57 Mixtures of Gaussians – Bayesian Statistics

Author

Oren Bochman

Keywords

Mixture Models, Homework

Exercise 57.1 True or False? A scale mixture of normals with density f(x) = \sum_{k=0}^{K} \omega_k \frac{1}{\sqrt{2 \pi}\sigma_k} e^{-\frac{x^2}{\sigma_k^2}} \qquad \tag{57.1}

is always unimodal?

Solution

True

We can see from the functional form that this is a sum of K Gaussian densities, with maximum at 0 and monotonically decreasing everywhere else.

Exercise 57.2 True or False? A scale mixture of normals with density as in Equation 57.1 is always symmetric?

Solution

True

If we inspect the functional form we can see that since x is squared in this function, it is symmetric around 0.

---
title : 'Mixtures of Gaussians'
subtitle : 'Bayesian Statistics: Mixture Models'
categories:
  - Bayesian Statistics
keywords:
  - Mixture Models
  - Homework
---



::: {#exr-1}
True or False? A scale mixture of normals with density
$$
f(x) = \sum_{k=0}^{K} \omega_k \frac{1}{\sqrt{2 \pi}\sigma_k} e^{-\frac{x^2}{\sigma_k^2}} \qquad
$$ {#eq-scale-mixture}

is always unimodal?

:::: {.callout-note collapse="true"}
## Solution

True

We can see from the functional form that this is a sum of K Gaussian densities, with maximum at 0 and monotonically decreasing everywhere else.

::::

:::

::: {#exr-2}

True or False? A scale mixture of normals with density as in @eq-scale-mixture is always symmetric?

:::: {.callout-note collapse="true"}
## Solution

True 

If we inspect the functional form we can see that since x is squared in this function, it is symmetric around 0.

::::

:::