1
$\begingroup$

I try to understand parameter estimation and learning problems at Graphical Models, especially in directed ones (Bayesian Networks). But first of all, I try to understand what exactly a parameter means in a Bayesian Network. The separation between a "variable" and a "parameter" becomes blurry when describing learning problems especially in Bayesian Learning, where the parameters are also variables.

So generally, we have a stochastic process, which is represented by the graph $G$. This graph consists of $S$ random variables, $X_1,X_2,...,X_S$ and the probability distribution over these variables is $P(X_1,X_2,...,X_S)=\prod_{i=1}^{S} P(X_i|parents(X_i))$. Now, where do the parameters fit here? From what I have understood so far:

  1. By the $P(X_1,X_2,...,X_S)=\prod_{i=1}^{S} P(X_i|parents(X_i))$ equation, we implicitly assume that a parameter set $\theta$ is already given and we actually mean $P(X_1,X_2,...,X_S|\theta_{1},\theta_{2},...,\theta_{S})=\prod_{i=1}^{S} P(X_i|parents(X_i),\theta_{i})$. If we run the process $G$, $N$ times in a i.i.d. fashion, then it is either: There is a single correct set for $\theta$ and each time we run the process, the variables are instantiated according to this one and only true $\theta$ (maximum likelihood view) or $\theta$ has a prior distribution of $P(\theta)$ and the "nature" draws a $\theta$ a priori and then our all $N$ previous and any future samples are generated with this $\theta$ (Bayesian view). Therefore, we integrate over all possible $\theta$ when making inference about a new sample. Is this thought pattern correct?
  2. If we think of this as a generative process, then can we say that for a variable $X_i$, an instantiation of its parents, $parents(X_{i})=\pi$ selects a subset of $\theta_{i}$ as $\theta_{i}^{\pi}$ and then generates $X_{i}$ according to the distribution $P(X_{i}|parents(X_{i})=\pi , \theta_{i}^{\pi})$? I think of this subset thing because $X_{i}$ has different distributions conditioned on different values of its parents and each of these distributions can have different parameters (or shared ones of course) among $\theta_{i}$. Again, is this correct or am I missing or misunderstood something completely?
  3. In the graphical model, we add for each $X_{i}$ a new node $\theta_{i}$ which is a new parent of $X_{i}$ and does not have any parents for itself. (There can be shared parameters among random variables, as well.) So, for each sample of the process $G$, we have a new directed graph $G_{n}$ whose nodes are connected to the corresponding $\theta_{i}$ nodes and given all $\theta$ nodes, these $N$ samples are independent from each other. (Of course this is from a theoretical point of view, this "plate notation" thingy is used for real applications as far as I know.) Finally, is this correct or wrong?

What I aim with this question is to verify or invalidate my current understanding of the "parameter" concept in Graphical Models. I need a solid understanding since I need to cope with advanced concepts like learning with the EM Algorithm, etc.

Thanks in advance.

$\endgroup$
  • 1
    $\begingroup$ Selam Ufuk, 1) and 2) sound correct. In 3), I did not quite get what you mean by "So, for each sample of the process G, we have a new directed graph Gn whose nodes are connected to the corresponding θi nodes and given all θ nodes, these N samples are independent from each other." $\endgroup$ – Zhubarb Dec 2 '13 at 9:11
  • $\begingroup$ Selamlar Zhubarb, both for you and @Daniel, what I meant in 3) was an understanding of how a Bayesian Network can be parameterized in a general way. For example if have a model like: Plate Model in which $A$ generates different $B$s and each $B$ then generates a series of $C$ and $D$s, which is a nested plate model, how can we determine the parameters? I have an intuition for a process which is repeated N times for example, but in the link, there are nested repetations in the model itself, as well. This is where I get confused. $\endgroup$ – Ufuk Can Bicici Dec 2 '13 at 11:13
1
$\begingroup$
  1. Yes: $$p(\theta| X) \propto p(X | \theta) p(\theta)$$
  2. Yes.
  3. You can create any model that you want. The question is that, is that a good model for the process you model or not?
| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.