To a problem asking to generalize the law of iterated expectations to show that
$$\mathbf E\,[Z\mid X]=\mathbf E\,[\mathbf E\,[Z\mid X,Y]\mid X]$$
, the solution proceeds as follows:
I can't quite grasp why the first underlined product of PMFs is equal to the second underlined PMF. With two variables, one can think visually about marginal and joint PDFs/PMFs as planes (surfaces?) and slices, but not with three or more variables.
Is there a good intuitive way to think about these relationships, or beyond two variables they can only be reasoned about analytically with formulas? Any help or advice would be much appreciated.
The intuition is that we can imagine we are working in the subspace $\{X=x\}.$ Probabilities then relativize to conditional probabilities on $X=x.$ In particular, a probability $P(A\mid Y=y)$ conditional on $\{Y=y\}$ will relativize to $P(A\mid Y=y, X=x).$ Similarly, the definition of conditional probability $A(A\mid B) = \frac{P(A,B)}{P(B)}$will become $P(A\mid B,C)= \frac{P(A,B\mid C) }{P(B\mid C)}$
More formally, if $A,B$ and $C$ are events then $$P(A\mid B\cap C) = \frac{P(A\cap B\cap C)}{P(B\cap C)} = \frac{P(A\cap B\mid C)P(C)}{P(B\mid C)P(C)} = \frac{P(A\cap B\mid C)}{P(B\mid C)},$$ so $$P(A\mid B\cap C) P(B\cap C) = P(A\cap B \mid C).$$ So, just let $C=\{X=x\},$ $B = \{Y=y\}$ and $A=\{Z=z\}.$