Answers.Body: <p>I believe you are correct. When Koller mentions a parameter, it is in reference to the values needed to express the probability distribution. For example, the total number of parameters for a Bernoulli distribution is one, the probability of success. You can watch <a href="https://youtu.be/Xhdpk9HZQuo?t=537" rel="nofollow noreferrer">Nando de Freitas's lecture on undergraduate machine learning 8: Inference in Bayesian networks and dynamic programming</a> for a detailed walkthrough of a binary Bayesian network where Nando also explains how many <em>parameters</em> are needed for each table in the network. In the lecture, he also briefly discussed some implementation details for Bayesian Networks, which also hints toward answering your question.</p>
  
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Body: <p>I just started to read the PGM book written by Daphne Koller.</p>
  <p>In the chapter of Bayesian Network Representation(Chapter 3), there are some descriptions about the standard parameterization of the joint distribution corresponding to n-trial coin tosses.</p>
  <p><a href="https://i.sstatic.net/9TElx.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9TElx.png" alt="enter image description here" /></a></p>
  <p>The book also says,</p>
  <p><a href="https://i.sstatic.net/YgM9g.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YgM9g.png" alt="enter image description here" /></a></p>
  <p>Here I'm very confused about the meaning of <em><strong><span class="math-container">$ 2^n parameters $</span></strong></em>. In terms of <strong>random variable</strong> or <strong>probability distribution</strong>, <em>parameter</em> means characteristic of the distribution. But <em>parameter</em> in this paragraph sounds like <span class="math-container">$O(2^n)$</span> space complexity. Because it also describes that we can reduce the space of all joint distribution to <span class="math-container">$n$</span>-dimension by using expression <span class="math-container">$ \prod_{i} \theta_{x_{i}} $</span>.</p>
  <p><a href="https://i.sstatic.net/w19f8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/w19f8.png" alt="enter image description here" /></a></p>
  <p>So, what's the meaning of <em><strong>parameter</strong></em> in this context? Does it mean space complexity for computation of the joint distribution?</p>
  
Tags: <terminology><bayesian-networks><books><probabilistic-graphical-models>
Title: In Probabilistic Graphical Model (written by Daphne Koller), what's the meaning of "parameter" in representation of the distribution?
UserDisplayName: Hongmin Yang
ThreadText: # In Probabilistic Graphical Model (written by Daphne Koller), what's the meaning of "parameter" in representation of the distribution?
  
  (Asked by: username_0 on 2021-03-25)
  
  I just started to read the PGM book written by Daphne Koller.
  
  In the chapter of Bayesian Network Representation(Chapter 3), there are some descriptions about the standard parameterization of the joint distribution corresponding to n-trial coin tosses.
  
  [![enter image description here](https://i.sstatic.net/9TElx.png)](https://i.sstatic.net/9TElx.png)
  
  The book also says,
  
  [![enter image description here](https://i.sstatic.net/YgM9g.png)](https://i.sstatic.net/YgM9g.png)
  
  Here I'm very confused about the meaning of ***$ 2^n parameters $***. In terms of **random variable** or **probability distribution**, *parameter* means characteristic of the distribution. But *parameter* in this paragraph sounds like $O(2^n)$ space complexity. Because it also describes that we can reduce the space of all joint distribution to $n$-dimension by using expression $ \prod\_{i} \theta\_{x\_{i}} $.
  
  [![enter image description here](https://i.sstatic.net/w19f8.png)](https://i.sstatic.net/w19f8.png)
  
  So, what's the meaning of ***parameter*** in this context? Does it mean space complexity for computation of the joint distribution?
  
  **Tags:** terminology, bayesian-networks, books, probabilistic-graphical-models
  
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  ## 1 answers
  
  ### Answer by username_1 on 2022-05-07. Score: 0
  
  I believe you are correct. When Koller mentions a parameter, it is in reference to the values needed to express the probability distribution. For example, the total number of parameters for a Bernoulli distribution is one, the probability of success. You can watch [Nando de Freitas's lecture on undergraduate machine learning 8: Inference in Bayesian networks and dynamic programming](https://youtu.be/Xhdpk9HZQuo?t=537) for a detailed walkthrough of a binary Bayesian network where Nando also explains how many *parameters* are needed for each table in the network. In the lecture, he also briefly discussed some implementation details for Bayesian Networks, which also hints toward answering your question.
  
  
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