[R] integration of two normal density

[Matt Shotwell]

On Fri, 2010-06-25 at 23:28 -0400, Carrie Li wrote:
> Hello everyone,
> 
> I have a question about integration of two density function
> Intuitively, I think the value after integration should be 1, but they are
> not. Am I missing something here ?
> 
> > t=function(y){dnorm(y, mean=3)*dnorm(y/2, mean=1.5)}
> > integrate(t, -Inf, Inf)
> 0.3568248 with absolute error < 4.9e-06

You've demonstrated (numerically) that the product of two normal density
functions, with means 3, and 1.5 respectively and variance 1, doesn't
result in a pdf. However, you could make a numerically normalized pdf by
multiplying by 1/0.3568248.

> K <- integrate(t, -Inf, Inf)$value
> Kt <- function(y) 1/K * dnorm(y, 3) * dnorm(y/2, 1.5)
> integrate(Kt, -Inf, Inf)
1 with absolute error < 1.4e-05

Hence, the quantity you computed (K) is the normalization constant, with
some small error. Note that this strategy _may_ not always work. Here's
a good homework question: Can the product of two pdfs with identical
support always be normalized to form a new pdf?

As for empirical multivariate integration, it's tough, especially if you
want to enumerate the "area under the surface", which is exactly the
strategy of functions like 'integrate' (search Wikipedia for numerical
integration). This problem becomes increasingly difficult in additional
dimensions; the dreaded "curse of dimensionality". On the bright side,
Bayesian statistical methods have to deal with this all the time, and we
have some good methods to compute numerical integrals. Check out Monte
Carlo integration, and Markov chain Monte Carlo methods.

-Matt

> 
> 
> Also, is there any R function or package could do multivariate integration ?
> 
> Thanks for any suggestions!
> 
> Carrie
> 
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> 
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-- 
Matthew S. Shotwell
Graduate Student
Division of Biostatistics and Epidemiology
Medical University of South Carolina
http://biostatmatt.com