I was reading this paper and the book "Introduction to Quantum State Estimation" by Yong Siah Teo and I am facing some issues trying to understand how the definition of the cost function given in these two references can match each other and how the likelihood function can rise from this definition.
The paper reads:
And the books says:
Although they seem the same, there are some differences that I cannot understand how to connect one with the other. I mean, the equation $(2)$ seems to be the same one as the $1.2.1$ from the book, except for the summation symbol when taking defining the average cost function. This summation symbol is defined later in the book as:
My issues are simply that:
- I cannot see how to connect these two definitions. This summation symbol that is defining "a conditional average over all possible measurement data" does not seem to enter in any way in the definition of the paper, which seems more plausible since minimizing a cost function is just taking the average "over all possible true configurations" which is the integral the author on the paper stated.
- why the likelihood function appears in the average?
- How to connect these two definitions considering this summation (average) with the likelihood?