Are components in a gaussian mixture independent?

In summary, the author is looking for a way to square a pdf corresponding to a mixture of gaussian distributions, but does not know how to prove or disprove the independence of the component normal pdfs.
  • #1
ConfusedCat
3
0
Hello all,

I have used Expectation Maximization algorithm to approximate a probability density function (pdf) using a mixture of gaussians.

I need to square the pdf corresponding to the mixture of gaussians( it is a weighted sum of normal distributions with non-identical parameters). Rather than square a sum of gaussians, I thought that if the constituent normal pdfs of the mixture were independent, they could be summed resulting in a single gaussian, which I could then square.

I don't know how to prove (or disprove) the independence of the components. For a regular joint distribution f(x,y), independence can be proved if f(x,y)=f(x)f(y). Here, I have a series of real values for each of the normal components in the distribution. How do I find the joint distribution?

Any thoughts would be welcome.

Cheers
 
Physics news on Phys.org
  • #2
ConfusedCat said:
Hello all,

I have used Expectation Maximization algorithm to approximate a probability density function (pdf) using a mixture of gaussians.

I need to square the pdf corresponding to the mixture of gaussians( it is a weighted sum of normal distributions with non-identical parameters). Rather than square a sum of gaussians, I thought that if the constituent normal pdfs of the mixture were independent, they could be summed resulting in a single gaussian, which I could then square.

I don't know how to prove (or disprove) the independence of the components. For a regular joint distribution f(x,y), independence can be proved if f(x,y)=f(x)f(y). Here, I have a series of real values for each of the normal components in the distribution. How do I find the joint distribution?

Any thoughts would be welcome.

Cheers

Hi ConfusedCat! Welcome to MHB! ;)

We would need to test for independence of 2 continuous variables.
For instance this link gives a couple of possible approaches.
 
  • #3
I like Serena said:
Hi ConfusedCat! Welcome to MHB! ;)

We would need to test for independence of 2 continuous variables.
For instance this link gives a couple of possible approaches.

Thank you for that link - it does look very useful.
 

1. What is a Gaussian mixture model?

A Gaussian mixture model is a statistical model used to represent a probability distribution of a dataset. It is composed of multiple Gaussian distributions, also known as normal distributions, that are combined to form a more complex probability distribution.

2. How are components in a Gaussian mixture model determined?

The components in a Gaussian mixture model are determined by fitting the model to a given dataset. This involves estimating the parameters of each Gaussian distribution, such as the mean and variance, and determining the weights of each component in the model.

3. Are the components in a Gaussian mixture model independent?

No, the components in a Gaussian mixture model are not independent. Each component is dependent on the other components in the model, as they are combined to form the overall probability distribution. However, the observations within each component are assumed to be independent.

4. How does the independence assumption affect the performance of a Gaussian mixture model?

The independence assumption in a Gaussian mixture model can affect the performance of the model if the data does not meet this assumption. If the data is not truly independent, the model may not accurately represent the underlying distribution and may not perform well in predicting future data.

5. Can the independence assumption be relaxed in a Gaussian mixture model?

Yes, the independence assumption can be relaxed in a Gaussian mixture model by using a more complex form of the model, such as a hierarchical Gaussian mixture model. This allows for dependencies between the components to be incorporated into the model, leading to potentially better performance on certain datasets.

Similar threads

I Linear regression and random variables
    • Set Theory, Logic, Probability, Statistics
Replies
30
Views
2K
I Fourier Transform of a Probability Distribution
    • Set Theory, Logic, Probability, Statistics
Replies
9
Views
3K
A Proving Lim F(x,y) is the Distribution Function for X
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
863
Error in summation of spectral components
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
MHB Two normal independent random variables
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
MLE estimator for mean always equal to the mean?
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
4K
I Generating samples on a 2-D composite binomial distribution
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
873
A Parameterizing conditional expectations (Gaussian case)
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
MHB Joint Probability Density from two Gaussian Distributions
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
A Combining model uncertainties with analytical uncertainties
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
846

Hot Threads

Recent Insights

Back
Top