#### 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 observationc_{k}, definethe quality of the informationprovided byo_{ku}for distinguishingh_{j}fromh_{i}(givenb·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 callh_{j}outcome-compatiblewithh_{i}on evidence streamc^{k}just whenfor each possible outcome sequencee^{k}ofc^{k}, ifP[e^{k}|h_{i}·b·c^{k}] > 0, thenP[e^{k}|h_{j}·b·c^{k}] > 0.

We also adopt the following convention:

IfP[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 outcomeo_{ku}has 0 probability of occurring givenh_{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= 1EQI[ 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= 1EQI[ 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= 1EQI[ c_{k}|h_{i}/h_{j}|b·c^{k−1}].