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We define a combination to be "**ratio-respected**" if it satisfies the following property: 

If there is a pair $(x,y)$ in the common support of $p_1, p_2$,  such that 
$$ \frac{p_2(x)}{p_1(x)}  =  \frac{p_2(y)}{p_1(y)}, $$
then for any intermediate distribution $p_{12}$, the density cannot distinguish $x$ and $y$, 

$$p_{12}(x)= p_{12}(y). $$


## Most common operators are indeed ratio-respected

First, since the second equation holds for any intermediate distribution, it will hold for the two endpoints $p_1$ and $p_2$. That is 

$$ p_{1}(x)= p_{1}(y), ~~ p_{2}(x)= p_{2}(y)$$

This is a strong condition, and we could verify that most combination operations will satisfy this property.

**Mixture**
$$p_{\lambda}(x)= \lambda p_1(x) + (1-\lambda )p_2(x).$$
Hence 
$$\frac{p_{\lambda}(x)}{p_{\lambda}(y)}= \frac{\lambda p_1(x) + (1-\lambda )p_2(x)}{\lambda p_1(y) + (1-\lambda )p_2(y)}$$
Using the fact that $p_{1}(x)= p_{1}(y), p_{2}(x)= p_{2}(y)$, this ratio is 1.

**Geometric bridge**
$$p_{\lambda}(x)= \frac{1}{z(\lambda)}  p_1^{\lambda}(x) p_2^{1-\lambda}(x).$$
Hence 
$$\frac{p_{\lambda}(x)}{p_{\lambda}(y)}= \frac{ \frac{1}{z(\lambda)}  p_1^{\lambda}(x) p_2^{1-\lambda}(x) }{ \frac{1}{z(\lambda)}  p_1^{\lambda}(y) p_2^{1-\lambda}(y)}$$
Using the fact that $p_{1}(x)= p_{1}(y), p_{2}(x)= p_{2}(y)$, this ratio is 1.



## Situation when an operator is not ratio-respected
1. **mixture with extra fixed bias terms**
$$p_{\lambda}(x) = 0.5( \lambda p_1(x) + (1-\lambda )p_2(x)) + 0.5 p_3(x).$$
2. **R.V. Transport.** 
Consider a joint distribution on $z_1, z_2$, such that marginally $\mathcal{L}(z_1)= p_1$ and  $\mathcal{L}(z_2)= p_2$. Then the law of  $\lambda z_1+  (1-\lambda) z_2$ is a valid bridge between $p_1$ and $p_2$. This bridge generally does not "respect ratio". 
To see this, consider a counterexample: both $p_1$ and $p_2$ are two symmetric densities on R and their means are 0. Then as long $x=-y$, we have $p_1(x)= p_1(y)$ and  $p_2(x)= p_1(y)$. If the intermediate r.v. $0.5 z_1 + 0.5 z_2$ satisfies the "ratio-respected" condition, then its law is also symmetric around 0. However, the sum of two symmetric random variables is generally not symmetric.



