{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Duality between confidence sets and hypothesis tests\n", "\n", "### We will observe $X \\sim \\mathbb P_\\mu$, where $\\mu \\in \\Theta$. \n", "+ $\\Theta$ is known\n", "+ $\\theta \\rightarrow \\mathbb P_\\theta$ is known\n", "+ $\\mu$ is unknown\n", "+ $X$ takes values in $\\mathcal X$.\n", "\n", "(We will ignore issues of measurability here: tacitly assume that for all $\\theta \\in \\Theta$, $A_\\eta$ is $\\mathbb P_\\theta$-measurable\n", "and that $\\mathcal I(X)$ is set-valued $\\mathbb P_\\theta$-measurable function.)\n", "\n", "