While I have always preferred the duplicate data approach described by
Martin, it still maintains the annoying problem of the original example
(i.e. that you are using the same data twice, once for calibration, and
once for model selection). This does have a noticable (though not huge)
impact on the distribution of the selection parameter (m) when the data
favors one model strongly over the other, since the distribution of yhat
is much better constrained by twice as many observations.
One obvious solution is to split the data into a calibration set
(duplicated as y1[] and y2[]) and a selection set. Cross-validation also
suggests itself (Bugs manual, page 42), but is more annoying to code,
especially if one wants to do one-out cross-validation.
If one is really committed to using the same data twice, you can counter
the artificial preference for the favored model by incorporating a y3[]
duplicate data set that is modeled by the unfavored model. This
additional data will constrain the unfavored model parameters, and will
artificially shift the model selection some distance towards the unfavored
parameter. If the posterior probability of that model is very small, then
this approach will provide two data sets that should equally (but still
artificially) constrain the parameters of the two models equally.
If the bayes factor is not far from one, then this approach will noticably
shift the bayes factor towards the unfavored model. One approach I have
used in these cases is to fiddle with the prior on m until the posterior
distribution is evenly distributed between the models, and then the bayes
factor is just one over this prior odds ratio. (yes its annoying, but it
seems to work).
Finn Krogstad
Colloge of Forest Resources
University of Washington
Seattle, WA 98195
http://students.washington.edu/fkrogsta
It is better to light a candle than to curse the darkness.
But who says you can't do both?
On Tue, 5 Jun 2001, Upsdell, Martin wrote:
> The earlier BUGS examples vol 2 (mine is 1995) had the Pines Example of
> computing a Bayes factor. It seems to have been left out of more recent
> versions.
> I have enclosed an example below, choosing between an exponential function
> or a broken stick function for fertilizer curves.
>
> Basically one includes a node, m below, which decides model to choose, 1 or
> 2. One difficulty is that when the node is choosing model 1 the other model
> doesn't have access to the y data which is supplying the information as to
> what the parameters for its model should be, so that when the choosing node
> m tries model 2, its parameters are not where they should be and hence has
> low probability which decreases its chances of being choosen which makes
> matters worse.
>
> The solution in the Pines example is to run the BUGS code for each model
> separately and use the output from each of the separate runs to produce very
> informative priors for the parameters of each model so that in the model
> comparing run, each model has a good idea of where its parameters should be.
>
>
> I have solved it another way by having multiple, identical, copies of the y
> data. Each model has its own private copy which provides information about
> its parameters whether or not it is the current favorite model, Y1 and Y2
> below. There is another copy, Ym below, which is used by the model choosing
> node to decide which model to choose.
>
> Martin Upsdell
> Statistics Section [log in to unmask]
> Ruakura Research Centre ph :64-7-838 5149
>
> Private Bag 3123, Hamilton fax:64-7-838 5012
> New Zealand
>
> -----Original Message-----
> From: Benjamin Chan [mailto:[log in to unmask]]
> Sent: Tuesday, June 05, 2001 5:34 AM
> To: [log in to unmask]
> Subject: Bayes factors and model selection
>
>
> BUGS users:
>
> Has anyone used BUGS to calculate Bayes factors for model selection? I would
> appreciate any references to the subject. Thanks.
>
> ~
> Benjamin K. S. Chan, M.S., Research Associate
> Division of Medical Informatics and Outcomes Research
> Oregon Health and Science University
> e-mail: [log in to unmask]; tel: 503 494 1607; fax: 503 494 4551
>
> ***** Bayes factor model *****
> model;
> {
> alpha[1] ~ dnorm( 0.0,1.0E-6)I( 0.0,)
> alpha[2] ~ dnorm( 0.0,1.0E-6)I( 0.0,)
> response[1] ~ dnorm( 0.0,1.0E-6)I( 0.0,)
> response[2] ~ dnorm( 0.0,1.0E-6)I( 0.0,)
> r ~ dunif(0,1)
> beta ~ dnorm( 0.0,1.0E-6)I( 0.8,)
> tauy[1] ~ dgamma(0.01,0.01)
> tauy[2] ~ dnorm( 0.0,1.0E-6)
> sigmay[1] <- 1 / sqrt(tauy[1])
> sigmay[2] <- 1 / sqrt(tauy[2])
> m ~ dcat(probm[])
> for( i in 1 : 37 ) {
> y1[i] ~ dnorm(yhat[1 , i],tauy[1])
> y2[i] ~ dnorm(yhat[2 , i],tauy[2])
> yhat[1 , i] <- alpha[1] + response[1] * ( 1.0 - pow(r,S[i]))
> yhat[2 , i] <- alpha[2] + min(response[2],beta * S[i])
> ym[i] ~ dnorm(yhat[m , i],tauy[m])
> }
> ry90[1] <- log( 0.1 + ( 0.1 * alpha[1]) / response[1]) / log(r)
> ry90[2] <- ( 0.9 * response[2] - 0.1 * alpha[2]) / beta
> ry90[3] <- ry90[m]
> }
>
> consts
> list(probm = c(0.5, 0.5) )
> inits
> list(alpha=c(20.0,20.0),response=c(80.0,80.0),r=0.95,beta=2.0,tauy=c(100.0,1
> 00.0),m=2)
>
> data
> S[] y1[] y2[] ym[]
> 59.3 75.5 75.5 75.5
> 9.3 31.7 31.7 31.7
> 108 90.6 90.6 90.6
> 125.2 94.3 94.3 94.3
> 32.4 84.4 84.4 84.4
> 60.3 72.8 72.8 72.8
> 108.5 78.3 78.3 78.3
> 55.4 91.2 91.2 91.2
> 132.1 100.3 100.3 100.3
> 70.1 96 96 96
> 59.7 98 98 98
> 61.5 97 97 97
> 53.1 97 97 97
> 66.4 93 93 93
> 39.1 94 94 94
> 93 92 92 92
> 59.1 91 91 91
> 83 97 97 97
> 52.8 94 94 94
> 57.2 97 97 97
> 70.9 97 97 97
> 39.7 78 78 78
> 46.5 73 73 73
> 99.9 94 94 94
> 92.5 94 94 94
> 84.7 84.3 84.3 84.3
> 45.1 78.5 78.5 78.5
> 74.8 85.3 85.3 85.3
> 26.4 54.9 54.9 54.9
> 11.5 63.6 63.6 63.6
> 17.8 59.4 59.4 59.4
> 12.6 58.6 58.6 58.6
> 33.1 84.2 84.2 84.2
> 50 94.8 94.8 94.8
> 22.9 66.1 66.1 66.1
> 14.2 11.9 11.9 11.9
> 47.8 56.7 56.7 56.7
>
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