Bayesian Networks

CS5804 Virginia Tech
Introduction to Artificial Intelligence

[fall]

For anyone that has trouble wrapping their head around why variable elimination is more efficient, writing out the explicit for loops to compute P(Y) was really helpful for me:

If we assume W, X, Y, Z each have K possible values then we need to compute K^4 values to fill out the complete table for P(Y). The naive triple sum has K^3 terms and we need to compute this triple sum for each of the K possible values of Y, giving us a total of K^4 values.

If we do variable elimination then we first compute f_W(x):

for each value of X, call this x:
f_W(x) = 0
for each value of W, call this w:
f_W(x) += P(w)*P(x|w)
Note:
- big capital letters denote the random variable, lower case letters denote a value of the corresponding random variable.
- f_W(x) is a table containing K numbers, one for each value of X
- the innermost operation is "constant time" because we are just looking up these values in a table.
- in total it takes K^2 operations to compute the f_W(x) table and then we store it away.

Next we compute f_X(y):
for each value of Y, call this y:
f_X(y) = 0
for each value of X, call this x:
f_X(y) += P(y|x) * f_W(x)
Note:
- f_X(y) is a table containing K numbers, one for each value of Y
- the innermost operation is "constant time" because we are just looking up these values in a table!
- in total it takes K^2 operations to compute the f_X(y) table and then we store it away.

Last we can now compute P(Y) for each value of y:
for each value of Y, call this y:
P(y) = 0
for each value of Z, call this z:
P(y) += P(z|y) * f_X(y)
Note:
- P(Y) is a table containing K numbers, one for each value of Y
- the innermost operation is "constant time" because we are just looking up these values in a table.
- in total it takes K^2 operations to compute this last table.

Computing P(Y) by variable elimination takes 3 * K^2 operations, which is much less than K^4 for large K!


Basically, by computing each of these tables in the right order we avoid repeating work that we already did.

[nathanielhoy]

Best explanation of probability I've received in my whole academic career, thank you

[theedmaster]

My professor for AI explained this so badly that I had no idea what was going on. Thanks for this in-depth and logical explanation of these topics

[jacobmoore]

By far the best explanation of variable elimination; thanks for motivating via brute force/enumeration. For the longest time, it wasn't clear to me that VE was about computational spend not about being the only possible mathematical solution to a problem.

[prasad]

I was struggling to understand this in my class. Glad I came here.

[CapsCtrl]

i have an assignment on this that i need to deliver in two hours and this video is saving me right now!

[joshuasegal]

Excellent video. You brought up a lot of small things that I was confused about and explained them

[ajit_edu]

12.23 doesn't c, r mean car wash AND ( not OR) RAIN as mentioned in lecture

[xXaylaXx]

Your explanation is brilliant, it gives a very good intuition for the theory. Thanks a ton

[bitvision-lgcl]

Very impressive, you make the model crystal clear, and I know that compute bayesian network is nothing than that to calculate a probability (for discrete variables), or a probability distribution (for continuous variables) efficiently.

[fupopanda]

Around 9:43 you simply say that P(S|W, R) is reduced to P(S|W) but you never give a more formal explanation of why. I know it's because of conditional independence.

You could have easily added clarity by stating that you started with the chain rule of probability and then applied conditional independence assumption. That would save anyone who has learned basic probability theory a few minutes of their time, instead of making them pause to think through what just happened there.

[jeffreyyoung]

This is a great video on Bayesian Network. Other people creating videos should take a note from this one.

[zeratulofaiur]

How come condition is "Rain or Carwash" not "Rain and Carwash"?

[cosmopaul]

Thanks for great video! Helped me a lot in understanding this stuff for my Uni course :)

[jallehansen]

Great video. Would love to see the code for that assigment.

[huangbinapple]

What's the difference between enumeration and variable elimination anyway, still think it's only a difference in notation.

[ShubhamSinghYoutube]

Came here searching for coal, found Gold ✌🏻✌🏻✌🏻✌🏻✌🏻

[rolfjohansen]

Does "variable-elimination" imply: "the overall network's functionality got changed"? thanks

[nulliusinverba]

For the slip node, can we say that the slip node is conditionally independent from rain? Or is it independent? Or is it still related indirectly?

Does the order of summations in variable elimination matter?

Also what are observed and unobserved variables? Ie are ancestor variables observed variables? Or are they the marginalized variables? Or something else?

[chingiskhant]

wtf is this how is it so simple. had it always been this simple. thanks