A Gaussian Mixture Model Is a Universal Approximator of Densities

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When discussing the concept of mixtures of distributions in my machine learning textbook, the authors state the following:




A Gaussian mixture model is a universal approximator of densities, in the sense that any smooth density can be approximated with any specific nonzero amount of error by a Gaussian mixture model with enough components.




I only have a basic background in probability and statistics, and I have absolutely no idea what this section is saying.



I would greatly appreciate it if people could please take the time to explain this in a way that is more understandable to someone of my level.










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  • I applied this idea once, very early in my career, to carry out quantum-mechanical calculations by using a basis of Gaussians to model bonds in organic molecules. After weeks of running into problems I realized, to my chagrin, that the densities I was trying to approximate were so sharply peaked that it was hopeless to develop a good approximation without using a ridiculously large number of Gaussians, which are too smooth for the job. Thus, one should always be cognizant of the distinction between a purely mathematical theorem and the practical needs of computation.
    – whuber♦
    Sep 3 at 13:33
















up vote
2
down vote

favorite
1












When discussing the concept of mixtures of distributions in my machine learning textbook, the authors state the following:




A Gaussian mixture model is a universal approximator of densities, in the sense that any smooth density can be approximated with any specific nonzero amount of error by a Gaussian mixture model with enough components.




I only have a basic background in probability and statistics, and I have absolutely no idea what this section is saying.



I would greatly appreciate it if people could please take the time to explain this in a way that is more understandable to someone of my level.










share|cite|improve this question





















  • I applied this idea once, very early in my career, to carry out quantum-mechanical calculations by using a basis of Gaussians to model bonds in organic molecules. After weeks of running into problems I realized, to my chagrin, that the densities I was trying to approximate were so sharply peaked that it was hopeless to develop a good approximation without using a ridiculously large number of Gaussians, which are too smooth for the job. Thus, one should always be cognizant of the distinction between a purely mathematical theorem and the practical needs of computation.
    – whuber♦
    Sep 3 at 13:33












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





When discussing the concept of mixtures of distributions in my machine learning textbook, the authors state the following:




A Gaussian mixture model is a universal approximator of densities, in the sense that any smooth density can be approximated with any specific nonzero amount of error by a Gaussian mixture model with enough components.




I only have a basic background in probability and statistics, and I have absolutely no idea what this section is saying.



I would greatly appreciate it if people could please take the time to explain this in a way that is more understandable to someone of my level.










share|cite|improve this question













When discussing the concept of mixtures of distributions in my machine learning textbook, the authors state the following:




A Gaussian mixture model is a universal approximator of densities, in the sense that any smooth density can be approximated with any specific nonzero amount of error by a Gaussian mixture model with enough components.




I only have a basic background in probability and statistics, and I have absolutely no idea what this section is saying.



I would greatly appreciate it if people could please take the time to explain this in a way that is more understandable to someone of my level.







machine-learning distributions gaussian-mixture density-estimation






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asked Sep 3 at 8:15









The Pointer

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  • I applied this idea once, very early in my career, to carry out quantum-mechanical calculations by using a basis of Gaussians to model bonds in organic molecules. After weeks of running into problems I realized, to my chagrin, that the densities I was trying to approximate were so sharply peaked that it was hopeless to develop a good approximation without using a ridiculously large number of Gaussians, which are too smooth for the job. Thus, one should always be cognizant of the distinction between a purely mathematical theorem and the practical needs of computation.
    – whuber♦
    Sep 3 at 13:33
















  • I applied this idea once, very early in my career, to carry out quantum-mechanical calculations by using a basis of Gaussians to model bonds in organic molecules. After weeks of running into problems I realized, to my chagrin, that the densities I was trying to approximate were so sharply peaked that it was hopeless to develop a good approximation without using a ridiculously large number of Gaussians, which are too smooth for the job. Thus, one should always be cognizant of the distinction between a purely mathematical theorem and the practical needs of computation.
    – whuber♦
    Sep 3 at 13:33















I applied this idea once, very early in my career, to carry out quantum-mechanical calculations by using a basis of Gaussians to model bonds in organic molecules. After weeks of running into problems I realized, to my chagrin, that the densities I was trying to approximate were so sharply peaked that it was hopeless to develop a good approximation without using a ridiculously large number of Gaussians, which are too smooth for the job. Thus, one should always be cognizant of the distinction between a purely mathematical theorem and the practical needs of computation.
– whuber♦
Sep 3 at 13:33




I applied this idea once, very early in my career, to carry out quantum-mechanical calculations by using a basis of Gaussians to model bonds in organic molecules. After weeks of running into problems I realized, to my chagrin, that the densities I was trying to approximate were so sharply peaked that it was hopeless to develop a good approximation without using a ridiculously large number of Gaussians, which are too smooth for the job. Thus, one should always be cognizant of the distinction between a purely mathematical theorem and the practical needs of computation.
– whuber♦
Sep 3 at 13:33










1 Answer
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The idea is that an arbitrary density on $mathbbR$, $f(cdot)$, can be approximated by a Gaussian mixture model
$$g_k(cdot;boldsymbolomega,mu,sigma)=sum_i=1^k omega_i varphi(cdot;mu_i,sigma_i)$$
in the sense that
$$mathfrakD(f(cdot),g_k(cdot;boldsymbolomega,mu,sigma)stackrelktoinftylongrightarrow0$$
for some specific functional distance $mathfrakD(cdot,cdot)$. This result only applies for weaker types of distance.






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  • 1




    Thanks. Your answer is clear.
    – The Pointer
    Sep 3 at 8:57










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
6
down vote



accepted










The idea is that an arbitrary density on $mathbbR$, $f(cdot)$, can be approximated by a Gaussian mixture model
$$g_k(cdot;boldsymbolomega,mu,sigma)=sum_i=1^k omega_i varphi(cdot;mu_i,sigma_i)$$
in the sense that
$$mathfrakD(f(cdot),g_k(cdot;boldsymbolomega,mu,sigma)stackrelktoinftylongrightarrow0$$
for some specific functional distance $mathfrakD(cdot,cdot)$. This result only applies for weaker types of distance.






share|cite|improve this answer
















  • 1




    Thanks. Your answer is clear.
    – The Pointer
    Sep 3 at 8:57














up vote
6
down vote



accepted










The idea is that an arbitrary density on $mathbbR$, $f(cdot)$, can be approximated by a Gaussian mixture model
$$g_k(cdot;boldsymbolomega,mu,sigma)=sum_i=1^k omega_i varphi(cdot;mu_i,sigma_i)$$
in the sense that
$$mathfrakD(f(cdot),g_k(cdot;boldsymbolomega,mu,sigma)stackrelktoinftylongrightarrow0$$
for some specific functional distance $mathfrakD(cdot,cdot)$. This result only applies for weaker types of distance.






share|cite|improve this answer
















  • 1




    Thanks. Your answer is clear.
    – The Pointer
    Sep 3 at 8:57












up vote
6
down vote



accepted







up vote
6
down vote



accepted






The idea is that an arbitrary density on $mathbbR$, $f(cdot)$, can be approximated by a Gaussian mixture model
$$g_k(cdot;boldsymbolomega,mu,sigma)=sum_i=1^k omega_i varphi(cdot;mu_i,sigma_i)$$
in the sense that
$$mathfrakD(f(cdot),g_k(cdot;boldsymbolomega,mu,sigma)stackrelktoinftylongrightarrow0$$
for some specific functional distance $mathfrakD(cdot,cdot)$. This result only applies for weaker types of distance.






share|cite|improve this answer












The idea is that an arbitrary density on $mathbbR$, $f(cdot)$, can be approximated by a Gaussian mixture model
$$g_k(cdot;boldsymbolomega,mu,sigma)=sum_i=1^k omega_i varphi(cdot;mu_i,sigma_i)$$
in the sense that
$$mathfrakD(f(cdot),g_k(cdot;boldsymbolomega,mu,sigma)stackrelktoinftylongrightarrow0$$
for some specific functional distance $mathfrakD(cdot,cdot)$. This result only applies for weaker types of distance.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 3 at 8:30









Xi'an

49.9k685333




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  • 1




    Thanks. Your answer is clear.
    – The Pointer
    Sep 3 at 8:57












  • 1




    Thanks. Your answer is clear.
    – The Pointer
    Sep 3 at 8:57







1




1




Thanks. Your answer is clear.
– The Pointer
Sep 3 at 8:57




Thanks. Your answer is clear.
– The Pointer
Sep 3 at 8:57

















 

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