Optimal transport and Wasserstein distances for causal models
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866917712398123008 |
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| author | Cheridito, Patrick Eckstein, Stephan |
| author_facet | Cheridito, Patrick Eckstein, Stephan |
| contents | In this paper, we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph $G$. Different graph structures lead to different specifications of the optimal transport problem. For instance, a fully connected graph yields standard optimal transport, a linear graph structure corresponds to causal optimal transport between the distributions of two discrete-time stochastic processes, and an empty graph leads to a notion of optimal transport related to CO-OT, Gromov-Wasserstein distances and factored OT. We derive different characterizations of $G$-causal transport plans and introduce Wasserstein distances between causal models that respect the underlying graph structure. We show that average treatment effects are continuous with respect to $G$-causal Wasserstein distances and small perturbations of structural causal models lead to small deviations in $G$-causal Wasserstein distance. We also introduce an interpolation between causal models based on $G$-causal Wasserstein distance and compare it to standard Wasserstein interpolation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_14085 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal transport and Wasserstein distances for causal models Cheridito, Patrick Eckstein, Stephan Statistics Theory Optimization and Control Probability In this paper, we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph $G$. Different graph structures lead to different specifications of the optimal transport problem. For instance, a fully connected graph yields standard optimal transport, a linear graph structure corresponds to causal optimal transport between the distributions of two discrete-time stochastic processes, and an empty graph leads to a notion of optimal transport related to CO-OT, Gromov-Wasserstein distances and factored OT. We derive different characterizations of $G$-causal transport plans and introduce Wasserstein distances between causal models that respect the underlying graph structure. We show that average treatment effects are continuous with respect to $G$-causal Wasserstein distances and small perturbations of structural causal models lead to small deviations in $G$-causal Wasserstein distance. We also introduce an interpolation between causal models based on $G$-causal Wasserstein distance and compare it to standard Wasserstein interpolation. |
| title | Optimal transport and Wasserstein distances for causal models |
| topic | Statistics Theory Optimization and Control Probability |
| url | https://arxiv.org/abs/2303.14085 |