$\newcommand\Om\Omega$Let $(\Om,F,P)$ be a probability space. For some natural $n$, let $A_1,\dots,A_n$ be events, that is, members of the $\sigma$-algebra $F$. For $k\in[n]:=\{1,\dots,n\}$, these events are said to be $k$-independent if
$$P\Big(\bigcap_{j\in J}A_j\Big)=\prod_{j\in J}P(A_j)\quad\text{for}\quad J\in\binom{[n]}{\le k},\tag{1}$$
where $\binom{[n]}{\le k}$ stands for the set of all subsets of cardinality $\le k$ of the set $[n]$.
Introduce the events
\begin{equation*}
C_K:=\Big(\bigcap_{j\in K}A_j\Big)\cap\Big(\bigcap_{j\in[n]\setminus K}(\Om\setminus A_j)\Big)
\end{equation*}
for $K\subseteq[n]$. These events constitute a partition of $\Om$. For $K\subseteq[n]$, let
$$x_K:=P(C_K).$$
So, the $x_K$'s are nonnegative real numbers such that
$$\sum_{K\subseteq[n]}x_K=1.\tag{2}$$
Moreover, clearly for each $J\subseteq[n]$
$$P\Big(\bigcap_{j\in J}A_j\Big)=\sum_{K:\,J\subseteq K\subseteq[n]}x_K.$$
So, equations (1) and (2) expressed in terms of the "variables" $x_K$, together with the inequalities $x_K>0$ for all $K\subseteq[n]$, define a semialgebraic set, say $I_k$, in $\mathbb R^{2^{n}}$. Of the $$N_k:=\sum_{m=0}^k \binom nm$$ equations (1), $1+n$ equations -- corresponding to the sets $J\in\binom{[n]}{\le k}$ of cardinalities $\le1$ -- are trivial: $P(\Om)=1$ for $J=\emptyset$ and $P(A_j)=P(A_j)$ for $J=\{j\}$ with any $j\in[n]$. Thus, of equations (1) and (2), there remain $N_k-(1+n)+1=N_k-n$ supposedly "nontrivial" and "algebraically independent'' equations.
This seems to make it plausible that the dimension of the semialgebraic set $I_k$ is $2^n-(N_k-n)$, that is, the dimension of the containing space minus the number of the "nontrivial" equations.
Is this conjecture indeed true? The cases of $k=2$ and $k=n$ are of particular interest.
