Let be a BN with joint , and let be an intervention. Then
1- is also a BN
And defining the energy function its representing
2- The difference of energy with the initial BN for a configuration is
Each correction is supported on only: the intervention compiles to one factor swap per intervened node, of order , and every energy term outside is bit-identical to the base model. In particular, sampling the Boltzmann distribution of recovers the post-intervention distribution exactly.
Proof
1- By Def. 8, keeps the graph of , and every CPT of is locally normalised — the untouched ones because is well-defined, the replaced ones by the local-normalisation requirement on the kernels . Hence is a well-defined Bayesian Network, and by the chain rule its joint is
Apply Prop. 1 to
Physical representability of infinite barriers is a separate, hardware-level question: it is exactly the zero-entry problem of note 2, and Corollary 3.1 below is one of its treatments. The Boltzmann distribution of the compiled model is, with no clamp, no limit, and no approximation involved.
2- Recall that
Computing the difference of energy and substituting by its expression, we get with values in : where the intervention creates a hard constraint (, e.g. the off-target states of a hard intervention), and where it lifts one ().
Some important special cases
Two special cases from Def. 8 are worth spelling out:
Parent-free kernel: every incoming edge of is inactive, the swapped factor becomes unary, and the incoming edges of vanish from the compiled factor graph. This is graph mutilation realised virtually, by compilation rather than by physical rewiring.
Hard intervention: it compiles like any other and still holds in the ideal model, but the point-mass kernel reintroduces zero entries ( off-target), so a compiled hard intervention inherits the zero-entry treatment of note 2 — -smoothing or , with the usual dynamic-range trade-off. Exactness at finite hardware scale is therefore a property of soft interventions with strictly positive kernels.
Consequences of intervention
Inactive edges and mutilation. Call an edge (with ) inactive under if is constant in . Deleting inactive edges leaves pointwise unchanged, so also factorises over the pruned graph. In particular, a parent-free kernel makes every incoming edge of inactive: graph mutilation is then recovered as a factorisation statement about , not imposed as a separate operation.
Since is again a BN-derived energy model with , all previous results apply verbatim to it: is a free correctness check for the compiled intervention, and Prop. 2 can be reused to clamp evidence on the intervened model, i.e. to estimate .
Corollary — -smoothing as a global soft intervention
The zero-entry treatment of -smoothing described in note 2 can now be recharacterised. It’s not an ad-hoc numerical fix: it is an **instance of Prop. 3 — the intervention of Def. 8, acting on every mechanism at once, whose replacement kernels are the -mixtures of the original CPTs with the uniform distribution on each node’s domain. For every no kernel is a point mass, so is soft.
Corollary 3.1 — -smoothing compiles as a soft intervention
Fix and let be the intervention with . Then:
(Well-formed) each is locally normalised and strictly positive — every entry lies in ;
(Exactness) the compiled energy satisfies and for every, with no infinite entries and no positivity caveat;
(Dynamic range) every energy entry lies in , so the required dynamic range is bounded by ;
(Smoothing bias), where .
Proof
(1) Each is a convex combination of two distributions normalised over for every fixed parent configuration, hence locally normalised. Every entry is bounded below by — in particular each zero CPT entry is lifted to a positive floor.
(2) By (1) the kernels are admissible, so is an intervention in the sense of Def. 8 and Prop. 3 applies with : and . Moreover, since every kernel entry is strictly positive, is finite on the entire state space and the compiled energy contains no infinite barrier.
Fix a configuration , abbreviate and .
(3) By definition of , we have
then, applying
The bound holds in particular for the largest compiled entry, so the required dynamic range is bounded by .
(4) Expand the product over the per-node binary choice (original vs uniform): Each is itself the chain-rule joint of a well-defined BN — the model under the intervention uniformising the mechanisms in — hence a normalised distribution, by Prop. 3 again is a convex mixture of the . Isolating the term (for which ), where is a mixture of the , . Total variation is affine in mixtures against a fixed reference: the last step by Bernoulli’s inequality.
Explanation
interpolates between the base model and maximum entropy. returns ; replaces every mechanism by the uniform kernel, whose joint is uniform over the full state space. Smoothing is a controlled flattening of the energy landscape.
Structure is preserved. Parent sets are untouched, so factor orders are -invariant; only table entries change. Strict positivity additionally puts in the Hammersley–Clifford regime: the smoothed model is a genuine Gibbs/MRF on all of state space.
Intervention vs Clamping
From Prop. 3, intervention semantics are obtained at compile time, not by physical forcing. Contrast with Prop. 2: clamping adds a penalty on top of the original landscape and recovers observational conditioning only in the limit. Intervention compilation instead replaces part of the landscape itself (mechanism replacement), so the sampled distribution equals the target exactly.
A finite-bias clamp is target-directed and approximate at finite ; is target-free and exact at every — the error it introduces is not leakage from an unconverged limit but bias in the target distribution itself: the compiled model samples , not .
The trade-off is now (3) versus (4). Dynamic range grows like while bias shrinks like : the two costs are monotone in opposite directions in the single parameter , and the compiler’s job is to place inside the feasible wedge of the 5 - Finite-Bias Epsilon and Dynamic-Range Analysis. Also, the burden moves from clamp strength to reconfiguration: the backend must accept a rewrite of energy tables between runs.
It explains the -floor under clamping. Evidence clamping applied after smoothing conditions , so no clamp strength can push the distribution-level error below the smoothing bias — the bias lives in the target factors, not in clamp leakage.
Numerical check — clamp vs do on the confounded graph
On the confounding example: the clamped marginal converges to the observational conditional at the exponential rate of Prop. 2 — TVD(clamp, obs) — while its distance to the do-reference flattens at a persistent floor: the confounding bias (TVD ≈ 0.11 here). No clamp strength closes that gap; it is structural (back-door path), not unconverged leakage. Clamping agrees with exactly when the graph already makes conditioning and intervention coincide ( parent-free; mechanism inactive in its parents; no back-door path) — verified on variant networks; the clamp never creates the equivalence.
Finetuning fallacy
TVD(clamp, do) is non-monotone in : on its way from the prior to the conditional, the clamped distribution happens to cross the do-value at one finite (the dip in the right panel). Tuning to that minimum would “reproduce” the intervention by coincidence — the crossing is query- and parameter-specific and carries no interventional semantics. Before convergence, a finite clamp equals neither semantics; apparent agreement with at a tuned proves nothing.
Commentary