text: # Travis: Segment 39
  
  ### To Calculate
  
  1. Suppose the domain of a model are the five integers $x = \{1,2,3,4,5\}$, and that your proposal distribution is: "When $x_1 = 2,3,4$, choose with equal probability $x_2 = x_1 \pm 1$. For $x_1=1$ always choose $x_2 =2$. For $x_1=5$ always choose $x_2 =4$. What is the ratio of $q$'s that goes into the acceptance probability $\alpha(x_1,x_2)$ for all the possible values of $x_1$ and $x_2$?
  
  $q(x2|x1) =$
  
   0 1 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 1 0
  
  and $q(x1|x2) = q(x2|x1)^{T}$
  
  So, the ratio of q's is just the element wise division of $\frac{q(x1|x2)}{q(x2|x1)} = \frac{q(x1|x2)}{q(x1|x2)^T}$ which is
  
   0 2 0 0 0 0.5 0 1 0 0 0 1 0 1 0 0 0 1 0 0.5 0 0 0 1 0
  
  2. Suppose the domain of a model is $-\infty < x < \infty$ and your proposal distribution is (perversely),
  
  $q(x_2|x_1) = \begin{cases}\tfrac{7}{2}\exp[-7(x_2-x_1)],\quad & x_2 \ge x_1 \\ \tfrac{5}{2}\exp[-5(x_1-x_2)],\quad & x_2 < x_1 \end{cases}$
  
  Sketch this distribution as a function of $x_2-x_1$. Then, write down an expression for the ratio of $q$'s that goes into the acceptance probability $\alpha(x_1,x_2)$.
explanation: The document contains a math problem with multiple questions and their respective solutions. The first question involves calculating the ratio of proposal distribution probabilities for a specific set of integers, and the solution is provided. The second question asks for a sketch of a proposal distribution and the expression for the ratio of $q$'s, but the solution is not provided.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Travis: Segment 39
  
  ### To Calculate
  
  1. Suppose the domain of a model are the five integers $x = \{1,2,3,4,5\}$, and that your proposal distribution is: "When $x_1 = 2,3,4$, choose with equal probability $x_2 = x_1 \pm 1$. For $x_1=1$ always choose $x_2 =2$. For $x_1=5$ always choose $x_2 =4$. What is the ratio of $q$'s that goes into the acceptance probability $\alpha(x_1,x_2)$ for all the possible values of $x_1$ and $x_2$?
  
  $q(x2|x1) =$
  
   0 1 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0 0 1 0
  
  and $q(x1|x2) = q(x2|x1)^{T}$
  
  So, the ratio of q's is just the element wise division of $\frac{q(x1|x2)}{q(x2|x1)} = \frac{q(x1|x2)}{q(x1|x2)^T}$ which is
  
   0 2 0 0 0 0.5 0 1 0 0 0 1 0 1 0 0 0 1 0 0.5 0 0 0 1 0
  
  2. Suppose the domain of a model is $-\infty < x < \infty$ and your proposal distribution is (perversely),
  
  $q(x_2|x_1) = \begin{cases}\tfrac{7}{2}\exp[-7(x_2-x_1)],\quad & x_2 \ge x_1 \\ \tfrac{5}{2}\exp[-5(x_1-x_2)],\quad & x_2 < x_1 \end{cases}$
  
  Sketch this distribution as a function of $x_2-x_1$. Then, write down an expression for the ratio of $q$'s that goes into the acceptance probability $\alpha(x_1,x_2)$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem with multiple questions and their respective solutions. The first question involves calculating the ratio of proposal distribution probabilities for a specific set of integers, and the solution is provided. The second question asks for a sketch of a proposal distribution and the expression for the ratio of $q$'s, but the solution is not provided.
  
  Label: multiple-question-and-answer