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Mastering Variable Elimination (VE) in Bayesian Networks: A Step-by-Step Guide ๐ŸŒŸ

Introduction ๐Ÿค–

Ever wondered how we can make sense of the vast probabilistic relationships in a Bayesian Network (BN)? The answer lies in Variable Elimination (VE)โ€”a key algorithm used for exact inference in BNs. ๐ŸŒ Whether you're trying to compute the probability of certain variables or infer hidden relationships from observed evidence, VE is your go-to tool. ๐ŸŽฏ
In this blog, we'll explore Variable Elimination from start to finish, breaking it down into manageable steps. We'll show how to compute query answers like
, all while keeping things as intuitive as possible. ๐Ÿ˜Ž

What is Variable Elimination? ๐Ÿ”

Variable Elimination (VE) is a graphical inference algorithm used to compute marginal distributions in Bayesian networks. Imagine you're working with a probabilistic model and you want to know the likelihood of certain events (query variables), given a set of observed evidence. VE helps us answer that by eliminating hidden variables one by one.

Inputs and Outputs ๐Ÿ› ๏ธ

Inputs:

  1. F: List of factors (conditional probability tables, or CPTs) describing the relationships between variables.
  1. Q: The list of query variables we're interested in.
  1. E: The list of observed variables (evidence).
  1. e: The observed values for the evidence E.
  1. ฯ: The elimination order of variables that are not in Q โˆช E. These are the latent (hidden) variables that we eliminate one by one.

Output:

  • The conditional probability of the query variables given the observed evidence.

Step-by-Step Process of VE ๐Ÿ“

1. Eliminate Variables in ฯ ๐Ÿš€

The core idea of VE is to eliminate hidden variables by summing them out. The order in which we eliminate them matters (that's why ฯ, the elimination order, is important). We eliminate each variable in ฯ until we only have the query variables left.
  • While ฯ is not empty:
      1. Eliminate the first variable Z from ฯ.
      1. Call the function eliminate(F, Z) to eliminate Z.
This process repeats until all hidden variables are eliminated. ๐Ÿง ๐Ÿ’ฅ

2. Product of All Factors โœจ

Once all variables in ฯ are eliminated, we multiply all remaining factors in the list F. These factors are the result of the elimination process.
This results in a joint distribution over the remaining variables in the network. ๐ŸŽฒ

3. Instantiate Observed Variables ๐Ÿงฉ

Now we need to incorporate the observed evidence. Given that observed variables (from E) are fixed, we instantly set them to their observed values in the product h.
For example, if you have an observation like S = s and R = r, you replace those variables in the factors with their observed values. This helps simplify the computation by reducing dependencies. ๐Ÿ› ๏ธ

4. Renormalization โš–๏ธ

After substituting the observed values, the resulting distribution might not sum to 1 (it's unnormalized). The next step is renormalization, where we divide the product h by the sum of all possible values for the query variables:
This ensures the resulting distribution is a valid probability distribution (summing to 1). ๐Ÿ”ขโœ…

5. Return the Final Result ๐ŸŽฏ

The final step is simple: after renormalizing, you return the marginal distribution for the query variables:
This is the answer to your query! You now know the probability of Q given the observed evidence E = e. ๐Ÿ†

How Does the Elimination Function Work? ๐Ÿ”„

Let's dive into the eliminate function that handles the actual elimination process. This is where the magic happens!

Inputs:

  • F: The list of current factors.
  • X: The variable we want to eliminate.

Outputs:

  • The modified list of factors, which no longer contains X.
Here's the step-by-step breakdown:

1. Remove Factors Involving X ๐Ÿ—‘๏ธ

First, we identify all the factors in F that involve the variable X. These are the factors we need to deal with in the elimination step.

2. Multiply the Remaining Factors ๐Ÿ”—

Next, we multiply all the factors that involve X together. This results in a new function g:
where fโ‚, fโ‚‚, ..., fโ‚– are the factors that involve X.

3. Sum Out X โž—

After multiplying the factors, we sum out X from the new function g to obtain a new function h:
This function h is the marginal distribution over the remaining variables, excluding X. ๐Ÿงฎ

4. Add the New Function โž•

Finally, we add the new function h back into the list of factors F, replacing the eliminated factors fโ‚, fโ‚‚, ..., fโ‚–.

Detailed Example ๐Ÿ‘ฉโ€๐Ÿ’ป

Let's look at a simple example with variables A, B, and C. The factors are:
  • P(A)
  • P(B|A)
  • P(C|B)
We want to compute P(A | C = c), i.e., the marginal probability of A given that C = c.

Step 1: Eliminate B

We start by eliminating B. Multiply the factors involving B:
Then sum out B:
Now, h is a new function involving only A and C. ๐ŸŽ‰

Step 2: Normalize

We then compute the marginal probability:
where P(A,C) is the product of P(A) and h, and the denominator ensures the final distribution sums to 1.

Conclusion ๐ŸŽ‰

Variable Elimination (VE) is a powerful algorithm that allows us to efficiently perform inference in Bayesian Networks. By systematically eliminating hidden variables in a chosen order and updating the factors, VE helps us compute the marginal distributions of query variables given observed evidence. ๐Ÿง 
In this blog, we covered:
  • The step-by-step process of VE.
  • How the eliminate function works to remove variables.
  • A detailed example demonstrating VE in action.
While VE is quite effective, it's important to remember that its performance can become computationally expensive for large networks with many variables. Nonetheless, understanding the process of exact inference with VE is a critical building block for anyone working with Bayesian networks. ๐ŸŒ
So, the next time you're faced with a probabilistic network and need to infer something about your variables, you'll be equipped with the tools to do so effectively! ๐Ÿ’ช