Body: <p>I have used Beta function to estimate the performance of the agent. I have failure and success data of the task that runs on the agent. 
  The parameter <span class="math-container">$\alpha$</span> is a number of successful tasks, while <span class="math-container">$\beta$</span> is the number of failures. Thus, I can estimate the performance by exploiting the expected value of Beta, as <span class="math-container">$$\mu = \frac{\alpha} {(\alpha+\beta)}$$</span></p>
  
  <p>So, I am looking for a similar model, such that its parameter can be estimated from the success and failure data. So far I found Dirichlet distribution. </p>
  
  <p>What is the expected value of Dirichlet distribution? How I can use the success and failure data to estimate parameters of this distribution?</p>
  
  <p>Let's check the following example:</p>
  
  <p>Suppose that we use a Dirichlet prior represented by <span class="math-container">$Dirichlet(1, 1, 1)$</span> and observe <span class="math-container">$13$</span> results with <span class="math-container">$8$</span> Successful, <span class="math-container">$2$</span> Missing, and <span class="math-container">$3$</span> Failures. Then we get the posterior to be <span class="math-container">$Dirichlet(1+8, 1+2, 1+3)$</span>. Then if you define the performance value <span class="math-container">$\alpha$</span> to be the expectation of <span class="math-container">$P(x=Successful)$</span>, 
  then <span class="math-container">$\alpha$</span> will be <span class="math-container">$(1+8)/[(1+8)+(1+2)+(1+3)] = 0.56$</span></p>
  
  <p>Now 
  Suppose that we use a Beta prior represented by <span class="math-container">$Beta(1,1)$</span> and observe <span class="math-container">$13$</span> results with <span class="math-container">$8$</span> Successful, and <span class="math-container">$3$</span> Failures. Then we get the posterior to be <span class="math-container">$Beta(1+8, 1+3)$</span>. Then if you define the performance value Pr to be the expectation of <span class="math-container">$P(x=Successful)$</span>, 
  then <span class="math-container">$\alpha = (1+8)/[(1+8)+(1+3)] = 0.69$</span></p>
  
  <p>Are my calculations and concept right?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-03-09T03:43:42.527Z
Comments.Text: Is this problem in the context of reinforcement learning? So, is your agent an RL agent?
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-03-09T20:52:26.877Z
Comments.Text: Actually no, but thanks for this good idea, seems interesting
Comments.UserDisplayName: jou
Tags: <probability-distribution>
Title: How can I use the success and failure data to estimate parameters of a Dirichlet distribution?
UserDisplayName: jou
ThreadText: # How can I use the success and failure data to estimate parameters of a Dirichlet distribution?
  
  (Asked by: username_0 on 2020-03-08)
  
  I have used Beta function to estimate the performance of the agent. I have failure and success data of the task that runs on the agent.
  The parameter $\alpha$ is a number of successful tasks, while $\beta$ is the number of failures. Thus, I can estimate the performance by exploiting the expected value of Beta, as $$\mu = \frac{\alpha} {(\alpha+\beta)}$$
  
  So, I am looking for a similar model, such that its parameter can be estimated from the success and failure data. So far I found Dirichlet distribution.
  
  What is the expected value of Dirichlet distribution? How I can use the success and failure data to estimate parameters of this distribution?
  
  Let's check the following example:
  
  Suppose that we use a Dirichlet prior represented by $Dirichlet(1, 1, 1)$ and observe $13$ results with $8$ Successful, $2$ Missing, and $3$ Failures. Then we get the posterior to be $Dirichlet(1+8, 1+2, 1+3)$. Then if you define the performance value $\alpha$ to be the expectation of $P(x=Successful)$,
  then $\alpha$ will be $(1+8)/[(1+8)+(1+2)+(1+3)] = 0.56$
  
  Now
  Suppose that we use a Beta prior represented by $Beta(1,1)$ and observe $13$ results with $8$ Successful, and $3$ Failures. Then we get the posterior to be $Beta(1+8, 1+3)$. Then if you define the performance value Pr to be the expectation of $P(x=Successful)$,
  then $\alpha = (1+8)/[(1+8)+(1+3)] = 0.69$
  
  Are my calculations and concept right?
  
  **Tags:** probability-distribution
  
  > **Comments:**
  > **username_1**: Is this problem in the context of reinforcement learning? So, is your agent an RL agent?  
  > **username_0**: Actually no, but thanks for this good idea, seems interesting  
  
  ---
  
  ## 0 answers
  
  