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

Question

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 $p_{ij}=\frac{p_{j|i}+p_{i|j}}{2n}$.

[TSNE paper, pg 2584]

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??

Answer 1

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]

Hence, although the notation looks similar to that in the other resources you've linked on joint distributions, it doesn't have the same meaning and those results may not be relevant here.