Why if we use independence and factorization, we cannot represent every joint distribution? (rigorous argument needed) I was reading Koller's Probabilistic Graphical models book and it says something like this:

Let $P(x_i) = \theta_i$.
Define:
$$P(x_1, \ldots , x_n) = \prod_{i=1}^n \theta_i$$
This representation is limited, and there are many distribution that
  we cannot capture by choosing values for $\theta_i, \ldots, \theta_n$.
  This fact is obvious not only from intuition, but also from a somewhat
  more formal perspective. The space of all join distributions is a $2^n - 1$
  dimensional subspace of $\mathbb{R}^{2^n}$ - the set $\{ (p_1, \ldots , p_{2^n}) \in \mathbb{R}^{2^n} : p_1 + \cdots + p_{2^n} = 1 \}$ . On
  the other hand, the space of all joint distributions specified in a
  facorized way as in equation above, is an n-dimensional manifold in
  $\mathbb{R}^{2^n}$

The thing that confuses me about this sentence is the last sentence: 


On the other hand, the space of all joint distributions specified in a
    facorized way as in equation above, is an n-dimensional manifold in
    $\mathbb{R}^{2^n}$


I do not understand what it means that the factorized version is an n-dimensional manifold in $\mathbb{R}^{2^n}$. I don't know if its the wording or I don't know what manifold means in this context (not sure if a description of a set would make the point more clear). But, the thing is that the set that it wrote initially describing a joint distribution made sense and I understand it (I think). Its just saying it can pick any vector in $\mathbb{R}^{2^n}$ that satisfies the normalization condition. However, I fail to understand rigorously what the limitation of the factorization version is.
 A: Some pieces of information are missing in the OP, and the notation is not very formal, however I think it is possible to reverse-engineer what's happening. 
You seem to have $n$ random variables each of which takes values over $\{0,1\}$, or alternatively you have $n$ probability distributions on $\{0,1\}$ given by $\theta_1,\dots,\theta_n$. Each such distribution is completely characterized by $q_i := \theta_i(1)$ since $\theta_i(0) = 1-q_i$ in such case. Now, we are interested in considering  a (joint) distribution $\theta$ over $\{0,1\}^n$. This set has $2^n$ points, so any joint distribution $\theta$ can be characterized as a $[0,1]^n\subseteq \Bbb R^n$ vector $p = (p_0,\dots,p_{2^n-1})$ satisfying
$$
  \sum_{j=0}^{2^n-1}p_j = 1. \tag{1}
$$
The equation $(1)$ defines the $(2^n-1)$-dimensional manifold in $\Bbb R$, each point on which represent a single joint distribution. Among such joint distributions, there are factorized (or product) distributions given by $\theta = \theta_1\otimes \theta_2\otimes\dots\otimes\theta_n$. Each such product distribution is determined uniquely by a sequence of $\theta_i$'s, and as we remember each $\theta_i$ is uniquely determined by a single real number $q_i$. As a result, each factorized joint distribution $\theta$ is uniquely determined by a collection of $n$ real numbers, which suggests why is that an $n$-dimensional manifold. In particular, if $[j]_i$ denotes the $i$-th coordinate of the binary representation of $j = 0,1,\dots,2^n-1$ then 
$$
  p_j = \prod_{i=1}^n \theta_i([j]_i) = \prod_{i=1}^n\left(q_i\cdot[j]_i + (1-q_i)(1 -[j]_i)\right).
$$
