# What is the justification for the use of this arbitrary post-processing vector?

I am reading through this paper which describes the use of a post-processing vector $$\vec{c}$$ with elements having values of $$(-1,0,1)$$. In equation 3 they give their solution as a linear combination of the measurement probabilities $$\vec{p}$$ with the post-processing vector $$\vec{c}$$. They justify this with the following:

"The measurement results give rise to an outcome probability vector $$\vec{p} = (p_1,...,p_l,...)$$. The desired output might be one of these probabilities $$p_l$$, or it might be some simple function of these probabilities. Hence, we allow for some simple classical post-processing of $$\vec{p}$$ in order to reveal the desired output."

It seems to me that they justify the use of $$\vec{c}$$ because it corrects the sign of the outcome probabilities. However, I am not satisfied with this explanation. I do believe that it is correct, but can someone provide a better justification for the use of this seemingly arbitrary post-processing method? Additionally, do you think there exists a derivation for how they arrive at the specific post-processing vector? Or, have they arrived at it empirically?

• I don't know the exact context, but if you measure a probability vector $\vec p$, and the desired outcome (thinking in a supervised learning context) is some $\vec x_{\rm target}$, then doesn't this amount to solving the linear system $\vec x_{\rm target}=A\vec p$ for a matrix/vector $A$? If you want to learn such relation, you can set up a system of the form $\vec x_k=A\vec p_k$. This is also equivalent to learning an observable (which is nothing but linear post-processing done on observed probabilities)
– glS
Aug 12 '21 at 20:49
• That makes sense in a supervised learning context where you know the desired solution. But what about the case when you don't know the exact solution, how do you determine the post-processing vector $\vec{c}$? Aug 12 '21 at 20:59
• well, you need to have a "desired outcome" specified somehow, no? How do they define it here? What's the precise task they are dealing with?
– glS
Aug 12 '21 at 21:18
• I am not so concerned about their exact problem, rather, I have an outcome probability vector for an entirely different problem which has elements that are correct in magnitude but which has some elements with opposite sign. I can't find an issue with my algorithm so I am looking for a justification to flip some of the signs as was done in the cited paper. It is easy to setup the linear system as you have done, however, I need to be able to justify the fact that the outcome probability is correct but that it requires this post-processing step. Aug 12 '21 at 21:34
• I would say that in general, for the problem to be well-defined, and assuming it is a "learning problem" of some kind, the correct answer oughts to be known or codified in a known way. It would help if you could provide more details on the context/problem setting you are thinking about
– glS
Aug 12 '21 at 21:37