# Some questions regarding independence requirements in the conditional likelihood

For iid random variables $$Y_1,\dots,Y_n$$ from a random sample that are mutually independent with pdf $$f(y|\theta)$$, my book defines the likelihood as $$L(y_1,\dots,y_n|\theta)= f(y_1,\dots,y_n|\theta) = f(y_1|\theta)\dots f(y_n|\theta)$$. For a random sample of the form $$(X_1, Y_1) \dots (X_n,Y_n)$$ with conditional densities $$f(y|x,\theta)$$, I've seen other sources define the conditional likelihoods as $$\Pi_{i=1}^{n} f(y_i|x_i,\theta)$$.

Based on how the likelihood is defined in my book, I interpret the conditional likelihood as a joint probability distribution $$f(y_1,\dots,y_n|\theta,x_1,\dots,x_n)$$ (written as $$L(\theta; y|x)$$ in some sources). If this is correct, is requiring that $$Y_1,\dots,Y_n$$ be mutually independent sufficient to define the likelihood as $$\Pi_{i=1}^{n} f(y_i|x_i, \theta)$$? I thought for the product be valid, $$f(y_i|x_i, \theta)$$ must equal $$f(y_i|x_1, \dots, x_n, \theta)$$ which implies that $$Y_i$$ and $$X_1,\dots,X_{i-1},X_{i+1},\dots,X_n$$ are conditionally independent given $$X_i$$? i.e. that $$f(y_i,x_1,\dots,x_{i-1},x_{i+1},x_n|x_i)=f(y_i|x_i)f(x_1,\dots,x_{i-1},x_{i+1},x_n|x_i)$$.

I haven't been able to find a source that explicitly state that $$Y_i$$ and $$X_1,\dots,X_{i-1},X_{i+1},\dots,X_n$$ be conditionally independent given $$X_i$$; is this implied or am I misunderstanding something?

Normally, iid assumption is for the pairs, not just outcome or feature RVs, i.e. $$(x_1,y_1)...(x_n, y_n)$$ are iid pairs in a dataset. This may not be explicitly stated in some sources. This assumption directly means $$Y_i$$ and $$X_{-i}$$, i.e. variables other than $$X_i$$, are conditionally independent given $$X_i$$.
• Based on the link, iid for pairs assumes $p(x_1,y_1,\dots,x_n,y_n)=f(x_1,y_1)\dots f(x_n,y_n)$ ($p$ and $f$ being joint densities)? If so, does this assumption mean that $Y_i$ and $X_{-i}$ are mutually independent and also that all the $X$'s are mutually independent from each other, which is stronger than just assuming $Y_i$ and $X_{-i}$ are conditionally independent given $X_i$ per this answer? Apr 5, 2020 at 16:51