Optimal worst-risk minimization in structural equation models with random coefficients
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916338383978496 |
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| author | Kennerberg, Philip Wit, Ernst |
| author_facet | Kennerberg, Philip Wit, Ernst |
| contents | The insight that causal parameters are particularly suitable for out-of-sample prediction has sparked a lot development of causal-like predictors. However, the connection with strict causal targets, has limited the development with good risk minimization properties, but without a direct causal interpretation. In this manuscript we derive the optimal out-of-sample risk minimizing predictor of a certain target $Y$ in a non-linear system $(X,Y)$ that has been trained in several within-sample environments. We consider data from an observation environment, and several shifted environments. Each environment corresponds to a structural equation model (SEM), with random coefficients and with its own shift and noise vector, both in $L^2$. Unlike previous approaches, we also allow shifts in the target value. We define a sieve of out-of-sample environments, consisting of all shifts $\tilde{A}$ that are at most $γ$ times as strong as any weighted average of the observed shift vectors. For each $β\in\mathbb{R}^p$ we show that the supremum of the risk functions $R_{\tilde{A}}(β)$ has a worst-risk decomposition into a (positive) non-linear combination of risk functions, depending on $γ$. We then define the set $\mathcal{B}_γ$, as minimizers of this risk. The main result of the paper is that there is a unique minimizer ($|\mathcal{B}_γ|=1$) that can be consistently estimated by an explicit estimator, outside a set of zero Lebesgue measure in the parameter space. A practical obstacle for the initial method of estimation is that it involves the solution of a general degree polynomials. Therefore, we prove that an approximate estimator using the bisection method is also consistent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15350 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal worst-risk minimization in structural equation models with random coefficients Kennerberg, Philip Wit, Ernst Statistics Theory Probability 62F10 The insight that causal parameters are particularly suitable for out-of-sample prediction has sparked a lot development of causal-like predictors. However, the connection with strict causal targets, has limited the development with good risk minimization properties, but without a direct causal interpretation. In this manuscript we derive the optimal out-of-sample risk minimizing predictor of a certain target $Y$ in a non-linear system $(X,Y)$ that has been trained in several within-sample environments. We consider data from an observation environment, and several shifted environments. Each environment corresponds to a structural equation model (SEM), with random coefficients and with its own shift and noise vector, both in $L^2$. Unlike previous approaches, we also allow shifts in the target value. We define a sieve of out-of-sample environments, consisting of all shifts $\tilde{A}$ that are at most $γ$ times as strong as any weighted average of the observed shift vectors. For each $β\in\mathbb{R}^p$ we show that the supremum of the risk functions $R_{\tilde{A}}(β)$ has a worst-risk decomposition into a (positive) non-linear combination of risk functions, depending on $γ$. We then define the set $\mathcal{B}_γ$, as minimizers of this risk. The main result of the paper is that there is a unique minimizer ($|\mathcal{B}_γ|=1$) that can be consistently estimated by an explicit estimator, outside a set of zero Lebesgue measure in the parameter space. A practical obstacle for the initial method of estimation is that it involves the solution of a general degree polynomials. Therefore, we prove that an approximate estimator using the bisection method is also consistent. |
| title | Optimal worst-risk minimization in structural equation models with random coefficients |
| topic | Statistics Theory Probability 62F10 |
| url | https://arxiv.org/abs/2307.15350 |