MMP.19:5 - Archetypal Grounding

These are constructed cases with supplied mechanisms or exact probability laws. They demonstrate the mathematical operations and limits, not empirical effectiveness.

MMP.19:5.1 - Change the route for an already observed processing case

A routing model gives latency in ticks as Y=2A+U. Route A=1 adds two ticks; residual latency U is 0 or 1 with equal probability. The recorded case used A=1 and had Y=3.

Under the factual equation only U=1 is compatible. Keeping that input and replacing the route by A=0 gives Y_0=1. Drawing a fresh U from the population instead gives mean latency 1/2. That is a new-case mean, not the requested alternative for the observed case.

If the record is noisy, use its likelihood rather than selecting U with certainty. If an exact Y=4 is reported under the unchanged model, neither admitted U is compatible; return the conflict instead of calculating a posterior with zero denominator. Whether the same residual condition would persist under a real route change is a subject premise of this model.

MMP.19:5.2 - Expose a coupling that experiments do not determine

Let a randomized binary action A be independent of the response type U=(Y_0,Y_1). Consider two possible type distributions:

Each action succeeds with probability 1/2 in both models. With a fair random assignment, all four observed (A,Y) combinations have probability 1/4 in each model.

For a case observed with A=1,Y=1, only types (0,1) and (1,1) remain. In S, the probability that it would also succeed under A=0 is (3/8)/(1/2)=3/4. In T it is (1/8)/(1/2)=1/4. Both models fit the supplied observational and intervention laws, so these laws do not identify the answer.

For a future choice scored only by expected success, the actions tie at 1/2 under both models. If A=1 adds a positive cost and no other consequence, A=0 is sufficient for that criterion. Resolving the counterfactual ambiguity would not improve this choice.

MMP.19:5.3 - Bound benefit without inventing a joint law

Suppose binary success probabilities under actions 1 and 0 are p1=7/10 and p0=2/5. Let b=P(Y_0=0,Y_1=1), the probability of succeeding only under action 1. The four type masses must be

P(0,1)=b                 P(1,1)=p1-b
P(1,0)=p0-p1+b           P(0,0)=1-p0-b.

Nonnegativity gives max(0,p1-p0) <= b <= min(p1,1-p0), hence 3/10 <= b <= 3/5. At b=3/10 the masses in order (00,01,10,11) are (3/10,3/10,0,2/5); at b=3/5 they are (0,3/5,3/10,1/10). Both attain the supplied marginals, establishing the bounds for this unrestricted response-type class.

Now suppose assignment A is independent of the response pair (Y_0,Y_1), P(A=1)>0, and a case is observed with A=1,Y=1. The probability that this case would fail under action 0 is P(Y_0=0 given A=1,Y=1)=b/p1, hence between 3/7 and 6/7. Conditioning retains the types with Y_1=1; the independent assignment probability cancels from numerator and denominator. The same endpoint distributions attain these conditional bounds, since their denominator is the fixed positive p1=7/10.

The mean effect is p1-p0=3/10 in every compatible model. An expected-success criterion with an action-1 cost of 1/10 therefore has net gain 1/5 without identifying b. A criterion that explicitly requires b to exceed 2/5 remains unsettled by these bounds.

If subject knowledge warrants that action 1 never changes a success into failure, P(1,0)=0 fixes b=3/10. That is an additional monotonicity assumption; it was not learned from the two marginal probabilities.

MMP.19:5.4 - State which mediated contrast is being calculated

Take the supplied mechanisms M=A+U and Y=3A+2M+AM, with U=0 or 1. Compare actions 0 and 1 for the same u. The total change is 6+u.

First retain the mediator at M_0=u while changing A: Y_(1,M_0)-Y_(0,M_0)=3+u. Then change the mediator to M_1=1+u while retaining A=1: Y_(1,M_1)-Y_(1,M_0)=3. The components sum to 6+u.

Reversing that decomposition gives mediator change 2 at A=0, followed by direct change 4+u at M_1. These components also sum to 6+u. The interaction AM makes the two decompositions different. A request for “the part caused through M” must select its intended contrast. Knowing the total effect alone supplies neither decomposition; observing data that identify it is a further question.