Formalization » Weak-ly Update

As everyone in who has shared an office with me knows, I spend a lot of time thinking about inverse problems. In particular, statistical inverse problems. I think this is important because fundamentally statistics is an inverse problem. Observe the following very general statistical problem. Suppose ${\mathcal{T}}$ is some separable Banach space. Suppose ${\Theta \subseteq \mathcal{T}}$ is a model space. Let ${(\phi_j) \subset \mathcal{T}^*}$ be a sequence of elements in the dual space of continuous linear functionals on ${\mathcal{T}}$ . Lastly, suppose ${n \in \mathbb{N}}$ , ${\sigma >0}$ is a positive number, and ${W}$ is a random variable defined on ${\mathbb{R}^n}$ . Then we can define the statisical problem as a mapping ${(\theta,\sigma,n) \mapsto \mathbb{P}_{\theta,\sigma,\phi}}$ where the observations are ${\phi_j(\theta) + \sigma W_j}$ for each ${1 \leq j \leq n}$ or equivalently ${\phi(\theta) + \sigma W}$ . Hence, our goal is to make some inference about ${\Theta}$ given an observation ${Y \sim \mathbb{P}_{\theta,\sigma,\phi}}$ .

Generally speaking, there is usually another layer of formalization that encapsulates what we mean by `some inference.’ In particular, suppose we have a space ${\mathcal{B}}$ and define a set of mappings ${\mathcal{G} = \{g : \mathcal{\Theta} \rightarrow \mathcal{B} \}}$ . Then, we consider the elements of ${\Theta}$ as models and either the space ${\mathcal{G}}$ ${\mathcal{B}}$ as the parameter space.

Now, the goal is to make a model ${\theta \in \Theta}$ , a set of functionals ${(\phi_j) \subset \mathcal{T}^*}$ , a parameter of interest ${g \in \mathcal{G}}$ and an observation ${Y \sim \mathbb{P}_{\theta,\sigma,\phi}}$ and develop some estimator ${\hat{g}}$ that is a ${\mathbb{P}_{\theta,\sigma,\phi}}$ - measureable function from ${\mathbb{R}^n}$ to ${\mathcal{B}}$ .

Statistics as an Inverse Problem

Inverse Problem Formalization (Working)

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