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Science Forum Index » Statistics - Math Forum » How do we choose covariance matrix for multivariate...
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| Maniaoh... |
 Posted: Thu Jul 17, 2008 2:43 am |
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Guest
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Hi there,
I have a question about Bayesian inference and a little bit related to
Markov chain Monte Carlo. Suppose that I have observed data D and I
want to have a model describing that data. I may test it by using
model f(x, theta) where theta = (a, b, c) (a vector with some
elements). We assume that (a, b, c) is multivariate normal so making
inference about them requires covariance matrix C, which is used in
MCMC method. The problem is that I do not know how C is built or
chosen, given the observed data D.
Please give me some instructions on this matter or refer me to any
related documents. Thank you very much in advance.
Iaoh. |
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| Posted: Thu Jul 17, 2008 5:55 pm |
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Guest
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On Jul 17, 7:43 am, Maniaoh <n.hoai... at (no spam) gmail.com> wrote:
Quote: Hi there,
I have a question about Bayesian inference and a little bit related to
Markov chain Monte Carlo. Suppose that I have observed data D and I
want to have a model describing that data. I may test it by using
model f(x, theta) where theta = (a, b, c) (a vector with some
elements). We assume that (a, b, c) is multivariate normal so making
inference about them requires covariance matrix C, which is used in
MCMC method. The problem is that I do not know how C is built or
chosen, given the observed data D.
Please give me some instructions on this matter or refer me to any
related documents. Thank you very much in advance.
Iaoh.
Since you're trying to be Bayesian, go ahead and treat the covariance
matrix as unknown and use a Normal-Wishart distribution to model your
data. This is a multivariate extension of the scalar problem of
estimating the parameters of a normal distribution with an unknown
mean and covariance.
Any upper-level/graduate textbook on Bayesian data analysis should
discuss this model. Bayesian Data Analysis by Gelman, Carlin, Stern
and Rubin is a good place to start.
Here is an early journal paper on the problem, too.
Bayesian Analysis of the Independent Multinormal Process. Neither Mean
Nor Precision Known by Albert Ando and G. M. Kaufman. Journal of the
American Statistical Association, Vol. 60, No. 309 (Mar., 1965), pp.
347-358
-Lucas |
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| Herman Rubin... |
| Posted: Fri Jul 18, 2008 9:12 am |
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Guest
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In article <a57857bf-8aaa-48b9-af04-113fbd8b45c2 at (no spam) w1g2000prd.googlegroups.com>,
<lscharen at (no spam) d.umn.edu> wrote:
Quote: On Jul 17, 7:43=A0am, Maniaoh <n.hoai... at (no spam) gmail.com> wrote:
Hi there,
I have a question about Bayesian inference and a little bit related to
Markov chain Monte Carlo. Suppose that I have observed data D and I
want to have a model describing that data. I may test it by using
model f(x, theta) where theta =3D (a, b, c) (a vector with some
elements). We assume that (a, b, c) is multivariate normal so making
inference about them requires covariance matrix C, which is used in
MCMC method. The problem is that I do not know how C is built or
chosen, given the observed data D.
Please give me some instructions on this matter or refer me to any
related documents. Thank you very much in advance.
Iaoh.
Since you're trying to be Bayesian, go ahead and treat the covariance
matrix as unknown and use a Normal-Wishart distribution to model your
data. This is a multivariate extension of the scalar problem of
estimating the parameters of a normal distribution with an unknown
mean and covariance.
The inverse Wishart distribution is the conjugate prior,
and thus easiest to compute. Not only is it not REQUIRED
to use a conjugate prior, but it is generally the case
that this prior produces estimates which can be improved
everywhere by a substantial amount.
Sound Bayesian practice requires that one use a prior which
is given by the person's experience and belief, NOT for
convenience. Robustness might justify the latter, but not
in this particular case.
Quote: Any upper-level/graduate textbook on Bayesian data analysis should
discuss this model. Bayesian Data Analysis by Gelman, Carlin, Stern
and Rubin is a good place to start.
Learn decision theory first. Cookbook procedures with
simplifying assumptions do not convey any understanding;
I am not that Rubin.
Quote: Here is an early journal paper on the problem, too.
Bayesian Analysis of the Independent Multinormal Process. Neither Mean
Nor Precision Known by Albert Ando and G. M. Kaufman. Journal of the
American Statistical Association, Vol. 60, No. 309 (Mar., 1965), pp.
347-358
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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