63 Homework The likelihood function M1L2HW3

Keywords

Mixture Models, Homework

Caution

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---
title : 'Homework The likelihood function M1L2HW3'
subtitle : 'Bayesian Statistics: Mixture Models'
categories:
  - Bayesian Statistics
keywords:
  - Mixture Models
  - Homework
---

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1. Consider a random sample $(−0.3,4.1,3.6,7.5,1.9,2.7)$ composed of $n=6$ observations form the mixture with density:
$$
f(x)=w_1 \frac{1}{\sqrt{2\pi}} \exp\left\{-\frac{x^2}{2}\right\} + w_2 \frac{1}{\sqrt{2\pi}} \exp\left\{-\frac{(x-2)^2}{2}\right\} + w_3 \frac{1}{\sqrt{2\pi}} \exp\left\{-\frac{(x-4)^2}{2}\right\}
$$

What is the complete-data likelihood associated with the indicator vector 
$(1,2,2,3,1,2)$

- [x] $w_1^2 w_2^2 w_3 \exp\{-11.705\}$
- [ ] $w_1^3 w_2^1 w_3^2 \exp\{-11.705\}$
- [ ] $w_1^2 w_2^3 w_3 \exp\{-23.41\}$
- [ ] $w_1^3 w_2 w_3^2 \exp\{-23.41\}$




2. True or False: Consider a location mixture of normals

$$
f(x)=\sum_{k=1}^{K} \omega_k \frac{1}{\sqrt{2\pi}} \sigma^{-1} \exp\left\{-\frac{(x-\mu_k)^2}{2\sigma^2}\right\}
$$

The following 3 constraints make all parameters fully identifiable:

1. The means  $\mu_1,\ldots,\mu_K$ should all be different.
2. No weight $\omega_k$ is allowed to be zero.
3. The component are ordered based on the values of their means, i.e., the component with the smallest  $\mu_k$ is labeled component 1, the one with the second smallest $\mu_k$ is labeled component 2, etc.


- [x] True
- [ ] False

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