Ambit Causality: Continuous Markov Blankets in Space–Time

What is Ambit Causality?

Ambit causality is a way to talk about cause and effect in continuous space–time stochastic fields. Instead of discrete nodes and arrows (as in DAG-based causality), we work with regions and supports: measurable parts of space–time where randomness and modulation actually flow. The key idea is that an ambit field’s geometry already encodes a causal structure — and this geometry yields a continuous analogue of a Markov blanket.

Consider an ambit field \[ Y(x,t) = \mu + \int_{A(x,t)} g(x,\xi,t,s)\,\sigma(\xi,s)\,L(d\xi\,ds) + Q(x,t) \] where \(A(x,t)\) is the ambit set (the region that can influence \(Y\)), \(L\) is a Lévy basis (independently scattered noise), \(g\) a deterministic kernel (propagator), \(\sigma\) a volatility field (modulation), and \(Q\) a drift/compensator. With the usual causality constraint \(A(x,t) \subseteq \{ s \leq t \}\), the field cannot depend on its future: acyclicity is baked in.

The Causal Ambit Blanket

For a fixed space–time point \((x,t)\), define the Causal Ambit Blanket (CAB) as the effective subset of the ambit set that transmits influence:

\[ B(x,t) := \operatorname{supp}(g(x,\,\cdot,\,t,\,\cdot)) \cap A(x,t) \]

To capture modulation and time deformation, augment it to \[ B^*(x,t) :=\; B(x,t) \;\cup\; (\operatorname{supp}(\sigma) \cap A(x,t)) \;\cup\; (\operatorname{supp}(T) \cap A(x,t)) \] where \(T\) is a “metatime” (random clock) used in extended subordination. Intuitively, \(g\) plays the role of “parents,” while \(\sigma\) and \(T\) route volatility or time-change — the “spouses/children” in the DAG analogy.

Local Conditional Independence

If the Lévy basis \(L\) is independently scattered and independent of any observation mechanism supported strictly outside \(B^*(x,t)\), and if \(\sigma\) and \(T\) are predictable with respect to the \(\sigma\)-algebra generated by \(L\) on \(B^*(x,t)\), then we obtain a continuous Markov blanket property:

\[ Y(x,t)\; \perp\!\!\!\perp\; O\;\big|\, \sigma(L|_{B^*(x,t)}) \]

In words: conditioning on the noise (and modulation) inside the blanket screens off everything outside. This is the continuous counterpart of d-separation, expressed through regions rather than nodes.

Key Concepts

• Ambit Set \(A(x,t)\): The causal domain of influence. Often a past light-cone–like region in \(\mathbb{R}^d \times \mathbb{R}\).
• Lévy Basis \(L\): An independently scattered random measure that drives the field; disjoint regions carry independent noise.
• Kernel \(g\): A deterministic propagator that shapes how influence travels from \((\xi, s)\) to \((x, t)\).
• Volatility \(\sigma\) and Metatime \(T\): Amplitude and timing modulators that thicken the causal “shield.”
• CAB \(B(x,t)\) and \(B^*(x,t)\): The minimal effective region that, when conditioned on, screens off outside influence.
• Predictability: Non-anticipativity with respect to the filtration generated by the restricted Lévy basis; ensures causal direction.

Interventions on Regions

Ambit models make Pearl-style interventions natural by working directly on regions:

• Noise intervention: \(L \Rightarrow L^{(\mathrm{do})}\) on \(R \subset B(x,t)\) — change the Lévy triplet locally.
• Volatility intervention: \(\sigma \Rightarrow \sigma^{(\mathrm{do})}\) on \(R \subset B^*(x,t)\) — alter amplitude modulation.
• Clock/metatime intervention: \(T \Rightarrow T^{(\mathrm{do})}\) on \(R \subset B^*(x,t)\) — deform the random clock.

This notion of intervention parallels the framework of Sokol & Hansen (2013), who defined post-intervention stochastic differential equations to give a causal semantics to continuous-time systems. Their approach interprets interventions as modifications of SDE coefficients (drift or diffusion), showing that such interventions correspond to limits of discrete structural-equation models. The ambit causality framework generalizes this idea spatially: interventions occur not only “in time” but on measurable regions of space–time where the field receives influence.

Multiscale “Triple-Integral” Blankets

Many systems add a third integration dimension, such as scale or frequency:

\[ Y(x,t) = \int_{s \leq t} \int_{\|\xi-x\| \leq c(t-s)} \int_{0}^\infty g(r)\,k(x,\xi,t,s;r)\,\sigma(\xi,s;r)\,L(d\xi\,ds\,dr) \]

The blanket now has a thickness in space–time–scale. Conditioning on this triple layer (noise + volatility + scale) yields a multiscale causal shield.

Applications and Influence

• Turbulence: Intermittency and energy cascades as multiscale blankets; interventions correspond to localized forcing.
• Finance: Volatility-modulated Volterra processes; the blanket mirrors memory kernels and roughness.
• Neural/Spatial Fields: Continuous-time active inference; blankets define sensory–internal separations in cortex-like media.
• SPDEs: Green’s function supports give causal cones; CAB aligns with domains of dependence.

Ambit Causality in Today’s World

The causal interpretation of SDEs developed by Sokol and Hansen (Annals of Statistics, 2013) showed that stochastic systems can admit well-defined post-intervention dynamics without departing from the Itô–Lévy framework. Ambit causality extends this insight to spatially extended fields: causation is encoded not in coefficients but in regions of space–time where noise and modulation propagate.

As datasets become high-frequency and spatially distributed, a field-based causal language is essential. Ambit causality offers a principled way to learn and manipulate where and how influence flows, enabling region-level interventions in physics, quantitative finance, climate, and neuroscience.

In short: instead of nodes and edges, we reason with regions and supports. Causality becomes geometry; stochastic integration becomes its calculus.