# Generalize bivariate conditional PDF to n-variate conditional PDF

This is an extension of my earlier question here Generalize 1 dimensional conditional PDF to n dimensional conditional PDF

Let there be the continuous random variables $$X$$ and $$Y$$.

Let there be the continuous random variables $$X_1$$, $$X_2$$, $$Y_1$$, and $$Y_2$$. This second set of random variables are distributed like...

$$X_1 \sim X$$

$$X_2 \sim X$$

$$Y_1 \sim Y$$

$$Y_2 \sim Y$$

$$Y_1 \text{ independent of } Y_2$$

$$X_1 \text{ independent of } Y_2$$

$$X_2 \text{ independent of } Y_1$$

We are given $$f_{X_1, X_2}(x_1, x_2 | Y_1 = y_1, Y_2 = y_2)$$. This conditional PDF is symmetric such that... $$f_{X_1, X_2}(x_1, x_2 | Y_1 = y_1, Y_2 = y_2) = f_{X_1, X_2}(x_2, x_1 | Y_1 = y_2, Y_2 = y_1)$$

Now let there also be the random variables $$X_3$$, $$X_4$$, ..., $$X_n$$ as well as $$Y_3$$, $$Y_4$$, ..., $$Y_n$$.

Like before... $$X_i \sim X$$ $$Y_i \sim Y$$

$$Y_i \text{ independent of } Y_j \text{ where } i \ne j$$

$$X_i \text{ independent of } Y_j \text{ where } i \ne j$$

We assume that all pairs $$X_i \text{ and } X_j \text{ (where } i \ne j \text{)}$$ move together according to a conditional PDF $$f_{X_i, X_j}(x_i, x_j | Y_i = y_i, Y_j = y_j)$$ with the same shape as $$f_{X_1, X_2}(x_1, x_2 | Y_1 = y_1, Y_2 = y_2)$$. In other words... $$\text{Given } i \ne j \text{ then } f_{X_i, X_j}(x_0, x_1 | Y_i = y_0, Y_j = y_1) = f_{X_1, X_2}(x_0, x_1 | Y_1 = y_0, Y_2 = y_1)$$

How can we write the n-variate case $$f_{X_1, X_2, ..., X_n}(x_1, x_2, ..., x_n | Y_1 = y_1, Y_2 = y_2, ..., Y_n = y_n)$$ in terms of the bivariate conditional PDF $$f_{X_i, X_j}(x_i, x_j | Y_i = y_i, Y_j = y_j)$$ and $$n$$?