# 27

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 c.

**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(\beta, c|data) = \frac{P(data|\beta,c)P(\beta,c)}{\int_0^1 \int_0^{\infty}P(data|\beta,c)P(\beta,c)} dc d\beta }**

Let us assume a uniform prior on c 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 \beta>}**
. More specifically, although **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 \epsilon (0, \infty) }**
, it is most likely no greater than 10,000, so we can place a uniform prior on **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>}**
in the range 0 to 10,000. Based on this uniform prior:

**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(\beta, c|data) \propto P(data|\beta, c)}**

**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(\beta, C|data) = \prod_{i} [P_{1i}C + P_{0i}(1-c)]P(\beta, c)}**

Where

**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_{1i} = \frac{1}{\beta}e^{-\frac{x}{\beta}}}**

**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_{0i} = \frac{2}{\pi(1+x^2)}}**

2. Calculate numerically the maximum likelihood values of \beta and c.

plugging into fmin in python:

**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 = 1.503 }**

**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 = .35956 }**

Code below:

import scipy as sc import scipy.optimize as opt import numpy as np import math

samples = np.loadtxt('Mixtureevals.txt')

def prob_1(data, beta): return (1./beta)*np.exp(-data/beta) def prob_2(data): return (2./sc.pi)/(1+np.power(data,2))

def likeHood(data, beta, c): probC = np.zeros(len(data)) probC = np.log(prob_1(data,beta)*c + prob_2(data)*(1.-c)) return np.sum(probC) def nl (params, data): beta, c = params return -likeHood(data, beta, c)

x0 = [1, 1]

beta, c = opt.fmin(nl, x0 , args=(samples,)) print beta print c

3. Estimate numerically the Bayes posterior distribution of \beta, marginalizing over c as a nuisance parameter. (You'll of course have to make some assumption about priors.)

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(\beta|data) = \int_0^1 P(\beta|data,c)P(c) dc }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(\beta|data,c) = \frac{P(C|data,\beta)*P(B)}{\int_0^{\infty} P(C|\beta, data)P(B)db} }

A uniform prior on P(c) is assumed.