what do these odds ratios represent?

Question

I am reading this article in which is given the matrix of the joint probabilities of two random variables, X=$(x_1,x_2)$ and Y=$(y_1,y_2)$. Let's say they are $(p_{1,1},p_{1,2},p_{2,1},p_{2,2})$. What does exactly the odds ratio $\alpha=\frac{p_{1,1}p_{2,2}}{p_{1,2}p_{2,1}}$ mean? What changes when it is <,>,= 1? In the article two more odds ratios are defined: $\beta=\frac{p_{1,1}p_{1,2}}{p_{2,1},p_{2,2}}$ and $\gamma=\frac{p_{1,1},p_{2,1}}{p_{1,2}p_{2,2}}$. How about these? What are they? Thank you in advance!

Answer 1

These odds are typically used in the context of classification where $Y$ is an evidence and $X$ the real situation. Thus, the result $(X,Y)=(1,1)$ and $(X,Y)=(2,2)$ are desired: the higher value of the odds is preferred. The odds means practically $$ \frac{\textrm{probability of properly classified}}{\textrm{probability of misclassified}} $$ in the other words, how many proper classifications are made for one misclassification. In this context $\beta$ is $$ \frac{\textrm{probability that X=1 occurs}}{\textrm{probability that X=2 occurs}} $$ i.e. how many times $X=1$ occurs more frequently than $X=2$. Similarly, $\gamma$ is $$ \frac{\textrm{probability that Y=1 observed}}{\textrm{probability that Y=2 observed}} $$ i.e. how many times the classification says $Y=1$ more than $Y=2$.