# Can the sum of two conditional probability distributions give a joint probability distribution?

A paper describing symmetric SNE for the conditional probability distribution

$$p_{j|i}=\frac{e^{-\left|x_i-x_j\right|^2/2\sigma_i}}{\displaystyle\sum_{k\neq i}e^{-\left|x_k-x_i\right|^2/2\sigma_i}}$$

says we can define

the joint probabilities $$p_{ij}$$ in the high-dimensional space to be the symmetrized conditional probabilities, that is, we set $$p_{ij}=\frac{p_{j|i}+p_{i|j}}{2n}$$.

However, every resource I can find on joint probability distributions shows them as a product of two other distributions, not their sum.

Can the sum of two conditional probability distributions generally produce a joint probability distribution, or is it some quirky feature of this particular conditional probability distribution??

The language and notation of the paper may be causing confusion here. Note that as defined on page 2581, $$p_{i|j}$$ in this paper does not mean "the probability of event $$i$$, conditional on event $$j$$". Instead, it's defined as:
The similarity of datapoint $$x_j$$ to datapoint $$x_i$$ is the conditional probability, $$p_{j|i}$$, that $$x_i$$ would pick $$x_j$$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at $$x_i$$ [with some chosen variance]