Statistical Inference

In this second lecture we will revisit the Hypothesis testing and parameter estimation scenarios discussed in the previous lecture, except we shall be invoking quantum theory. In the case of quantum Hypothesis testing I shall review Holevo’s optimality theorem which establishes the conditions for achieving the maximum probability of success in a hypothesis testing scenario involving quantum states and measurements. We shall also cast these conditions in terms of semi-definite programs and apply them to obtain the optimal measurement (aka the Helstrom measurement) for the case of binary hypothesis testing. We will then consider quantum parameter estimation, review the quantum version of the Cramér-Rao bound and the Quantum Fisher Information (QFI). These concepts will be applied to the simple case of interferometry where we will encounter the Standard and Heisenberg limits in estimation precision.

Statistical inference is an important branch of decision theory and deals with taking optimal decisions in the face of uncertainty. For instance, the decision of whether a new drug combats a given disease or not is taken based on samples taken from medical trials and forms the primary example of classical hypothesis testing. On the other hand, determining the precision in the estimation of a certain physical constant—such as the value of the gravitational constant g—is an example of parameter estimation. In this talk we will go through both these tasks from the point of view of statistical decision theory. In the case of hypothesis testing we will derive the optimal decision rule for minimizing the probability of erroneously identifying the likely hypothesis, as well as several variants of symmetric and asymmetric hypothesis testing (the famous Neyman-Pearson theorem). For parameter estimation we will derive the celebrated Cramér-Rao theorem which sets a lower bound on the precision of any unbiased estimator, and we will connect this to the traditional error propagation formula we all know and love (or hate as the case may be).

Use the following link (https://gloton.ugr.es/owncloud/index.php/s/p6avxB0j8lBz3Qa) to download the notes for this seminar