# Difference between revisions of "Eleisha's Segment 27: Mixture Models"

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1. In problem 3, above, you assumed some definite prior for <math>c</math> . What if <math> c </math> is itself drawn (just once for the whole data set) from a distribution <math> \text{Beta}(\mu,\nu) </math>, with unknown hyperparameters <math> \mu,\nu </math> . How would you now estimate the Bayes posterior distribution of <math> \beta </math> , marginalizing over everything else? | 1. In problem 3, above, you assumed some definite prior for <math>c</math> . What if <math> c </math> is itself drawn (just once for the whole data set) from a distribution <math> \text{Beta}(\mu,\nu) </math>, with unknown hyperparameters <math> \mu,\nu </math> . How would you now estimate the Bayes posterior distribution of <math> \beta </math> , marginalizing over everything else? | ||

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+ | <b> Back To: </b> [[Eleisha Jackson]] |

## Latest revision as of 10:51, 3 April 2014

** To Calculate: **

The file Media:Mixturevals.txt contains 1000 values, each drawn either with probability c from the distribution **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 \text{Exponential}(\beta) }**
(for some constant **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 \beta }**
), or otherwise (with 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 1-c }**
) from the distribution **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 p(x) = (2/\pi)/(1+x^2),\; x>0 }**
.

1. Write down an expression for the probability of the file's data given some values for the parameters **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 \beta }**
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 c }**
.

2. Calculate numerically the maximum likelihood 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 \beta }**
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 c }**
.

3. Estimate numerically the Bayes posterior distribution of

** To Think About: **

1. In problem 3, above, you assumed some definite prior 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 c}**
. What if **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 c }**
is itself drawn (just once for the whole data set) from a distribution **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 \text{Beta}(\mu,\nu) }**
, with unknown hyperparameters **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 \mu,\nu }**
. How would you now estimate the Bayes posterior distribution of

** Back To: ** Eleisha Jackson