I have an expression like $P(C|SR)$, and I know $S$ and $R$ are conditionally independent given $C$ (EDIT they are also independent given $C^{\complement}$). I want to simplify it into a certain form.
It must be expressed solely in terms of 3 other quantities
- $P(S|C)/P(S|C^{\complement})$ (call this $a$)
- $P(R|C)/P(R|C^{\complement})$ (call this $b$)
- $P(C)$ (call this $c$)
I spent a while on it and was surprised to find that this was possible. I proved it below.
My question: is it possible to do the same for $P(C|SR^{\complement})$? (i.e. represent it in terms of $a$,$b$, & $c$) My intuition says yes, but I have been laboring under this for a while.
I included my attempt below, but I get stuck because I can't simplify the $b$ terms. It seems that there is no way to express $\frac{1 - P(R|C^{\complement})}{1 - P(R|C)}$ in terms of $\frac{P(R|C)}{P(R|C^{\complement})}$ alone, but perhaps it can be combined with the other terms in some way to cancel the issue?
To give some background (perhaps not relevant to answer the question), the reason I want to express these quantities in terms of $a$ and $b$ is because they come from the Cornfield Inequalities. I am trying to understand the minimal amount of assumptions needed to come up with a confounder. I need to be able to express $P(C|SR)$, $P(C|SR^{\complement})$, $P(C|S^{\complement}R)$ and $P(C|S^{\complement}R^{\complement})$ in terms of these few assumptions so that I can sample from them to create a new dataset that includes the confounder.
It is not generally possible to express $P(C|S,R^c)$ in terms of only $a,b,c$ since, as you have shown, the quantity depends on
$$\frac{P(R^c|C)}{P(R^c|C^c)}=\frac{1-P(R|C)}{1-P(R|C^c)}=\frac{\frac{1}{P(R|C^c)}-b}{\frac{1}{P(R|C^c)}-1},$$
which is not uniquely determined by $b$ (unless $b=1$). We see it is expressible in terms of $a,b,c,$ and $P(R|C^c).$
One thing to point out: in your LOTP step, it appears you assume conditional independence of $R,S$ given $C^c$. This is an added assumption that is not implied by conditional independence of $R,S$ given $C$ (see here).