Small Markov models problem

Hi everyone.

Assume that I have a Markov model. There are two states a, and b.
Assume that I know the long term state probabilities p(a) and p(b), and
that these are also the probability distribution for the initial state.
Given that I know these, I can describe my model with the
probabilities:

p(a_i & a_i-1 ) = p1
p(a_i & b_i-1 ) = p2
p(b_i & a_i-1 ) = p3
p(b_i & b_i-1 ) = p4

since p(a|b) = p(a & b ) / p( b )

Now, let's say that I have a different Markov model which has exactly
the same initial and long term state probabilities p(a) and p(b), but
potentially different conditional probabilities. I can describe this
model as:

p2(a_i & a_i-1 ) = p5
p2(a_i & b_i-1 ) = p6
p2(b_i & a_i-1 ) = p7
p2(b_i & b_i-1 ) = p8

OK, I'm having a go here at asking my question in a simple form,
simpler than my real problem. I can create a new Markov model which is
a weighted mixture of the two models. For example, assuming that I want
"0.7" of the first model, and "0.3" of the second model, then I can
define the model:

p3(a_i & a_i-1) = 0.7 * p1 + 0.3 * p5
p3(a_i & b_i-1) = 0.7 * p2 + 0.3 * p6
p3(b_i & a_i-1) = 0.7 * p3 + 0.3 * p7
p3(b_i & b_i-1) = 0.7 * p4 + 0.3 * p8

Trivial algebra will show that the third network will have the same
long term state probabilities p(a) and p(b) for any combination of the
two original networks where the proportions add up to 1.0.

Now, my real problem is this: My actual models are two extremes. One is
calculated with an independence assumption between two variables in
underlying details which I haven't explained. The second is calculated
with an assumption that these same two variables are equivalent. I
would like to claim that the "intermediate" models show varying degrees
of dependence between the two variables, from 0 (full independence) to
1.0 (full dependence/equivalence).

However, given the seat of the pants solution I have, I cannot label
the intermediate states exactly. If I have a 0.7, 0.3 mix as above,
then what do I call that? Does the simple weighted sum of the
probabilities give plausible intermediate states between full
dependence and full independence? Are the details I give woefully
inadequate to answer the question.

In terms of what I call the intermediate networks, should I just take
the networks I create, calculate the mutual information between the two
variables, and use that on the x axis of my graph?

Cheers,

Ross-c

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