What does the expression $\frac{P(X=1 \mid Y_1, Y_2) - P(X=1 \mid Y_1)}{P(X=1 \mid Y_1)}$ imply from a probabilistic standpoint?

Question

Suppose we have a binary random variable $X \in \{0,1\}$ and two other random variables $Y_1, Y_2$. Suppose we have the expression below defined as the net change between $a = P\left(X=1 \mid Y_1, Y_2\right)$ and $b = P\left(X=1 \mid Y_1\right)$.

$$ \Delta\left(a, b\right) = \dfrac{P\left(X=1 \mid Y_1, Y_2\right) - P\left(X=1 \mid Y_1\right)}{P\left(X=1 \mid Y_1\right)} $$

I have seen such an expression many times before in the literature and wondering what this means from a probabilistic interpretation. For example, is it the marginal change of conditioning on more information, which is $Y_2$? I don't multiply by 100 so it's technically not a percentage, and so I am not sure how to interpret it correctly, and if there are additional insights to be taken. Thanks!

Answer 1

Let us have a coin toss experiment. $Y_1$ is a binary variable, if the coin is biased towards heads then $Y_1=1$, if the coin is biased towards tails then $Y_1=0$ (the coin is biased towards heads or tails with equal probability, it cannot be fair). Now, We can also assume that the outcome of the toss i.e $X_1$($0$ for tails and $1$ for heads) depends on start position given by $Y_2$. The start position is chosen at random as well. The coin is more likely to land on the same position as start position.The following table describes the probability distribution over all such cases.

\begin{array} {|r|r|}\hline Y_1(Start-position) & Y_2(Bias) & X_1(Outcome) & Probability \\ \hline 0 & 0 & 0 & 0.2 \\ \hline 0 & 0 & 1 & 0.05 \\ \hline 0 & 1 & 0 & 0.075 \\ \hline 0 & 1 & 1 & 0.175 \\ \hline 1 & 0 & 0 & 0.175 \\ \hline 1 & 0 & 1 & 0.075 \\ \hline 1 & 1 & 0 & 0.05 \\ \hline 1 & 1 & 1 & 0.2 \\ \hline \end{array}

Now, lets say you saw a coin toss, and you saw that the start position of the coin was Heads, i.e $Y_1 = 1$,now what is the belief in your head of the coin turning heads ? $$P(X_1 = 1| Y_1=1) = \frac{P(X_1=1,Y_1=1)}{P(Y_1=1)} = 0.55 $$ While the coin is in the air some one shouts that the coin is biased towards tails i.e $Y_2 = 0$, what is your belief about coin turning heads now? $$P(X_1 = 1| Y_1=1,Y_2=0) = \frac{P(X_1 = 1,Y_1=1,Y_2=0)}{P(Y_1=1,Y_2=0)} = 0.3$$ Now what is the relative change in your belief ? A nice way to define it would be following $$\Delta = \frac{Prior Belief - Belief With New Evidence }{Prior Belief}$$

The magnitude of $\Delta$ tells you, how much your belief changed with respect to your prior belief. The sign of $\Delta$ tells you in which direction (positive for stronger belief in the prior belief and negative for weakening of your prior belief). In this case $\Delta = \frac{0.3 - 0.55}{0.55} = - 0.45$

Answer 2

This Δ(,) is a probability taken over two-step of the random experiment. While P(=1∣1,2) is a posteriori probability =(=1∣1) is a a priori probability. Δ(,) is the deviation from one of the quotients of a=P(=1∣1,2) to =(=1∣1). It does not expect the quotient to be smaller than one. This can be a test or indicator for the bias between two steps of the random experiment. Whether 1 or 0 are prefered. Prior refers only to the time before the second step. The second step delivers new evidence has to be verified or proven.