C.28.MR:5 - Archetypal Grounding

C.28.MR:5.1 - Observing an alarm and forcing its output

Let H mean binary high load and S mean a binary alarm. Take independent uniform inputs U_H,U_S on [0,1] and the supplied mechanisms

H = 1[U_H < 0.5]

S = 1[U_S < 0.1 + 0.8 H].

Here 1[condition] is one when the condition holds and zero otherwise. In high-load cases, the alarm sounds with probability 0.9. In low-load cases, it sounds with probability 0.1.

The observational probability of no alarm is 0.5*0.1 + 0.5*0.9 = 0.5. High load with no alarm has probability 0.5*0.1 = 0.05. Thus P(H=1 | S=0)=0.05/0.5=0.1: the observation changes what we infer about the load.

For do(S=0), replace the second mechanism by S=0. The equation for H and the input law are retained. Therefore P(H=1 | do(S=0))=0.5. Forcing this indicator removes its information about H without changing H in the supplied model.

The result answers a current-load question under the stated absence of an S-to-H influence. If alarm output controls subsequent cooling, a later-load query needs that later mechanism and the duration of the override. The current calculation remains usable for the current question; the new temporal question has an additional required input.

C.28.MR:5.2 - Change a display or a supply command

Consider an ideal regulated supply and a resistive load. C is the command in volts, V the delivered voltage, D the displayed voltage, R a fixed resistance, I the load current and W its power. In the operating region being modeled,

C=12 V; V=C; D=V; R=6 ohm; I=V/R; W=V I.

The natural result is I=2 A and W=24 W.

Replacing the display equation by D=6 V leaves V=12 V; current and power remain 2 A and 24 W. Replacing the command equation by C=6 V instead gives V=6 V, I=1 A and W=6 W. The two changes share a numeral but replace different mechanisms.

Suppose the supply’s current limit is now relevant. Use the revised positive-command model V=min(C, I_max R) with I_max=1.5 A. The 12 V command now gives 9 V at the load and 13.5 W. The 6 V command still gives 6 W. Relative to natural behavior, the calculated power reduction is now 7.5 W, instead of 18 W in the earlier operating-region model.

This is a model comparison. Using this result for a physical supply requires an account of its regulation, load and limiting behavior. Altering a graphical interface value is a physical intervention only when the interface actually controls the modeled command.

C.28.MR:5.3 - A definite intervention answer with an indefinite baseline

Take real-valued variables with mechanisms X=Y and Y=X. Every equal pair satisfies the baseline equations. Without another condition, the model does not determine the baseline value of Y.

Replace the first mechanism by X=1. The retained equation now gives Y=1. That intervention consequence is determined. A difference from the baseline remains undetermined because the baseline’s value or distribution has not been supplied.

If the requested result is only Y under the intervention, return one. If the requested result is its change from natural behavior, return the missing baseline condition. Resolving the first question does not require inventing that condition.