# A Note on the Estimation Method of Intervention Effects based on Statistical Decision Theory

## TL;DR

This paper formulates the estimation of intervention effects within statistical decision theory, deriving optimal Bayesian methods for cases where the causal diagram is known or unknown, and demonstrates their effectiveness via simulations.

## Contribution

It introduces a novel Bayesian decision-theoretic approach to estimate intervention effects, improving upon traditional methods by deriving optimal estimators.

## Key findings

- Proposes Bayesian optimal estimation methods for intervention effects.
- Demonstrates improved estimation accuracy through numerical simulations.
- Addresses both known and unknown causal diagram scenarios.

## Abstract

In this paper, we deal with the problem of estimating the intervention effect in the statistical causal analysis using the structural equation model and the causal diagram. The intervention effect is defined as a causal effect on the response variable $Y$ when the causal variable $X$ is fixed to a certain value by an external operation and is defined based on the causal diagram. The intervention effect is defined as a function of the probability distributions in the causal diagram, however, generally these probability distributions are unknown, so it is required to estimate them from data. In other words, the steps of the estimation of the intervention effect using the causal diagram are as follows: 1. Estimate the causal diagram from the data, 2. Estimate the probability distributions in the causal diagram from the data, 3. Calculate the intervention effect. However, if the problem of estimating the intervention effect is formulated in the statistical decision theory framework, estimation with this procedure is not necessarily optimal. In this study, we formulate the problem of estimating the intervention effect for the two cases, the case where the causal diagram is known and the case where it is unknown, in the framework of statistical decision theory and derive the optimal decision method under the Bayesian criterion. We show the effectiveness of the proposed method through numerical simulations.