Answers.Body: <p>The main difference between a Bayesian network and a Markov chain is not that a Markov Chain is not directional, it is that the graph of the Bayesian network is not trivial whereas the graph of a Markov chain would be somewhat trivial, as all the previous <span class="math-container">$k$</span> nodes would just point to the current node. To illustrate further why this would be trivial, we let each node represent a random variable <span class="math-container">$X_i$</span>. Then the nodes representing <span class="math-container">$X_i$</span> for <span class="math-container">$ t-k \leq i &lt; t$</span> would be connected by a directed edge to <span class="math-container">$X_t$</span>. That is, the edges <span class="math-container">$(X_i, X_t) \in E$</span> for <span class="math-container">$ t-k \leq i &lt; t$</span> and where <span class="math-container">$E$</span> is the set of edges of the graph. </p>
  
  <p>To illustrate this please see the examples below. </p>
  
  <p>Assume that we have a <span class="math-container">$k$</span>th order Markov chain, then by definition we have <span class="math-container">$\forall t &gt; k$</span> <span class="math-container">$\mathbb{P}(X_t = x | X_0,...,X_{t-1}) = \mathbb{P}(X_t = x | X_{t-k},...,X_{t-1})$</span>. </p>
  
  <p>The main difference between the above definition and the definition of a Bayesian Network is that due to the direction of the graph we can have different dependencies for each <span class="math-container">$X_t$</span>. Consider the Bayesian Network in the Figure below 
  <a href="https://i.sstatic.net/dsyLj.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/dsyLj.png" alt="Bayesian Network"></a></p>
  
  <p>We would get that <span class="math-container">$\mathbb{P}(X_4 = x| X_1, X_2, X_3) = \mathbb{P}(X_4 = x | X_2, X_3)$</span> and <span class="math-container">$\mathbb{P}(X_5 = x| X_1, X_2, X_3, X_4) = \mathbb{P}(X_5 = x | X_3)$</span>. </p>
  
  <p>So, the past events that the current random variable depends on don't have to have the same 'structure' in a Bayesian Network as in a Markov Chain. </p>
  
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Answers.CreationDate: 2020-05-18T17:57:25.787Z
Answers.LastActivityDate: 2020-05-19T13:37:08.703Z
Answers.LastEditDate: 2020-05-19T13:37:08.703Z
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Body: <p>I am trying to understand the difference between a Bayesian Network and a Markov Chain.</p>
  
  <p>When I search for this one the web, the unanimous solution seems to be that a Bayesian Network is directional (i.e. it's a DAG) and a Markov Chain is not directional.</p>
  
  <p>However, often a Markov Chain example is overtime, where the weather today is impacting the weather tomorrow, but the weather tomorrow is not (obviously) impacting the weather today. So I am quite confused how is a Markov Chain not directional?</p>
  
  <p>I seem to be missing something here. Can someone please help me understand?</p>
  
Tags: <markov-chain><bayesian-networks>
Title: What is the difference between a Bayesian Network and a Markov Chain?
UserDisplayName: Newskooler
ThreadText: # What is the difference between a Bayesian Network and a Markov Chain?
  
  (Asked by: username_0 on 2020-05-18)
  
  I am trying to understand the difference between a Bayesian Network and a Markov Chain.
  
  When I search for this one the web, the unanimous solution seems to be that a Bayesian Network is directional (i.e. it's a DAG) and a Markov Chain is not directional.
  
  However, often a Markov Chain example is overtime, where the weather today is impacting the weather tomorrow, but the weather tomorrow is not (obviously) impacting the weather today. So I am quite confused how is a Markov Chain not directional?
  
  I seem to be missing something here. Can someone please help me understand?
  
  **Tags:** markov-chain, bayesian-networks
  
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  ## 1 answers
  
  ### Answer by username_1 on 2020-05-18. Score: 1
  
  The main difference between a Bayesian network and a Markov chain is not that a Markov Chain is not directional, it is that the graph of the Bayesian network is not trivial whereas the graph of a Markov chain would be somewhat trivial, as all the previous $k$ nodes would just point to the current node. To illustrate further why this would be trivial, we let each node represent a random variable $X\_i$. Then the nodes representing $X\_i$ for $ t-k \leq i < t$ would be connected by a directed edge to $X\_t$. That is, the edges $(X\_i, X\_t) \in E$ for $ t-k \leq i < t$ and where $E$ is the set of edges of the graph.
  
  To illustrate this please see the examples below.
  
  Assume that we have a $k$th order Markov chain, then by definition we have $\forall t > k$ $\mathbb{P}(X\_t = x | X\_0,...,X\_{t-1}) = \mathbb{P}(X\_t = x | X\_{t-k},...,X\_{t-1})$.
  
  The main difference between the above definition and the definition of a Bayesian Network is that due to the direction of the graph we can have different dependencies for each $X\_t$. Consider the Bayesian Network in the Figure below
  [![Bayesian Network](https://i.sstatic.net/dsyLj.png)](https://i.sstatic.net/dsyLj.png)
  
  We would get that $\mathbb{P}(X\_4 = x| X\_1, X\_2, X\_3) = \mathbb{P}(X\_4 = x | X\_2, X\_3)$ and $\mathbb{P}(X\_5 = x| X\_1, X\_2, X\_3, X\_4) = \mathbb{P}(X\_5 = x | X\_3)$.
  
  So, the past events that the current random variable depends on don't have to have the same 'structure' in a Bayesian Network as in a Markov Chain.
  
  
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