The Central Idea
A Bayesian network consists of:
- a directed acyclic graph;
- a probability distribution for each node conditioned on its parents.
The network converts a large joint probability model into smaller local pieces.
For variables
\[ X_1,X_2,\ldots,X_n, \]the joint distribution factors as
\[ \boxed{ P(X_1,\ldots,X_n) = \prod_{i=1}^{n} P(X_i\mid \operatorname{Pa}(X_i)) }. \]Directed Acyclic Graph
The graphical part of a Bayesian network is a directed acyclic graph, often abbreviated
\[ \boxed{\text{DAG}}. \]“Directed” means the edges have arrows.
“Acyclic” means there is no directed path that returns to its starting node.
An Allowed Directed Structure
The chain
\[ A\longrightarrow B\longrightarrow C \]is acyclic.
Following the arrows forward never returns to \(A\).
A Directed Cycle Is Not Allowed
The structure
\[ A\longrightarrow B\longrightarrow C\longrightarrow A \]contains a directed cycle.
It is therefore not a Bayesian-network DAG.
Nodes Represent Random Variables
Each node represents a random variable.
Examples include:
- weather state;
- traffic level;
- equipment condition;
- test result;
- customer type;
- default status;
- delivery delay.
The variables may be binary, categorical, or numerical, depending on the model.
Edges Represent Local Conditional Structure
If
\[ A\longrightarrow B, \]then \(A\) is a parent of \(B\), and the local probability model for \(B\) is conditioned on \(A\):
\[ P(B\mid A). \]The arrow does not by itself provide the numerical values of this conditional distribution.
Root Nodes
A root node has no parents.
Its probability distribution is unconditional.
If \(A\) is a root node, the network includes a factor
\[ \boxed{P(A)}. \]Non-Root Nodes
A non-root node is modeled conditionally on its parents.
If
\[ A\longrightarrow B \]and
\[ C\longrightarrow B, \]then the local model for \(B\) has the form
\[ \boxed{ P(B\mid A,C) }. \]Local Conditional Probability Tables
For discrete variables, local conditional models are often stored in conditional probability tables.
Suppose \(A\) and \(B\) are binary and
\[ A\longrightarrow B. \]A table might specify:
| \(A\) | \(P(B=1\mid A)\) | \(P(B=0\mid A)\) |
|---|---|---|
| 0 | 0.20 | 0.80 |
| 1 | 0.75 | 0.25 |
Each Conditional Row Must Sum to One
For each fixed parent configuration, the child probabilities form a probability distribution.
Thus,
\[ P(B=1\mid A=a)+P(B=0\mid A=a)=1. \]This rule applies separately for every parent state.
The Bayesian-Network Factorization
Suppose the network contains
\[ A\longrightarrow B\longrightarrow C. \]The factorization is
\[ \boxed{ P(A,B,C) = P(A)P(B\mid A)P(C\mid B) }. \]Each node contributes one local probability factor.
Why the Factorization Works
The general chain rule of probability gives
\[ P(A,B,C) = P(A)P(B\mid A)P(C\mid A,B). \]The network structure asserts the conditional independence
\[ C\perp A\mid B. \]Therefore,
\[ P(C\mid A,B)=P(C\mid B), \]giving
\[ P(A,B,C) = P(A)P(B\mid A)P(C\mid B). \]A Common-Cause Network
Consider
\[ A\longleftarrow B\longrightarrow C. \]The factorization is
\[ \boxed{ P(A,B,C) = P(B)P(A\mid B)P(C\mid B) }. \]The root node \(B\) supplies the unconditional factor.
A Common-Effect Network
Consider
\[ A\longrightarrow C\longleftarrow B. \]If \(A\) and \(B\) are root nodes, the factorization is
\[ \boxed{ P(A,B,C) = P(A)P(B)P(C\mid A,B) }. \]The child \(C\) is conditioned on both parents.
Example: Weather, Traffic, and Delay
Consider
\[ W\longrightarrow T\longrightarrow D, \]where:
- \(W\) = bad weather;
- \(T\) = heavy traffic;
- \(D\) = delivery delay.
The joint distribution factors as
\[ \boxed{ P(W,T,D) = P(W)P(T\mid W)P(D\mid T) }. \]Specify the Root Probability
Suppose
\[ P(W=1)=0.20. \]Therefore,
\[ P(W=0)=0.80. \]Specify the Traffic Model
Suppose
\[ P(T=1\mid W=1)=0.80 \]and
\[ P(T=1\mid W=0)=0.30. \]Then
\[ P(T=0\mid W=1)=0.20 \]and
\[ P(T=0\mid W=0)=0.70. \]Specify the Delay Model
Suppose
\[ P(D=1\mid T=1)=0.70 \]and
\[ P(D=1\mid T=0)=0.10. \]The network now has enough local information to calculate any complete joint assignment.
Calculate One Joint Probability
Find
\[ P(W=1,T=1,D=1). \]Use the factorization:
\[ P(W=1)P(T=1\mid W=1)P(D=1\mid T=1). \]Therefore,
\[ 0.20(0.80)(0.70) = \boxed{0.112}. \]Another Joint Probability
Find
\[ P(W=0,T=0,D=1). \]Then
\[ P(W=0)=0.80, \] \[ P(T=0\mid W=0)=0.70, \] \[ P(D=1\mid T=0)=0.10. \]Therefore,
\[ 0.80(0.70)(0.10) = \boxed{0.056}. \]Joint Probabilities for Complete Assignments
A complete assignment specifies one state for every variable.
For a binary three-node network, examples include
\[ (W,T,D)=(1,1,1), \] \[ (1,1,0), \] \[ (1,0,1), \]and so forth.
The probabilities of all complete assignments sum to \(1\).
Marginalization
Often we want the probability of one variable without specifying the others.
This requires summing over the unobserved variables.
This operation is called marginalization.
Example: Marginal Probability of Traffic
In the weather-traffic model,
\[ P(T=1) = P(W=1)P(T=1\mid W=1) + P(W=0)P(T=1\mid W=0). \]Therefore,
\[ P(T=1) = 0.20(0.80)+0.80(0.30). \]Thus,
\[ \boxed{ P(T=1)=0.40 }. \]Example: Marginal Probability of Delay
Since
\[ P(T=1)=0.40, \]we have
\[ P(T=0)=0.60. \]Therefore,
\[ P(D=1) = 0.40(0.70)+0.60(0.10). \]Thus,
\[ \boxed{ P(D=1)=0.34 }. \]Inference from Evidence
A Bayesian network becomes especially useful when some node values are observed.
The observed values are called evidence.
We can then ask for conditional probabilities of unobserved nodes.
For example:
\[ P(W=1\mid D=1). \]Bayesian Inference Can Move Backward
The network has arrows
\[ W\longrightarrow T\longrightarrow D. \]Yet after observing a delay, we may infer backward toward weather:
\[ P(W\mid D). \]The arrow directions define the factorization, not the only direction in which inference may occur.
Find the Joint Probability of Weather and Delay
To calculate
\[ P(W=1,D=1), \]sum over the possible traffic states:
\[ P(W=1,D=1) = \sum_t P(W=1,T=t,D=1). \]Thus,
\[ P(W=1,D=1) = 0.20(0.80)(0.70) + 0.20(0.20)(0.10). \]Complete the Weather-and-Delay Calculation
The two path probabilities are
\[ 0.112 \]and
\[ 0.004. \]Therefore,
\[ \boxed{ P(W=1,D=1)=0.116 }. \]Infer Weather from an Observed Delay
We already found
\[ P(D=1)=0.34. \]Therefore,
\[ P(W=1\mid D=1) = \frac{P(W=1,D=1)}{P(D=1)}. \]Thus,
\[ P(W=1\mid D=1) = \frac{0.116}{0.34} \approx \boxed{0.3412}. \]Observing a delay raises the probability of bad weather from \(20\%\) to about \(34.12\%\).
Conditional Independence Reduces the Number of Parameters
In
\[ W\longrightarrow T\longrightarrow D, \]the network assumes
\[ D\perp W\mid T. \]Therefore, the delay model needs
\[ P(D\mid T), \]not a separate probability for every combination of \(W\) and \(T\).
Why Factorization Saves Work
A fully unrestricted joint distribution for several discrete variables may require many parameters.
A Bayesian network uses conditional-independence structure to replace one large table with smaller local tables.
This can greatly reduce the amount of information needed to specify the model.
Example: Three Binary Variables Without Structure
Three binary variables have
\[ 2^3=8 \]possible joint states.
Since the eight probabilities must sum to \(1\), an unrestricted joint table requires
\[ 8-1=7 \]independent probability parameters.
Example: Three-Node Chain with Structure
For
\[ A\longrightarrow B\longrightarrow C, \]with binary variables, we need:
- one parameter for \(P(A=1)\);
- two parameters for \(P(B=1\mid A)\);
- two parameters for \(P(C=1\mid B)\).
That is
\[ \boxed{5} \]free parameters rather than \(7\).
Larger Networks Can Save Much More
The parameter savings become more important as the number of variables grows.
Sparse dependency structure can make a probability model practical when a full joint table would be too large to specify directly.
Example: Four-Node Network
Consider
\[ A\longrightarrow B, \] \[ A\longrightarrow C, \] \[ B\longrightarrow D, \qquad C\longrightarrow D. \]The factorization is
\[ \boxed{ P(A,B,C,D) = P(A)P(B\mid A)P(C\mid A)P(D\mid B,C) }. \]Read the Parent Sets
In the four-node network,
\[ \operatorname{Pa}(A)=\varnothing, \] \[ \operatorname{Pa}(B)=\{A\}, \] \[ \operatorname{Pa}(C)=\{A\}, \] \[ \operatorname{Pa}(D)=\{B,C\}. \]Each factor conditions only on the node's own parent set.
The Local Markov Idea
Informally, a Bayesian network treats each node as conditionally independent of non-descendant variables once its parents are known.
This principle underlies the factorization
\[ P(X_1,\ldots,X_n) = \prod_iP(X_i\mid\operatorname{Pa}(X_i)). \]A more complete graphical treatment belongs to later probability theory.
Common Cause in a Bayesian Network
Consider
\[ X\longleftarrow Z\longrightarrow Y. \]The factorization is
\[ P(X,Y,Z) = P(Z)P(X\mid Z)P(Y\mid Z). \]The model implies
\[ X\perp Y\mid Z. \]Common Effect in a Bayesian Network
Consider
\[ X\longrightarrow Z\longleftarrow Y. \]The factorization is
\[ P(X,Y,Z) = P(X)P(Y)P(Z\mid X,Y) \]if \(X\) and \(Y\) are root nodes with no connecting dependence in the model.
Observing a Common Effect Can Connect Its Parents
In
\[ X\longrightarrow Z\longleftarrow Y, \]the root nodes \(X\) and \(Y\) may be independent before \(Z\) is observed.
Conditioning on \(Z\) can make them dependent.
Bayesian networks preserve the same collider behavior introduced in dependency diagrams.
Example: Alarm Network
Suppose an alarm \(A\) can be triggered by smoke \(S\) or a sensor fault \(F\):
\[ S\longrightarrow A\longleftarrow F. \]Suppose the priors are
\[ P(S)=0.02, \qquad P(F)=0.01. \]The local alarm model must specify
\[ P(A\mid S,F) \]for all four combinations of the two parent states.
A Conditional Probability Table with Two Parents
| \(S\) | \(F\) | \(P(A=1\mid S,F)\) |
|---|---|---|
| 0 | 0 | 0.01 |
| 0 | 1 | 0.80 |
| 1 | 0 | 0.95 |
| 1 | 1 | 0.99 |
The child distribution depends jointly on both parents.
Calculate One Alarm-Network Joint Probability
Find
\[ P(S=1,F=0,A=1). \]Since the factorization is
\[ P(S)P(F)P(A\mid S,F), \]we obtain
\[ 0.02(0.99)(0.95). \]Therefore,
\[ \boxed{ P(S=1,F=0,A=1)=0.01881 }. \]Bayesian Networks Support Diagnostic Inference
After observing
\[ A=1, \]we may ask:
\[ P(S=1\mid A=1), \] \[ P(F=1\mid A=1), \]or even
\[ P(S=1\mid A=1,F=0). \]These are posterior probabilities obtained by conditioning the joint model on evidence.
Prediction and Diagnosis
In a Bayesian network, two broad inference directions are common.
Predictive inference moves from upstream variables toward downstream variables:
\[ \text{cause-like state} \longrightarrow \text{possible observation}. \]Diagnostic inference uses observed downstream variables to update upstream possibilities.
Inference Can Combine Both Directions
In a larger network, evidence can appear at several locations.
A posterior may depend on both upstream and downstream observations.
The network structure helps organize how all of those pieces enter the calculation.
Observed Variables
An observed variable is one whose value is known.
If
\[ D=1 \]is observed in the weather-traffic-delay network, that value becomes evidence.
Probabilities for unobserved variables are then conditioned on that evidence.
Hidden Variables
A hidden or latent variable is part of the model but is not directly observed.
Bayesian networks can estimate probabilities of hidden states using observed descendants or related variables.
Example: Hidden Equipment State
Suppose
\[ H\longrightarrow M_1, \qquad H\longrightarrow M_2, \]where \(H\) is hidden equipment condition and \(M_1,M_2\) are sensor readings.
The factorization is
\[ P(H,M_1,M_2) = P(H)P(M_1\mid H)P(M_2\mid H). \]Conditional Independence of Measurements
The previous model assumes
\[ M_1\perp M_2\mid H. \]Once the hidden state \(H\) is known, the two measurements are modeled as independent.
Marginally, however, the measurements may be associated because both depend on \(H\).
The Network Is a Model, Not the Data Itself
A Bayesian network is a mathematical representation of assumptions and probabilities.
Data may be used to estimate the network's local probability tables, but the graph and probabilities together constitute the model.
Graph Structure May Come from Different Sources
A Bayesian-network structure may be proposed from:
- domain knowledge;
- process knowledge;
- temporal ordering;
- statistical modeling assumptions;
- data-driven structure-learning methods.
The interpretation depends on how the network was constructed.
Arrows Do Not Automatically Prove Causation
A Bayesian network may be used purely as a probabilistic factorization.
In that setting,
\[ A\longrightarrow B \]means that \(B\)'s local distribution is conditioned on \(A\).
A causal interpretation requires additional assumptions.
Bayesian Networks Are Not Decision Trees
A decision tree usually organizes choices and consequences.
A Bayesian network organizes random variables and their conditional dependencies.
The two structures can be used together in decision analysis, but they are not the same mathematical object.
Bayesian Networks Are Not Probability Trees
A probability tree expands sequential branches into complete paths.
A Bayesian network represents variables and local dependence relations.
The network may be much more compact because it does not need to draw every complete outcome path explicitly.
Bayesian Networks Can Represent Many Joint States Compactly
A network with several variables implicitly represents a probability for every complete joint assignment.
The local conditional tables determine those joint probabilities through multiplication.
Marginalization and conditioning then recover other probabilities of interest.
Three Fundamental Operations
Introductory Bayesian-network calculations rely heavily on three operations:
- factorization — multiply local conditional factors;
- marginalization — sum over unobserved variables;
- conditioning — restrict the model using observed evidence.
Factorization
Factorization converts the graph into a joint distribution:
\[ P(X_1,\ldots,X_n) = \prod_iP(X_i\mid\operatorname{Pa}(X_i)). \]This is the core structural equation of a Bayesian network.
Marginalization
If a variable is unobserved and not part of the requested event, sum over its possible values.
For example,
\[ P(A,C) = \sum_bP(A,b,C). \]Conditioning
If evidence \(E=e\) is observed, posterior probabilities are computed from
\[ \boxed{ P(H\mid E=e) = \frac{P(H,E=e)} {P(E=e)} }. \]Bayesian-network structure helps calculate the numerator and denominator efficiently.
Normalization
When several candidate states remain after evidence is observed, their unnormalized weights can be divided by their total.
This is the same normalization idea used in Bayes' theorem.
Example: Normalize Two Hidden States
Suppose evidence \(E\) produces unnormalized weights
\[ 0.12 \]for \(H=1\) and
\[ 0.08 \]for \(H=0\).
The total is
\[ 0.20. \]Therefore,
\[ P(H=1\mid E)=\frac{0.12}{0.20}=\boxed{0.60}. \]Topological Order
Because a Bayesian network is acyclic, its nodes can be ordered so that each parent appears before its children.
Such an ordering is called a topological order.
Example of a Topological Order
For
\[ A\longrightarrow B, \qquad A\longrightarrow C, \qquad B\longrightarrow D, \qquad C\longrightarrow D, \]one valid ordering is
\[ \boxed{A,B,C,D}. \]Another valid ordering is
\[ \boxed{A,C,B,D}. \]Topological Order Supports the Factorization
In a valid ordering, every node's parent values are conceptually available before the node's local conditional distribution is applied.
This aligns naturally with the chain-rule construction of the joint probability.
The Network Encodes More Than One Conditional Independence
A larger graph can imply many conditional-independence relationships.
The graph therefore serves as a compact language for describing how information is structured across many variables.
A full graphical criterion for reading all such relationships is beyond this introductory lesson.
Markov Blanket: A Gentle Introduction
For a node \(X\), a particularly important local set consists of:
- its parents;
- its children;
- the other parents of its children.
This set is called the Markov blanket of \(X\).
Why the Markov Blanket Matters
In a Bayesian network, once the Markov blanket of \(X\) is known, variables outside that blanket provide no additional information about \(X\) under the model.
This is a useful local conditional-independence idea.
Detailed Markov-blanket methods belong to more advanced graphical-model study.
Example: Markov Blanket in a Small Network
Suppose
\[ A\longrightarrow X, \] \[ X\longrightarrow C, \] \[ B\longrightarrow C. \]Then the Markov blanket of \(X\) contains:
- parent \(A\);
- child \(C\);
- co-parent \(B\).
Common Error: Multiplying Every Probability in Sight
The network factorization uses exactly one local factor for each node:
\[ P(X_i\mid\operatorname{Pa}(X_i)). \]Extra conditional probabilities should not be inserted arbitrarily.
Common Error: Conditioning a Node on All Earlier Variables
The full chain rule could condition a variable on every earlier variable.
A Bayesian network uses graph structure to reduce that conditioning set to the node's parents.
For
\[ A\longrightarrow B\longrightarrow C, \]use
\[ P(C\mid B), \]not automatically
\[ P(C\mid A,B). \]Common Error: Assuming No Edge Means Complete Independence
Two nodes without a direct edge may still be dependent through another path.
The graph must be interpreted through its full conditional structure.
Common Error: Forgetting to Sum over Hidden Variables
If a requested probability does not specify an intermediate hidden node, that node's possible states must usually be marginalized out.
For example,
\[ P(A,C) = \sum_bP(A,b,C). \]Common Error: Forgetting Normalization after Evidence
Joint weights such as
\[ P(H,E) \]are not automatically posterior probabilities.
They must be divided by
\[ P(E). \]Common Error: Treating the Graph as Numerical Data
The graph provides dependency structure.
Conditional probability tables or other local distributions provide the numerical parameters.
Both are required for a complete Bayesian network.
Common Error: Assuming Every Bayesian Network Is Causal
Bayesian networks can be used purely as probabilistic models.
Arrows should receive causal interpretation only when additional causal assumptions justify that interpretation.
Common Error: Allowing Directed Cycles
A standard Bayesian network must be acyclic.
A structure such as
\[ A\longrightarrow B\longrightarrow C\longrightarrow A \]violates the DAG requirement.
A Reliable Procedure for Reading a Bayesian Network
- Identify every node and its possible states.
- Identify the parents of each node.
- Check that the graph is directed and acyclic.
- Write one local probability factor for each node.
- Multiply the local factors to obtain complete joint probabilities.
- Sum over unobserved variables when marginal probabilities are needed.
- Condition on observed evidence when posterior probabilities are needed.
- Use normalization when comparing remaining hidden states.
- Interpret conditional-independence assumptions carefully.
A Reliable Procedure for Factorization
- Choose any topological ordering of the nodes.
- For each root node, write its unconditional probability.
- For each non-root node, write its probability conditioned on its parent set.
- Multiply all node factors together.
- Check that every node appears exactly once as the child variable of a factor.
Example: Full Factorization Workflow
Consider
\[ A\longrightarrow B, \] \[ A\longrightarrow C, \] \[ B\longrightarrow D, \qquad C\longrightarrow D. \]The parent sets are
\[ \operatorname{Pa}(A)=\varnothing, \] \[ \operatorname{Pa}(B)=\{A\}, \] \[ \operatorname{Pa}(C)=\{A\}, \] \[ \operatorname{Pa}(D)=\{B,C\}. \]Therefore,
\[ \boxed{ P(A,B,C,D) = P(A)P(B\mid A)P(C\mid A)P(D\mid B,C) }. \]Example: Full Numerical Workflow
Suppose
\[ A\longrightarrow B \]with binary variables and
\[ P(A=1)=0.30, \] \[ P(B=1\mid A=1)=0.80, \] \[ P(B=1\mid A=0)=0.20. \]Find
\[ P(A=1,B=1). \]The factorization gives
\[ P(A=1,B=1) = 0.30(0.80) = \boxed{0.24}. \]Find the Marginal Probability of \(B=1\)
Use both possible values of \(A\):
\[ P(B=1) = P(A=1)P(B=1\mid A=1) + P(A=0)P(B=1\mid A=0). \]Therefore,
\[ P(B=1) = 0.30(0.80)+0.70(0.20). \]Thus,
\[ \boxed{ P(B=1)=0.38 }. \]Infer \(A\) after Observing \(B\)
Bayes' theorem gives
\[ P(A=1\mid B=1) = \frac{P(A=1,B=1)}{P(B=1)}. \]Therefore,
\[ P(A=1\mid B=1) = \frac{0.24}{0.38} = \boxed{\frac{12}{19}} \approx \boxed{0.6316}. \]The posterior probability of \(A=1\) rises from \(30\%\) to about \(63.16\%\).
Practice 1: Identify the Factorization
For
\[ A\longrightarrow B\longrightarrow C, \]write the Bayesian-network factorization of
\[ P(A,B,C). \]Practice 2: Common Cause
For
\[ A\longleftarrow B\longrightarrow C, \]write the factorization of
\[ P(A,B,C). \]Practice 3: Common Effect
For
\[ A\longrightarrow C\longleftarrow B, \]assume \(A\) and \(B\) are root nodes. Write the factorization.
Practice 4: One Joint Probability
Suppose
\[ A\longrightarrow B, \] \[ P(A=1)=0.40, \]and
\[ P(B=1\mid A=1)=0.70. \]Find
\[ P(A=1,B=1). \]Practice 5: Marginalization
Suppose
\[ P(A=1)=0.40, \] \[ P(B=1\mid A=1)=0.70, \] \[ P(B=1\mid A=0)=0.20. \]Find
\[ P(B=1). \]Practice 6: Posterior Inference
Using the probabilities from Practice 5, find
\[ P(A=1\mid B=1). \]Practice 7: Parent Set
Suppose
\[ A\longrightarrow D, \qquad B\longrightarrow D, \qquad C\longrightarrow D. \]Write the local probability factor associated with \(D\).
Practice 8: Acyclicity
Is
\[ A\longrightarrow B\longrightarrow C\longrightarrow A \]a valid Bayesian-network structure? Explain.
Practice 9: Conditional Independence
In the chain
\[ A\longrightarrow B\longrightarrow C, \]what conditional-independence statement is encoded between \(A\) and \(C\) once \(B\) is known?
Practice 10: Identify the Error
A student sees
\[ A\longrightarrow B \]and writes
\[ P(A,B)=P(A)P(B). \]Explain the error and give the correct factorization.
Practice Answers
Practice 1
\[ \boxed{ P(A,B,C) = P(A)P(B\mid A)P(C\mid B) }. \]Practice 2
\[ \boxed{ P(A,B,C) = P(B)P(A\mid B)P(C\mid B) }. \]Practice 3
\[ \boxed{ P(A,B,C) = P(A)P(B)P(C\mid A,B) }. \]Practice 4
\[ P(A=1,B=1) = 0.40(0.70) = \boxed{0.28}. \]Practice 5
Since
\[ P(A=0)=0.60, \]total probability gives
\[ P(B=1) = 0.40(0.70)+0.60(0.20). \]Therefore,
\[ P(B=1) = 0.28+0.12 = \boxed{0.40}. \]Practice 6
The joint probability is
\[ P(A=1,B=1)=0.28. \]The marginal is
\[ P(B=1)=0.40. \]Therefore,
\[ P(A=1\mid B=1) = \frac{0.28}{0.40} = \boxed{0.70}. \]Practice 7
The parent set is
\[ \operatorname{Pa}(D)=\{A,B,C\}. \]Therefore, the local factor is
\[ \boxed{ P(D\mid A,B,C) }. \]Practice 8
No. The structure contains the directed cycle
\[ A\longrightarrow B\longrightarrow C\longrightarrow A. \]A Bayesian network must be a directed acyclic graph.
Practice 9
\[ \boxed{ A\perp C\mid B }. \]Practice 10
The arrow indicates that \(B\) is modeled conditionally on \(A\), so independence should not be assumed.
The correct factorization is
\[ \boxed{ P(A,B)=P(A)P(B\mid A) }. \]What to Remember
A Bayesian network consists of a directed acyclic graph and local conditional probability models.
Its central factorization is
\[ \boxed{ P(X_1,\ldots,X_n) = \prod_{i=1}^{n} P(X_i\mid\operatorname{Pa}(X_i)) }. \]- Each node represents a random variable.
- Each directed edge identifies a parent-child relationship.
- Root nodes have unconditional distributions.
- Non-root nodes are conditioned on their parents.
- The graph must be acyclic.
- Conditional independence allows the joint distribution to factor into smaller pieces.
- Joint probabilities are found by multiplying local factors.
- Marginal probabilities are found by summing over unobserved variables.
- Posterior probabilities are found by conditioning on observed evidence.
- Inference can move in either direction even though the factorization follows the arrows.
- A Bayesian network is not automatically a causal model.
The central habit is: use the graph to identify the local factors, multiply to build joint probabilities, and then sum or condition to answer the question being asked.
Where We Go Next
Bayesian networks show how a structured collection of conditional probabilities can support inference across many connected variables. The final lesson in this sequence focuses on what happens when evidence arrives repeatedly over time.
The next lesson, Sequential Updating, develops repeated Bayesian updates, shows how one posterior becomes the next prior, and examines the conditions under which successive pieces of evidence can be combined consistently.