# What can be estimated? Identifiability, estimability, causal inference and ill-posed inverse problems

arXiv: https://arxiv.org/abs/1904.02826 · HTML: https://mlsanity.com/paper/1904.02826/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2019.

## Built on by (most cited first, 11 of 11)

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- [DRCFS: Doubly Robust Causal Feature Selection](https://mlsanity.com/paper/2306.07024/) (2023)
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- [Identification of Average Causal Effects in Confounded Additive Noise Models](https://mlsanity.com/paper/2407.10014/) (2024)
- [Generating Hypotheses of Dynamic Causal Graphs in Neuroscience: Leveraging Generative Factor Models of Observed Time Series](https://mlsanity.com/paper/2505.20697/) (2025)
- [Deep Learning to Estimate Permeability using Geophysical Data](https://mlsanity.com/paper/2110.10077/) (2021)
- [Manually Selecting The Data Function for Supervised Learning of small datasets](https://mlsanity.com/paper/2303.03894/) (2023)
- [A tutorial on the Bayesian statistical approach to inverse problems](https://mlsanity.com/paper/2304.07610/) (2023)
- [Neural Networks with Causal Graph Constraints: A New Approach for Treatment Effects Estimation](https://mlsanity.com/paper/2404.12238/) (2024)
- [From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven models](https://mlsanity.com/paper/2606.08956/) (2026)
- [Two Layers of Instability in Causal Estimation](https://mlsanity.com/paper/2606.21185/) (2026)

## Abstract (arXiv)

We consider basic conceptual questions concerning the relationship between statistical estimation and causal inference. Firstly, we show how to translate causal inference problems into an abstract statistical formalism without requiring any structure beyond an arbitrarily-indexed family of probability models. The formalism is simple but can incorporate a variety of causal modelling frameworks, including 'structural causal models', but also models expressed in terms of, e.g., differential equations. We focus primarily on the structural/graphical causal modelling literature, however. Secondly, we consider the extent to which causal and statistical concerns can be cleanly separated, examining the fundamental question: 'What can be estimated from data?'. We call this the problem of estimability. We approach this by analysing a standard formal definition of 'can be estimated' commonly adopted in the causal inference literature -- identifiability -- in our abstract statistical formalism. We use elementary category theory to show that identifiability implies the existence of a Fisher-consistent estimator, but also show that this estimator may be discontinuous, and thus unstable, in general. This difficulty arises because the causal inference problem is, in general, an ill-posed inverse problem. Inverse problems have three conditions which must be satisfied to be considered well-posed: existence, uniqueness, and stability of solutions. Here identifiability corresponds to the question of uniqueness; in contrast, we take estimability to mean satisfaction of all three conditions, i.e. well-posedness. Lack of stability implies that naive translation of a causally identifiable quantity into an achievable statistical estimation target may prove impossible. Our article is primarily expository and aimed at unifying ideas from multiple fields, though we provide new constructions and proofs.