c_{k}·(c^{k−1}·e^{ k−1})] = 0, since the outcome o_{ku}has 0 probability of occurring given h_{i}·b·c_{k}·(c^{k−1}·e^{k−1}).

Now, for h_{j} *outcome-compatible* with
h_{i} on c^{k}, we define

EQI[c_{k}| h_{i}/h_{j}| b·(c^{k−1}·e^{k−1})] = ∑_{u}QI[o_{ku}| h_{i}/h_{j}| b·c_{k}·(c^{k−1}·e^{k−1})]· P[o_{ku}| h_{i}·b·c_{k}·(c^{k−1}·e^{k−1})];

and define

EQI[c_{k}| h_{i}/h_{j}| b·c^{k−1}] = ∑_{{ek−1}}EQI[c_{k}| h_{i}/h_{j}| b·(c^{k−1}·e^{k−1})] · P[e^{k−1}| h_{i}·b·c^{k−1}].

Also define

EQI[c^{n}| h_{i}/h_{j}| b] = ∑_{{en}}QI[e^{n}| h_{i}/h_{j}| b] · P[e^{n}| h_{i}·b·c^{n}].

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

Theorem: The EQI Decomposition Theorem:

EQI[c^{n}| h_{i}/h_{j}| b]

=n

∑

k=1

EQI[c_{k}|h_{i}/h_{j}| b·c^{k−1}].

Proof:EQI[c

^{n}| h_{i}/h_{j}| b]

= ∑ _{{en}}QI[e^{n}| h_{i}/h_{j}| b·c^{n}] · P[e^{n}| h_{i}·b·c^{n}]= ∑ _{{en}}log[P[e^{n}| h_{i}·b·c^{n}]/P[e^{n}| h_{j}·b·c^{n}]] · P[e^{n}| h_{i}·b·c^{n}]= ∑ _{{en−1}}∑_{{en}}(log[P[e_{n}| h_{i}·b·c_{n}·(c^{n−1}·e^{n−1})]/P[e_{n}| h_{j}·b·c_{n}·(c^{n−1}·e^{n−1})]] + log[P[e^{n−1}| h_{i}·b·c^{n−1}]/P[e^{n−1}| h_{j}·b·c^{n−1}]]) ·

P[e_{n}| h_{i}·b·c_{n}·(c^{n−1}·e^{n−1})] · P[e^{n−1}| h_{i}·b·c^{n−1}]= (∑ _{{en−1}}∑_{{en}}log[P[e_{n}| h_{i}·b·c_{n}·(c^{n−1}·e^{n−1})]/P[e_{n}| h_{j}·b·c_{n}·(c^{n−1}·e^{n−1})]] · P[e_{n}| h_{i}·b·c_{n}·(c^{n−1}·e^{n−1})] · P[e^{n−1}| h_{i}·b·c^{n−1}]) + (∑_{{en−1}}log[P[e^{n−1}| h_{i}·b·c^{n−1}]/P[e^{n−1}| h_{j}·b·c^{n−1}]] · P[e^{n−1}| h_{i}·b·c^{n−1}] · ∑_{{en}}P[e_{n}| h_{i}·b·c_{n}·(c^{n−1}·e^{n−1})])= EQI[c _{n}| h_{i}/h_{j}| b·c^{n−1}] + EQI[c^{n−1}| h_{i}/h_{j}| b]= …

=

n

∑

k=1

EQI[c_{k}| h_{i}/h_{j}| b·c^{k−1}].

The *average expected quality of information* then
becomes:

Definition:The Average Expected Quality of Information

EQI[c ^{n }| h_{i}/h_{j}| b]= EQI[c ^{n}| h_{i}/h_{j}| b] ÷ n

=

(1/n) ·n

∑

k=1

EQI[c_{k}| h_{i}/h_{j}| b·c^{k−1}].