# Difference between revisions of "Eleisha's Segment 38: Mutual Information"

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<b> To Calculate </b> | <b> To Calculate </b> | ||

− | 1. Suppose the domain of a model are the five integers <math> x = \{1,2,3,4,5\} </math>, and that your proposal distribution is: "When <math>x_1 = 2,3,4 </math>, choose with equal probability <math> x_2 = x_1 \pm 1 </math>. For <math> x_1=1 </math> always choose <math>x_2 =2 </math>. For <math>x_1=5 </math> always choose <math>x_2 =4 </math>. What is the ratio of q's that goes into the acceptance probability <math> \alpha(x_1,x_2) </math> for all the possible values of <math> x_1 </math> and <math>x_2 </math>? | + | 1. Suppose the domain of a model are the five integers <math> x = \{1,2,3,4,5\} </math>, and that your proposal distribution is: "When <math>x_1 = 2,3,4 </math>, choose with equal probability <math> x_2 = x_1 \pm 1 </math>. For <math> x_1=1 </math> always choose <math>x_2 =2 </math>. For <math>x_1=5 </math> always choose <math>x_2 =4 </math>. What is the ratio of <math>q </math>'s that goes into the acceptance probability <math> \alpha(x_1,x_2) </math> for all the possible values of <math> x_1 </math> and <math>x_2 </math>? |

2. Suppose the domain of a model is <math> -\infty < x < \infty </math> and your proposal distribution is (perversely), | 2. Suppose the domain of a model is <math> -\infty < x < \infty </math> and your proposal distribution is (perversely), |

## Revision as of 21:56, 6 April 2014

** To Calculate **
1. Suppose the domain of a model are the five integers **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \{1,2,3,4,5\} }**
, and that your proposal distribution is: "When **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 = 2,3,4 }**
, choose with equal probability **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 = x_1 \pm 1 }**
. For **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1=1 }**
always choose **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 =2 }**
. For **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1=5 }**
always choose **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 =4 }**
. What is the ratio of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q }**
's that goes into the acceptance probability **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(x_1,x_2) }**
for all the possible values of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1 }**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2 }**
?

2. Suppose the domain of a model is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty < x < \infty }**
and your proposal distribution is (perversely),

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2-x_1 }**
. Then, write down an expression for the ratio of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q }**
's that goes into the acceptance probability **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha(x_1,x_2) }**
.

**To Think About **

1. Suppose an urn contains 7 large orange balls, 3 medium purple balls, and 5 small green balls. When balls are drawn randomly, the larger ones are more likely to be drawn, in the proportions large:medium:small = 6:4:3. You want to draw exactly 6 balls, one at a time without replacement. How would you use Gibbs sampling to learn: (a) How often do you get 4 orange plus 2 of the same (non-orange) color? (b) What is the expectation (mean) of the product of the number of purple and number of green balls drawn?

2. How would you do the same problem computationally but without Gibbs sampling?

3. How would you do the same problem non-stochastically (e.g., obtain answers to 12 significant figures)? (Hint: This is known as the Wallenius non-central hypergeometric distribution.)

[Answers: 0.155342 and 1.34699]