From 6c8491b97d65ce4c499ce2f73dd528edb7b30a77 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Wed, 15 Dec 2021 17:23:38 +0330 Subject: [PATCH 01/16] Initial commit --- .gitignore | 1 + .../assets/.gitkeep | 0 .../index.md | 12 ++++++++ .../matadata.yml | 29 +++++++++++++++++++ notebooks/index.yml | 2 ++ 5 files changed, 44 insertions(+) create mode 100644 .gitignore create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/.gitkeep create mode 100644 notebooks/10_inference_in_bayesian_networks_1/index.md create mode 100644 notebooks/10_inference_in_bayesian_networks_1/matadata.yml diff --git a/.gitignore b/.gitignore new file mode 100644 index 00000000..44a09645 --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +*.ini \ No newline at end of file diff --git a/notebooks/10_inference_in_bayesian_networks_1/assets/.gitkeep b/notebooks/10_inference_in_bayesian_networks_1/assets/.gitkeep new file mode 100644 index 00000000..e69de29b diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md new file mode 100644 index 00000000..c8d6205d --- /dev/null +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -0,0 +1,12 @@ +# Inference in Bayes Nets 1 + +## Table of Contents +- [Introduction](#introduction) +- [Conclusions](#conclusions) +- [References](#references) + +## Introduction + +## Conclusions + +## References diff --git a/notebooks/10_inference_in_bayesian_networks_1/matadata.yml b/notebooks/10_inference_in_bayesian_networks_1/matadata.yml new file mode 100644 index 00000000..9852f9d9 --- /dev/null +++ b/notebooks/10_inference_in_bayesian_networks_1/matadata.yml @@ -0,0 +1,29 @@ +title: Inference in Bayes Nets 1 + +header: + title: Inference in Bayes Nets 1 # title of your notebook + description: An Introduction to Inference in Bayesian Networks Part 1 (Up to Sampling) + +authors: + label: + position: top + content: + - name: Parsa Mohammadian + role: Author + contact: + - link: https://github.com/parsa2820 + icon: fab fa-github + - name: Sara Azarnoush + role: Author + contact: + - link: https://github.com/saaz742 + icon: fab fa-github + - name: Kasra Amani + role: Author + contact: + - link: https://github.com/iTskAsra + icon: fab fa-github + +comments: + label: false + kind: comments \ No newline at end of file diff --git a/notebooks/index.yml b/notebooks/index.yml index 1b2133c0..640fc438 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -54,4 +54,6 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ + kind: S2021, LN, Notebook + - notebook: notebooks/10_inference_in_bayesian_networks_1/ kind: S2021, LN, Notebook \ No newline at end of file From 9cf6d2a0e7948e0be2bc7524c843660fa86c9182 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Wed, 15 Dec 2021 17:53:55 +0330 Subject: [PATCH 02/16] Add intro --- notebooks/10_inference_in_bayesian_networks_1/index.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index c8d6205d..a18c5384 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -6,7 +6,10 @@ - [References](#references) ## Introduction +The basic task of a Bayesian network is to compute the posterior probability distributions for a set of query variables,given an observation of a set of evidence variables. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. [[1]](###1) + ## Conclusions ## References +### [1] [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) \ No newline at end of file From a5e25a85738b3b22b385d2299d446c2af835e684 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Wed, 15 Dec 2021 19:36:51 +0330 Subject: [PATCH 03/16] Add enumeration details --- .../index.md | 47 ++++++++++++++++++- 1 file changed, 45 insertions(+), 2 deletions(-) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index a18c5384..d8c5a417 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -2,14 +2,57 @@ ## Table of Contents - [Introduction](#introduction) +- [Inference by Enumeration](#inference-by-enumeration) + - [Algorithm Explanation](#algorithm-explanation) + - [Algorithm Steps](#algorithm-steps) + - [Algorithm Example](#algorithm-example) - [Conclusions](#conclusions) - [References](#references) ## Introduction -The basic task of a Bayesian network is to compute the posterior probability distributions for a set of query variables,given an observation of a set of evidence variables. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. [[1]](###1) +The basic task of a Bayesian network is to compute the posterior probability distributions for a set of query variables, given an observation of a set of evidence variables. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. In this lecture, we discuss the exact inference method. Approximate (Sampling) method will be discussed in the next lecture. + +## Inference by Enumeration +- exact +- expnential complexity + +### Algorithm Explanation + +The enumeration algorithm is a simple, brute-force algorithm for computing the distribution of a variable in a Bayes net. In this algorithm we partition all Bayes net variables into three groups: + +1. evidence variables +1. hidden variables +1. query variables + +This algorithm takes query variables and evidence variables as input, and outputs the distribution of query variables. + +The evidence e is whatever values you already know about the variables in the Bayes net. Evidence simplifies your work because instead of having to consider those variables’ whole distributions, you can assign them particular values, so they are no longer variables, they are constants. In the most general case, there is no evidence. + +This algorithm has to compute a distribution over $X$, which, because $X$ is a discrete variable, means computing the probability that $X$ takes on each of its possible values (the values in its domain). The algorithm does this simply by looping through all of the possible values, and computing the probability for each one. Possible values are the result of marginalizing out hidden variables from entries which are consistent with given evidence. Note that if there is no evidence, then it is literally just computing the probabilities $P(X=x_i)$ for each xi in $X$’s domain. If there is evidence, then it is computing $P(e, X=xi)$ for each xi in $X$’s domain – that is, it is computing the probability that $X$ has the given value $x_i$ and the evidence is true – so in that case, we use the law of conditional probability, which says that $P(X=x_i | e) = \frac{P(e, X=x_i)}{P(e)}$. Once we have computed $P(e, X=x_i)$ for all $x_i$, we can just normalize those values to get the correct distribution $P(X | e)$. + +### Algorithm Steps +We can summurize the explained algorithm in the following steps: + +1. Select the entries consistent with the evidence. +2. Sum out the hidden variables to get joint distribution of query and evidence variables. +3. Normalize the distribution to get the distribution of query variables. + +### Algorithm Example + + + + +## Variable Elimination +- exact +- worst case exponential complexity, often better + +## + + ## Conclusions ## References -### [1] [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) \ No newline at end of file +- [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) +- [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) \ No newline at end of file From 736bd0b1837eaa60f5b94963837182cf54392f88 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Thu, 16 Dec 2021 03:55:21 +0330 Subject: [PATCH 04/16] Add VE --- .../assets/intro.png | Bin 0 -> 54225 bytes .../index.md | 131 ++++++++++++++++-- 2 files changed, 118 insertions(+), 13 deletions(-) create mode 100644 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Early)](#inference-by-variable-elimination-marginalizing-early) + - [Algorithm Explanation](#algorithm-explanation-1) + - [Algorithm Steps](#algorithm-steps-1) + - [Algorithm Pseudocode](#algorithm-pseudocode-1) + - [Algorithm Time Complexity](#algorithm-time-complexity-1) + - [Algorithm Example](#algorithm-example-1) - [Conclusions](#conclusions) - [References](#references) +
+ ## Introduction -The basic task of a Bayesian network is to compute the posterior probability distributions for a set of query variables, given an observation of a set of evidence variables. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. In this lecture, we discuss the exact inference method. Approximate (Sampling) method will be discussed in the next lecture. +The basic task of a Bayesian network is to compute the posterior probability distributions for a set of query variables, given an observation of a set of evidence variables. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. In this lecture, we discuss the exact inference method. Approximate (Sampling) method will be discussed in the next lecture. ## Inference by Enumeration -- exact -- expnential complexity - -### Algorithm Explanation The enumeration algorithm is a simple, brute-force algorithm for computing the distribution of a variable in a Bayes net. In this algorithm we partition all Bayes net variables into three groups: @@ -24,35 +30,134 @@ The enumeration algorithm is a simple, brute-force algorithm for computing the d 1. hidden variables 1. query variables -This algorithm takes query variables and evidence variables as input, and outputs the distribution of query variables. +This algorithm takes query variables and evidence variables as input, and outputs the distribution of query variables. The evidence $e$ is whatever values you already know about the variables in the Bayes net. Evidence simplifies your work because instead of having to consider those variables’ whole distributions, you can assign them particular values, so they are no longer variables, they are constants. In the most general case, there is no evidence. -The evidence e is whatever values you already know about the variables in the Bayes net. Evidence simplifies your work because instead of having to consider those variables’ whole distributions, you can assign them particular values, so they are no longer variables, they are constants. In the most general case, there is no evidence. +### Algorithm Explanation -This algorithm has to compute a distribution over $X$, which, because $X$ is a discrete variable, means computing the probability that $X$ takes on each of its possible values (the values in its domain). The algorithm does this simply by looping through all of the possible values, and computing the probability for each one. Possible values are the result of marginalizing out hidden variables from entries which are consistent with given evidence. Note that if there is no evidence, then it is literally just computing the probabilities $P(X=x_i)$ for each xi in $X$’s domain. If there is evidence, then it is computing $P(e, X=xi)$ for each xi in $X$’s domain – that is, it is computing the probability that $X$ has the given value $x_i$ and the evidence is true – so in that case, we use the law of conditional probability, which says that $P(X=x_i | e) = \frac{P(e, X=x_i)}{P(e)}$. Once we have computed $P(e, X=x_i)$ for all $x_i$, we can just normalize those values to get the correct distribution $P(X | e)$. +This algorithm has to compute a distribution over $X$, which, because $X$ is a discrete variable, means computing the probability that $X$ takes on each of its possible values (the values in its domain). The algorithm does this simply by looping through all of the possible values, and computing the probability for each one. Note that if there is no evidence, then it is literally just computing the probabilities $P(X=x_i)$ for each $x_i$ in $X$’s domain. If there is evidence, then it is computing $P(e, X=x_i)$ for each $x_i$ in $X$’s domain – that is, it is computing the probability that $X$ has the given value $x_i$ and the evidence is true – so in that case, we use the law of conditional probability, which says that $$P(X=x_i | e) = \frac{P(e, X=x_i)}{P(e)}$$. Once we have computed $P(e, X=x_i)$ for all $x_i$, we can just normalize those values to get the correct distribution $P(X | e)$. ### Algorithm Steps + We can summurize the explained algorithm in the following steps: 1. Select the entries consistent with the evidence. 2. Sum out the hidden variables to get joint distribution of query and evidence variables. 3. Normalize the distribution to get the distribution of query variables. +### Algorithm Pseudocode +```python +def enumeration_ask(X, e, bn): + """ + Input: + X: query variable + e: observed values for all variables + bn: given Bayes net + Output: + P(X | e) + """ + q = ProbDist(X) # a ditribution over X, where q(x) = P(X=x) + for xi in X.domain: + q[xi] = enumerate_all(e + [(X, xi)], bn.vars) + return q.normalize() + +def enumerate_all(e, vars): + """ + Input: + e: observed values for all variables plus a new variable X assignment + vars: list of all variables + Output: + P(e[-1] | e[:-1]) + """ + if not vars.any(): + return 1.0 + Y = vars[0] + if e.contains(Y): + return probability_condition_parents(Y, e, vars[1:]) * enumerate_all(e, vars[1:]) + else: + sum = 0.0 + for yi in Y.domain: + sum += probability_condition_parents(Y, e + [(Y, yi)], vars[1:]) * enumerate_all(e + [(Y, yi)], vars[1:]) + return sum +``` + +### Algorithm Time Complexity + +In worst case, we have no evidence, so we have to for loop through all possible values of all variables. Hence this algorithm has a complexity of $O(d^n)$ where $d$ is the size of the domain of the variables and $n$ is the number of variables. + ### Algorithm Example +[Here](https://youtu.be/BrK7X_XlGB8) is a video of the algorithm running on a simple example. + +## Inference by Enumeration is Slow + +In enumeration, first we find the whole joint distribution and then we marginalize out the hidden variables. Hence it is too slow because of the big joint distributio we need to compute. + +If we marginalize out the hidden variables in partial joint distribution, we can get a much faster algorithm. This method is called Infrence by Variable Elimination. +## Inference by Variable Elimination (Marginalizing Early) +The point of the variable-elimination algorithm is that it is more bottom-up than top-down. Instead of figuring out the probabilities we need to compute and then computing all the other probabilities that each one depends on, we try to compute probabilities and then compute the other terms that depend on them, and repeatedly simplify the expression until we have something that is in terms of only the variable we’re looking for. -## Variable Elimination -- exact -- worst case exponential complexity, often better +The variable-elimination algorithm uses things called factors. A factor is basically a CPT, except that the entries are not necessarily probabilities (but they would be if you normalized them). You can think of a factor as a matrix with a dimension for each variable, where $Factor[VAL1][VAL2][...]$ is (proportional to) a probability such as $$P(VAR1=VAL1, VAR2=VAL2, ...)$$ or you can think of it as a table with one row for each possible combination of assignments of values to the variables. -## +We also define two operations on factors: +1. Join +2. Eliminate +Join is used to combine two factors. For example, if we have two factors $F_1$ and $F_2$, we can compute the joint distribution of $F_1$ and $F_2$ by multiplying their probabilities together, corresponding to variables value in factors. +Elimate is used to eliminate a variable from a factor. For example, if we have a factor $F$ and we want to eliminate $X$ from $F$, we can group rows by non-$X$ values and sum probabilities. This is exactly marginalization. +### Algorithm Explanation + +This algorithm, like previous one, takes a variable X and returns a distribution over X, given some evidence e. First it initializes the list of factors; prior to any simplification, this is just the conditional probability tables for each variable given the evidence e. Then, it joins each factors with it. The summing-out process takes all the factors that depend on a given variable and replaces them with a single new factor that does not depend on that variable (by summing over all possible values of the variable). By the end of the loop, all the variables have been summed out except the query variable X, so then we can just multiply the factors together and normalize to get the distribution. + +### Algorithm Steps + +1. Initialize the list of factors which are local CPTs instantiated by the evidence. +2. While there are any hidden variables: + - Join all the factors containing the hidden variable. + - Eliminate the hidden variable. +3. Join all remaining factors. +4. Normalize the resulting factor. + +### Algorithm Pseudocode + +```python +def elimination_ask(X, e, bn): + """ + Input: + X: query variable + e: observed values for all variables + bn: given Bayes net + Output: + P(X | e) + """ + factors = [t.marginalize(X) for t in bn.cpt] + for var in bn.vars: + if var not in e and var != X: + relevant_factors = [f for f in factors if var in f.vars] + for f in relevant_factors: + factors.remove(f) + factors.append(f.eliminate(var, join(relevant_factors))) + return join(factors).normalize() +``` + +### Algorithm Time Complexity + +In worst case, this algorithm has exponential complexity too. But variable elimination ordering can greatly affect efficiency. + +### Ordering Polytree Variables for VE + +### Algorithm Example + +[Here](https://youtu.be/w4sJ8SazmFo) is a video of elimination of a variable from a set of factors. ## Conclusions +We reviewed two major exact inference algorithms, the enumeration algorithm and the variable elimination algorithm. The enumeration algorithm is a simple algorithm that is easy to understand and implement, but it is not very efficient. On the other hand, the variable elimination algorithm is a more complex algorithm that is more efficient, but it is also harder to understand and implement. For both introduced algorithms, the worst-case time complexity is exponential. So in practice, using sampling is usually a better choice, which you will learn more about in the next lecture. + ## References - [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) -- [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) \ No newline at end of file +- [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) +- [Variable Elimination](https://ermongroup.github.io/cs228-notes/inference/ve/) From 87c1c4890d7827580e3277bee8636ebbb3294581 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 17 Dec 2021 21:53:01 +0330 Subject: [PATCH 05/16] Add ordering for VE --- .../index.md | 21 ++++++++++++++++++- 1 file changed, 20 insertions(+), 1 deletion(-) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index fe63062a..1cdba407 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -108,6 +108,9 @@ Join is used to combine two factors. For example, if we have two factors $F_1$ a Elimate is used to eliminate a variable from a factor. For example, if we have a factor $F$ and we want to eliminate $X$ from $F$, we can group rows by non-$X$ values and sum probabilities. This is exactly marginalization. +> - Join is exactly like SQL join. +> - Eliminate is like a group by and SUM aggregation function in SQL. + ### Algorithm Explanation This algorithm, like previous one, takes a variable X and returns a distribution over X, given some evidence e. First it initializes the list of factors; prior to any simplification, this is just the conditional probability tables for each variable given the evidence e. Then, it joins each factors with it. The summing-out process takes all the factors that depend on a given variable and replaces them with a single new factor that does not depend on that variable (by summing over all possible values of the variable). By the end of the loop, all the variables have been summed out except the query variable X, so then we can just multiply the factors together and normalize to get the distribution. @@ -145,10 +148,24 @@ def elimination_ask(X, e, bn): ### Algorithm Time Complexity -In worst case, this algorithm has exponential complexity too. But variable elimination ordering can greatly affect efficiency. +The computational and space complexity of variable elimination is determined by the largest factor. In worst case, this algorithm has exponential complexity like the enumeration algorithm. But variable elimination ordering can greatly affect the largest factor. There is no general oredering that provides only small factors. ### Ordering Polytree Variables for VE +Polytree is a directed graph which has no undirected cycles. In this special kind of graph we can introduce an algorithm for ordering nodes in order to achieve small factors, and efficiently compute the joint distribution of the variables. + +1. Drop edge directions +2. Pick an arbitrary node as the root +3. Do depth first search from the root +4. Sort the resulting nodes in topological order +5. Reverse the order + +Know if we eliminate variables with this order, we would never get a factor larger than the original factors. This makes the VE algorithm linear time complexity. + +### Cut-set Conditioning + +We can cut the bayes net at an instantiated variable, and this can transform a multi connected graph into a polytree, for which we can find the order of elimination. If these variables are not actually known, we can set them to each of their possible values and then solve the problem with the polytree. + ### Algorithm Example [Here](https://youtu.be/w4sJ8SazmFo) is a video of elimination of a variable from a set of factors. @@ -158,6 +175,8 @@ In worst case, this algorithm has exponential complexity too. But variable elimi We reviewed two major exact inference algorithms, the enumeration algorithm and the variable elimination algorithm. The enumeration algorithm is a simple algorithm that is easy to understand and implement, but it is not very efficient. On the other hand, the variable elimination algorithm is a more complex algorithm that is more efficient, but it is also harder to understand and implement. For both introduced algorithms, the worst-case time complexity is exponential. So in practice, using sampling is usually a better choice, which you will learn more about in the next lecture. ## References + - [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) - [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) - [Variable Elimination](https://ermongroup.github.io/cs228-notes/inference/ve/) +- [Bayesian Networks - Inference (Part II)](https://ccc.inaoep.mx/~esucar/Clases-mgp/Notes/c7-bninf-p2.pdf) \ No newline at end of file From 0dde737a727019b1f3e0ee51f802e43241b68caa Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 17 Dec 2021 22:04:15 +0330 Subject: [PATCH 06/16] Update TOC --- notebooks/10_inference_in_bayesian_networks_1/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index 1cdba407..0b73a575 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -13,6 +13,8 @@ - [Algorithm Steps](#algorithm-steps-1) - [Algorithm Pseudocode](#algorithm-pseudocode-1) - [Algorithm Time Complexity](#algorithm-time-complexity-1) + - [Ordering Polytree Variables for VE](#ordering-polytree-variables-for-ve) + - [Cut-set Conditioning](#cut-set-conditioning) - [Algorithm Example](#algorithm-example-1) - [Conclusions](#conclusions) - [References](#references) From 05bf0c216d27031170758ff4e56fbec4743eabaf Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 17 Dec 2021 22:25:02 +0330 Subject: [PATCH 07/16] Add conditioning example --- .../assets/conditioning-example.png | Bin 0 -> 201015 bytes .../index.md | 11 +++++++++-- 2 files changed, 9 insertions(+), 2 deletions(-) create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/conditioning-example.png diff --git 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This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. In this lecture, we discuss the exact inference method. Approximate (Sampling) method will be discussed in the next lecture. ## Inference by Enumeration @@ -47,6 +49,7 @@ We can summurize the explained algorithm in the following steps: 3. Normalize the distribution to get the distribution of query variables. ### Algorithm Pseudocode + ```python def enumeration_ask(X, e, bn): """ @@ -100,7 +103,9 @@ If we marginalize out the hidden variables in partial joint distribution, we can The point of the variable-elimination algorithm is that it is more bottom-up than top-down. Instead of figuring out the probabilities we need to compute and then computing all the other probabilities that each one depends on, we try to compute probabilities and then compute the other terms that depend on them, and repeatedly simplify the expression until we have something that is in terms of only the variable we’re looking for. -The variable-elimination algorithm uses things called factors. A factor is basically a CPT, except that the entries are not necessarily probabilities (but they would be if you normalized them). You can think of a factor as a matrix with a dimension for each variable, where $Factor[VAL1][VAL2][...]$ is (proportional to) a probability such as $$P(VAR1=VAL1, VAR2=VAL2, ...)$$ or you can think of it as a table with one row for each possible combination of assignments of values to the variables. +The variable-elimination algorithm uses things called factors. A factor is basically a CPT, except that the entries are not necessarily probabilities (but they would be if you normalized them). You can think of a factor as a matrix with a dimension for each variable, where $Factor[VAL1][VAL2][...]$ is (proportional to) a probability such as +$$P(VAR1=VAL1, VAR2=VAL2, ...)$$ +or you can think of it as a table with one row for each possible combination of assignments of values to the variables. We also define two operations on factors: 1. Join @@ -166,7 +171,9 @@ Know if we eliminate variables with this order, we would never get a factor larg ### Cut-set Conditioning -We can cut the bayes net at an instantiated variable, and this can transform a multi connected graph into a polytree, for which we can find the order of elimination. If these variables are not actually known, we can set them to each of their possible values and then solve the problem with the polytree. +We can cut the bayes net at an instantiated variable, and this can transform a multi connected graph into a polytree, for which we can find the order of elimination. If these variables are not actually known, we can set them to each of their possible values and then solve the problem with the polytree. You can see an example of this below. + +

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b/notebooks/10_inference_in_bayesian_networks_1/index.md index 62f42543..f42c5343 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -20,7 +20,7 @@ - [Conclusions](#conclusions) - [References](#references) -
+
## Introduction @@ -173,7 +173,7 @@ Know if we eliminate variables with this order, we would never get a factor larg We can cut the bayes net at an instantiated variable, and this can transform a multi connected graph into a polytree, for which we can find the order of elimination. If these variables are not actually known, we can set them to each of their possible values and then solve the problem with the polytree. You can see an example of this below. -
+
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z>~5@L!$6Qj`Dg%YQNW@2KVf)~fsvR~c+% VS~|A* +# Inference in Bayes Nets 1 + +## Table of Contents + +- [Introduction](#introduction) +- [Inference by Enumeration](#inference-by-enumeration) + - [Algorithm Explanation](#algorithm-explanation) + - [Algorithm Steps](#algorithm-steps) + - [Algorithm Pseudocode](#algorithm-pseudocode) + - [Algorithm Time Complexity](#algorithm-time-complexity) + - [Algorithm Example](#algorithm-example) +- [Inference by Variable Elimination (Marginalizing Early)](#inference-by-variable-elimination-marginalizing-early) + - [Algorithm Explanation](#algorithm-explanation-1) + - [Algorithm Steps](#algorithm-steps-1) + - [Algorithm Pseudocode](#algorithm-pseudocode-1) + - [Algorithm Time Complexity](#algorithm-time-complexity-1) + - [Ordering Polytree Variables for VE](#ordering-polytree-variables-for-ve) + - [Cut-set Conditioning](#cut-set-conditioning) + - [Algorithm Example](#algorithm-example-1) +- [Conclusions](#conclusions) +- [References](#references) + +

    + +## Introduction + +The basic task of a Bayesian network is to compute the posterior probability distributions for a set of query variables, given an observation of a set of evidence variables. This process is known as inference, but is also called Bayesian updating, belief updating or reasoning. There are two ways to approach this, either exact or approximate. Both approaches are worst-case NP-hard. An exact method obviously gives an exact result, while an approximate method tries to approach the correct outcome as close as possible. In this lecture, we discuss the exact inference method. Approximate (Sampling) method will be discussed in the next lecture. + +## Inference by Enumeration + +The enumeration algorithm is a simple, brute-force algorithm for computing the distribution of a variable in a Bayes net. In this algorithm, we partition all Bayes net variables into three groups: + +1. evidence variables +2. hidden variables +3. query variables + +This algorithm takes query variables and evidence variables as input, and outputs the distribution of query variables. The evidence $e$ is whatever values you already know about the variables in the Bayes net. Evidence simplifies your work because instead of having to consider those variables’ whole distributions, you can assign them particular values, so they are no longer variables, they are constants. In the most general case, there is no evidence. + +### Algorithm Explanation + +This algorithm has to compute a distribution over $X$, which, because $X$ is a discrete variable, means computing the probability that $X$ takes on each of its possible values (the values in its domain). The algorithm does this simply by looping through all the possible values, and computing the probability for each one. Note that if there is no evidence, then it is literally just computing the probabilities $P(X=x_i)$ for each $x_i$ in $X$’s domain. If there is evidence, then it is computing $P(e, X=x_i)$ for each $x_i$ in $X$’s domain – that is, it is computing the probability that $X$ has the given value $x_i$ and the evidence is true – so in that case, we use the law of conditional probability, which says that $$P(X=x_i | e) = \frac{P(e, X=x_i)}{P(e)}$$. Once we have computed $P(e, X=x_i)$ for all $x_i$, we can just normalize those values to get the correct distribution $P(X | e)$. + +### Algorithm Steps + +We can summarize the explained algorithm in the following steps: + +1. Select the entries consistent with the evidence. +2. Sum out the hidden variables to get joint distribution of query and evidence variables. +3. Normalize the distribution to get the distribution of query variables. + +### Algorithm Pseudocode + +```python +def enumeration_ask(X, e, bn): + """ + Input: + X: query variable + e: observed values for all variables + bn: given Bayes net + Output: + P(X | e) + """ + q = ProbDist(X) # a ditribution over X, where q(x) = P(X=x) + for xi in X.domain: + q[xi] = enumerate_all(e + [(X, xi)], bn.vars) + return q.normalize() + +def enumerate_all(e, vars): + """ + Input: + e: observed values for all variables plus a new variable X assignment + vars: list of all variables + Output: + P(e[-1] | e[:-1]) + """ + if not vars.any(): + return 1.0 + Y = vars[0] + if e.contains(Y): + return probability_condition_parents(Y, e, vars[1:]) * enumerate_all(e, vars[1:]) + else: + sum = 0.0 + for yi in Y.domain: + sum += probability_condition_parents(Y, e + [(Y, yi)], vars[1:]) * enumerate_all(e + [(Y, yi)], vars[1:]) + return sum +``` + +### Algorithm Time Complexity + +In the worst case, we have no evidence, so we have to for loop through all possible values of all variables. Hence, this algorithm has a complexity of $O(d^n)$ where $d$ is the size of the domain of the variables and $n$ is the number of variables. + +### Algorithm Example + +[Here](https://youtu.be/BrK7X_XlGB8) is a video of the algorithm running on a simple example. + +## Inference by Enumeration is Slow + +In enumeration, first we find the whole joint distribution, and then we marginalize out the hidden variables. Hence, it is too slow because of the big joint distribution we need to compute. + +If we marginalize out the hidden variables in the partial joint distribution, we can get a much faster algorithm. This method is called Inference by Variable Elimination. + +

    + +## Inference by Variable Elimination (Marginalizing Early) + +The point of the variable-elimination algorithm is that it is more bottom-up than top-down. Instead of figuring out the probabilities we need to compute and then computing all the other probabilities that each one depends on, we try to compute probabilities and then compute the other terms that depend on them, and repeatedly simplify the expression until we have something that is in terms of only the variable we’re looking for. + +The variable-elimination algorithm uses things called factors. A factor is basically a CPT, except that the entries are not necessarily probabilities (but they would be if you normalized them). You can think of a factor as a matrix with a dimension for each variable, where $Factor[VAL1][VAL2][…]$ is (proportional to) a probability such as +$$P(VAR1=VAL1, VAR2=VAL2, …)$$ +or you can think of it as a table with one row for each possible combination of assignments of values to the variables. + +We also define two operations on factors: +1. Join +2. Eliminate + +Join is used to combine two factors. For example, if we have two factors $F_1$ and $F_2$, we can compute the joint distribution of $F_1$ and $F_2$ by multiplying their probabilities together, corresponding to the variable's value in factors. + +Eliminate is used to eliminate a variable from a factor. For example, if we have a factor $F$, and we want to eliminate $X$ from $F$, we can group rows by non-$X$ values and sum probabilities. This is exactly marginalization. + +> - Join is exactly like SQL join. +> - Eliminate is like a group by and SUM aggregation function in SQL. + +### Algorithm Explanation + +This algorithm, like the previous one, takes a variable X and returns a distribution over X, given some evidence e. First, it initializes the list of factors; prior to any simplification, this is just the conditional probability tables for each variable given the evidence e. Then, it joins each factors with it. The summing-out process takes all the factors that depend on a given variable and replaces them with a single new factor that does not depend on that variable (by summing over all possible values of the variable). By the end of the loop, all the variables have been summed out except the query variable X, so then we can just multiply the factors together and normalize to get the distribution. + +### Algorithm Steps + +1. Initialize the list of factors which are local CPTs instantiated by the evidence. +2. While there are any hidden variables: + - Join all the factors containing the hidden variable. + - Eliminate the hidden variable. +3. Join all remaining factors. +4. Normalize the resulting factor. + +### Algorithm Pseudocode + +```python +def elimination_ask(X, e, bn): + """ + Input: + X: query variable + e: observed values for all variables + bn: given Bayes net + Output: + P(X | e) + """ + factors = [t.marginalize(X) for t in bn.cpt] + for var in bn.vars: + if var not in e and var != X: + relevant_factors = [f for f in factors if var in f.vars] + for f in relevant_factors: + factors.remove(f) + factors.append(f.eliminate(var, join(relevant_factors))) + return join(factors).normalize() +``` + +### Algorithm Time Complexity + +The computational and space complexity of variable elimination is determined by the largest factor. In the worst case, this algorithm has exponential complexity, like the enumeration algorithm. But variable elimination ordering can greatly affect the largest factor. For example, in the following Bayes net, assuming the query is $P(X_n | Y_1, …, Y_n)$, the largest factor for following orders are different: + +- $Z, X_1, …, X_n \rightarrow 2^{n+1}$ +- $X_1, …, X_n, Z \rightarrow 2^{2}$ + +

    + +There is no general ordering that provides only small factors. + +### Ordering Polytree Variables for VE + +Polytree is a directed graph which has no undirected cycles. In this special kind of graph we can introduce an algorithm for ordering nodes in order to achieve small factors, and efficiently compute the joint distribution of the variables. + +1. Drop edge directions +2. Pick an arbitrary node as the root +3. Do depth first search from the root +4. Sort the resulting nodes in topological order +5. Reverse the order + +Now, if we eliminate variables with this order, we would never get a factor larger than the original factors. This makes the VE algorithm linear time complexity. + +### Cut-set Conditioning + +We can cut the bayes net at an instantiated variable, and this can transform a multi connected graph into a polytree, for which we can find the order of elimination. If these variables are not actually known, we can set them to each of their possible values and then solve the problem with the polytree. You can see an example of this below. + +

    + +### Algorithm Example + +[Here](https://youtu.be/w4sJ8SazmFo) is a video of elimination of a variable from a set of factors. + +## Conclusions + +We reviewed two major exact inference algorithms, the enumeration algorithm and the variable elimination algorithm. The enumeration algorithm is a simple algorithm that is easy to understand and implement, but it is not very efficient. On the other hand, the variable elimination algorithm is a more complex algorithm that is more efficient, but it is also harder to understand and implement. For both introduced algorithms, the worst-case time complexity is exponential. So in practice, using sampling is usually a better choice, which you will learn more about in the next lecture. + +## References + +- [Class Presentation](http://ce.sharif.edu/courses/99-00/1/ce417-2/resources/root/Slides/PDF/Session%2013_14.pdf) +- [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) +- [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) +- [Variable Elimination](https://ermongroup.github.io/cs228-notes/inference/ve/) +- [Bayesian Networks - Inference (Part II)](https://ccc.inaoep.mx/~esucar/Clases-mgp/Notes/c7-bninf-p2.pdf) + diff --git a/notebooks/10_inference_in_bayesian_networks_1/matadata.yml b/notebooks/10_inference_in_bayesian_networks_1/matadata.yml new file mode 100644 index 00000000..9852f9d9 --- /dev/null +++ b/notebooks/10_inference_in_bayesian_networks_1/matadata.yml @@ -0,0 +1,29 @@ +title: Inference in Bayes Nets 1 + +header: + title: Inference in Bayes Nets 1 # title of your notebook + description: An Introduction to Inference in Bayesian Networks Part 1 (Up to Sampling) + +authors: + label: + position: top + content: + - name: Parsa Mohammadian + role: Author + contact: + - link: https://github.com/parsa2820 + icon: fab fa-github + - name: Sara Azarnoush + role: Author + contact: + - link: https://github.com/saaz742 + icon: fab fa-github + - name: Kasra Amani + role: Author + contact: + - link: https://github.com/iTskAsra + icon: fab fa-github + +comments: + label: false + kind: comments \ No newline at end of file diff --git a/notebooks/index.yml b/notebooks/index.yml index 1b2133c0..640fc438 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -54,4 +54,6 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ + kind: S2021, LN, Notebook + - notebook: notebooks/10_inference_in_bayesian_networks_1/ kind: S2021, LN, Notebook \ No newline at end of file From 3042536a47fa275578007d1423117c11638a3c78 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 31 Dec 2021 18:42:11 +0330 Subject: [PATCH 15/16] Replace notebook entry with md --- notebooks/index.yml | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 640fc438..d5df8afd 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,5 +55,5 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - notebook: notebooks/10_inference_in_bayesian_networks_1/ - kind: S2021, LN, Notebook \ No newline at end of file + - md: notebooks/10_inference_in_bayesian_networks_1/index.md + kind: S2021, LN, Notebook From c823435b89fb4329fc6796019f4fbe5dcaca3770 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Sun, 30 Jan 2022 14:02:46 +0330 Subject: [PATCH 16/16] Fix title heading --- notebooks/10_inference_in_bayesian_networks_1/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index da510797..2b623880 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -1,4 +1,5 @@
    + # Inference in Bayes Nets 1 ## Table of Contents
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zaS#72f5$ERQD)p-mO_sG{qIf2uq~AaPyjE`4RH#x1*mL6(?2@7Gd`m%M(a_r-PnJA l)PKL{|KtA~njc{P literal 0 HcmV?d00001 diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index f42c5343..653ce027 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -95,10 +95,12 @@ In worst case, we have no evidence, so we have to for loop through all possible ## Inference by Enumeration is Slow -In enumeration, first we find the whole joint distribution and then we marginalize out the hidden variables. Hence it is too slow because of the big joint distributio we need to compute. +In enumeration, first we find the whole joint distribution and then we marginalize out the hidden variables. Hence it is too slow because of the big joint distribution we need to compute. If we marginalize out the hidden variables in partial joint distribution, we can get a much faster algorithm. This method is called Infrence by Variable Elimination. +
    + ## Inference by Variable Elimination (Marginalizing Early) The point of the variable-elimination algorithm is that it is more bottom-up than top-down. Instead of figuring out the probabilities we need to compute and then computing all the other probabilities that each one depends on, we try to compute probabilities and then compute the other terms that depend on them, and repeatedly simplify the expression until we have something that is in terms of only the variable we’re looking for. From 4289178f1485468c0a8f3be05b35421a29df0706 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 17 Dec 2021 22:50:13 +0330 Subject: [PATCH 10/16] Fix some language mistakes --- .../index.md | 32 +++++++++---------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index 653ce027..9a3770f5 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -28,21 +28,21 @@ The basic task of a Bayesian network is to compute the posterior probability dis ## Inference by Enumeration -The enumeration algorithm is a simple, brute-force algorithm for computing the distribution of a variable in a Bayes net. In this algorithm we partition all Bayes net variables into three groups: +The enumeration algorithm is a simple, brute-force algorithm for computing the distribution of a variable in a Bayes net. In this algorithm, we partition all Bayes net variables into three groups: 1. evidence variables -1. hidden variables -1. query variables +2. hidden variables +3. query variables This algorithm takes query variables and evidence variables as input, and outputs the distribution of query variables. The evidence $e$ is whatever values you already know about the variables in the Bayes net. Evidence simplifies your work because instead of having to consider those variables’ whole distributions, you can assign them particular values, so they are no longer variables, they are constants. In the most general case, there is no evidence. ### Algorithm Explanation -This algorithm has to compute a distribution over $X$, which, because $X$ is a discrete variable, means computing the probability that $X$ takes on each of its possible values (the values in its domain). The algorithm does this simply by looping through all of the possible values, and computing the probability for each one. Note that if there is no evidence, then it is literally just computing the probabilities $P(X=x_i)$ for each $x_i$ in $X$’s domain. If there is evidence, then it is computing $P(e, X=x_i)$ for each $x_i$ in $X$’s domain – that is, it is computing the probability that $X$ has the given value $x_i$ and the evidence is true – so in that case, we use the law of conditional probability, which says that $$P(X=x_i | e) = \frac{P(e, X=x_i)}{P(e)}$$. Once we have computed $P(e, X=x_i)$ for all $x_i$, we can just normalize those values to get the correct distribution $P(X | e)$. +This algorithm has to compute a distribution over $X$, which, because $X$ is a discrete variable, means computing the probability that $X$ takes on each of its possible values (the values in its domain). The algorithm does this simply by looping through all the possible values, and computing the probability for each one. Note that if there is no evidence, then it is literally just computing the probabilities $P(X=x_i)$ for each $x_i$ in $X$’s domain. If there is evidence, then it is computing $P(e, X=x_i)$ for each $x_i$ in $X$’s domain – that is, it is computing the probability that $X$ has the given value $x_i$ and the evidence is true – so in that case, we use the law of conditional probability, which says that $$P(X=x_i | e) = \frac{P(e, X=x_i)}{P(e)}$$. Once we have computed $P(e, X=x_i)$ for all $x_i$, we can just normalize those values to get the correct distribution $P(X | e)$. ### Algorithm Steps -We can summurize the explained algorithm in the following steps: +We can summarize the explained algorithm in the following steps: 1. Select the entries consistent with the evidence. 2. Sum out the hidden variables to get joint distribution of query and evidence variables. @@ -87,7 +87,7 @@ def enumerate_all(e, vars): ### Algorithm Time Complexity -In worst case, we have no evidence, so we have to for loop through all possible values of all variables. Hence this algorithm has a complexity of $O(d^n)$ where $d$ is the size of the domain of the variables and $n$ is the number of variables. +In the worst case, we have no evidence, so we have to for loop through all possible values of all variables. Hence, this algorithm has a complexity of $O(d^n)$ where $d$ is the size of the domain of the variables and $n$ is the number of variables. ### Algorithm Example @@ -95,9 +95,9 @@ In worst case, we have no evidence, so we have to for loop through all possible ## Inference by Enumeration is Slow -In enumeration, first we find the whole joint distribution and then we marginalize out the hidden variables. Hence it is too slow because of the big joint distribution we need to compute. +In enumeration, first we find the whole joint distribution, and then we marginalize out the hidden variables. Hence, it is too slow because of the big joint distribution we need to compute. -If we marginalize out the hidden variables in partial joint distribution, we can get a much faster algorithm. This method is called Infrence by Variable Elimination. +If we marginalize out the hidden variables in the partial joint distribution, we can get a much faster algorithm. This method is called Inference by Variable Elimination.
    @@ -105,24 +105,24 @@ If we marginalize out the hidden variables in partial joint distribution, we can The point of the variable-elimination algorithm is that it is more bottom-up than top-down. Instead of figuring out the probabilities we need to compute and then computing all the other probabilities that each one depends on, we try to compute probabilities and then compute the other terms that depend on them, and repeatedly simplify the expression until we have something that is in terms of only the variable we’re looking for. -The variable-elimination algorithm uses things called factors. A factor is basically a CPT, except that the entries are not necessarily probabilities (but they would be if you normalized them). You can think of a factor as a matrix with a dimension for each variable, where $Factor[VAL1][VAL2][...]$ is (proportional to) a probability such as -$$P(VAR1=VAL1, VAR2=VAL2, ...)$$ +The variable-elimination algorithm uses things called factors. A factor is basically a CPT, except that the entries are not necessarily probabilities (but they would be if you normalized them). You can think of a factor as a matrix with a dimension for each variable, where $Factor[VAL1][VAL2][…]$ is (proportional to) a probability such as +$$P(VAR1=VAL1, VAR2=VAL2, …)$$ or you can think of it as a table with one row for each possible combination of assignments of values to the variables. We also define two operations on factors: 1. Join 2. Eliminate -Join is used to combine two factors. For example, if we have two factors $F_1$ and $F_2$, we can compute the joint distribution of $F_1$ and $F_2$ by multiplying their probabilities together, corresponding to variables value in factors. +Join is used to combine two factors. For example, if we have two factors $F_1$ and $F_2$, we can compute the joint distribution of $F_1$ and $F_2$ by multiplying their probabilities together, corresponding to the variable's value in factors. -Elimate is used to eliminate a variable from a factor. For example, if we have a factor $F$ and we want to eliminate $X$ from $F$, we can group rows by non-$X$ values and sum probabilities. This is exactly marginalization. +Eliminate is used to eliminate a variable from a factor. For example, if we have a factor $F$, and we want to eliminate $X$ from $F$, we can group rows by non-$X$ values and sum probabilities. This is exactly marginalization. > - Join is exactly like SQL join. > - Eliminate is like a group by and SUM aggregation function in SQL. ### Algorithm Explanation -This algorithm, like previous one, takes a variable X and returns a distribution over X, given some evidence e. First it initializes the list of factors; prior to any simplification, this is just the conditional probability tables for each variable given the evidence e. Then, it joins each factors with it. The summing-out process takes all the factors that depend on a given variable and replaces them with a single new factor that does not depend on that variable (by summing over all possible values of the variable). By the end of the loop, all the variables have been summed out except the query variable X, so then we can just multiply the factors together and normalize to get the distribution. +This algorithm, like the previous one, takes a variable X and returns a distribution over X, given some evidence e. First, it initializes the list of factors; prior to any simplification, this is just the conditional probability tables for each variable given the evidence e. Then, it joins each factors with it. The summing-out process takes all the factors that depend on a given variable and replaces them with a single new factor that does not depend on that variable (by summing over all possible values of the variable). By the end of the loop, all the variables have been summed out except the query variable X, so then we can just multiply the factors together and normalize to get the distribution. ### Algorithm Steps @@ -157,7 +157,7 @@ def elimination_ask(X, e, bn): ### Algorithm Time Complexity -The computational and space complexity of variable elimination is determined by the largest factor. In worst case, this algorithm has exponential complexity like the enumeration algorithm. But variable elimination ordering can greatly affect the largest factor. There is no general oredering that provides only small factors. +The computational and space complexity of variable elimination is determined by the largest factor. In the worst case, this algorithm has exponential complexity, like the enumeration algorithm. But variable elimination ordering can greatly affect the largest factor. There is no general ordering that provides only small factors. ### Ordering Polytree Variables for VE @@ -169,7 +169,7 @@ Polytree is a directed graph which has no undirected cycles. In this special kin 4. Sort the resulting nodes in topological order 5. Reverse the order -Know if we eliminate variables with this order, we would never get a factor larger than the original factors. This makes the VE algorithm linear time complexity. +Now, if we eliminate variables with this order, we would never get a factor larger than the original factors. This makes the VE algorithm linear time complexity. ### Cut-set Conditioning @@ -190,4 +190,4 @@ We reviewed two major exact inference algorithms, the enumeration algorithm and - [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) - [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) - [Variable Elimination](https://ermongroup.github.io/cs228-notes/inference/ve/) -- [Bayesian Networks - Inference (Part II)](https://ccc.inaoep.mx/~esucar/Clases-mgp/Notes/c7-bninf-p2.pdf) \ No newline at end of file +- [Bayesian Networks - Inference (Part II)](https://ccc.inaoep.mx/~esucar/Clases-mgp/Notes/c7-bninf-p2.pdf) From eff9764c412b5ef1a4973508822bcb837855d28f Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 17 Dec 2021 23:30:43 +0330 Subject: [PATCH 11/16] Add ordering image --- .../assets/ve-ordering.png | Bin 0 -> 176235 bytes .../index.md | 9 ++++++++- 2 files 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z{Mo1;%O?v)>Idx0?la%J7mr89DpYh4chU*{((8M1$!ncMNBP{G<(QiUyQNP1q%Ixh z&fYO;S0Y`@lR+^@n*_L1r^JL5U7cY~N--{`+FVY{atm3i)n+oH5qIy=Y%`!pdhvx? zK9MCl#IcnIWtK2F8ydxCC~Moqk0+S z|9Qv^!Gy>xUtjt75FR0bSr!OL{gqh!>x}L(!F~zF5#jFN_Q&!;_i*sPqhN^u*5>@b z>~5@L!$6Qj`Dg%YQNW@2KVf)~fsvR~c+% VS~|A* + +There is no general ordering that provides only small factors. ### Ordering Polytree Variables for VE From ea46ec2768f7ff27c55ea96cfff4d9edb5c72179 Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Fri, 17 Dec 2021 23:34:30 +0330 Subject: [PATCH 12/16] Add class slide to references --- notebooks/10_inference_in_bayesian_networks_1/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index 0120ef12..07f97347 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -194,6 +194,7 @@ We reviewed two major exact inference algorithms, the enumeration algorithm and ## References +- [Class Presentation](http://ce.sharif.edu/courses/99-00/1/ce417-2/resources/root/Slides/PDF/Session%2013_14.pdf) - [Visualizing Inference in Bayesian Networks](http://www.kbs.twi.tudelft.nl/Publications/MSc/2006-JRKoiter-Msc.html) - [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) - [Variable Elimination](https://ermongroup.github.io/cs228-notes/inference/ve/) From 6e9ddcbffd1a8ea7401fd01d6f8d7ef8d335f3db Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Sun, 26 Dec 2021 15:52:47 +0330 Subject: [PATCH 13/16] Fix images size --- notebooks/10_inference_in_bayesian_networks_1/index.md | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/notebooks/10_inference_in_bayesian_networks_1/index.md b/notebooks/10_inference_in_bayesian_networks_1/index.md index 07f97347..da510797 100644 --- a/notebooks/10_inference_in_bayesian_networks_1/index.md +++ b/notebooks/10_inference_in_bayesian_networks_1/index.md @@ -1,3 +1,4 @@ +
    # Inference in Bayes Nets 1 ## Table of Contents @@ -20,7 +21,7 @@ - [Conclusions](#conclusions) - [References](#references) -
    +

    ## Introduction @@ -99,7 +100,7 @@ In enumeration, first we find the whole joint distribution, and then we marginal If we marginalize out the hidden variables in the partial joint distribution, we can get a much faster algorithm. This method is called Inference by Variable Elimination. -
    +

    ## Inference by Variable Elimination (Marginalizing Early) @@ -162,7 +163,7 @@ The computational and space complexity of variable elimination is determined by - $Z, X_1, …, X_n \rightarrow 2^{n+1}$ - $X_1, …, X_n, Z \rightarrow 2^{2}$ -
    +

    There is no general ordering that provides only small factors. @@ -182,7 +183,7 @@ Now, if we eliminate variables with this order, we would never get a factor larg We can cut the bayes net at an instantiated variable, and this can transform a multi connected graph into a polytree, for which we can find the order of elimination. If these variables are not actually known, we can set them to each of their possible values and then solve the problem with the polytree. You can see an example of this below. -
    +

    ### Algorithm Example @@ -199,3 +200,4 @@ We reviewed two major exact inference algorithms, the enumeration algorithm and - [Exact Inference in Bayes Nets](http://courses.csail.mit.edu/6.034s/handouts/spring12/bayesnets-pseudocode.pdf) - [Variable Elimination](https://ermongroup.github.io/cs228-notes/inference/ve/) - [Bayesian Networks - Inference (Part II)](https://ccc.inaoep.mx/~esucar/Clases-mgp/Notes/c7-bninf-p2.pdf) +
    From 9f7b2c49712638d778b096057af1f50aa83f0b0a Mon Sep 17 00:00:00 2001 From: Parsa Mohammadian Date: Mon, 27 Dec 2021 09:15:24 +0330 Subject: [PATCH 14/16] Lecturenote/10 bayes net inference (#2) * Initial commit * Add intro * Add enumeration details * Add VE * Add ordering for VE * Update TOC * Add conditioning example * Change images size * Add comparison image * Fix some language mistakes * Add ordering image * Add class slide to references * Fix images size --- .gitignore | 1 + .../assets/.gitkeep | 0 .../assets/comparison.png | Bin 0 -> 296547 bytes .../assets/conditioning-example.png | Bin 0 -> 201015 bytes .../assets/intro.png | Bin 0 -> 41567 bytes .../assets/ve-ordering.png | Bin 0 -> 176235 bytes .../index.md | 203 ++++++++++++++++++ .../matadata.yml | 29 +++ notebooks/index.yml | 2 + 9 files changed, 235 insertions(+) create mode 100644 .gitignore create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/.gitkeep create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/comparison.png create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/conditioning-example.png create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/intro.png create mode 100644 notebooks/10_inference_in_bayesian_networks_1/assets/ve-ordering.png create mode 100644 notebooks/10_inference_in_bayesian_networks_1/index.md create mode 100644 notebooks/10_inference_in_bayesian_networks_1/matadata.yml diff --git a/.gitignore b/.gitignore new file mode 100644 index 00000000..44a09645 --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +*.ini \ No newline at end of file diff --git a/notebooks/10_inference_in_bayesian_networks_1/assets/.gitkeep b/notebooks/10_inference_in_bayesian_networks_1/assets/.gitkeep new file mode 100644 index 00000000..e69de29b diff --git a/notebooks/10_inference_in_bayesian_networks_1/assets/comparison.png b/notebooks/10_inference_in_bayesian_networks_1/assets/comparison.png new file mode 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