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Everything you ever wanted to know about BUGS, R2winBUGS, and Adaptive Rejection Sampling

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Everything you ever wanted to know about BUGS, R2winBUGS, and Adaptive Rejection Sampling A Presentation by Keith Betts About BUGS Bayesian analysis Using Gibbs ... – PowerPoint PPT presentation

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Title: Everything you ever wanted to know about BUGS, R2winBUGS, and Adaptive Rejection Sampling


1
Everything you ever wanted to know about BUGS,
R2winBUGS, and Adaptive Rejection Sampling
  • A Presentation by
  • Keith Betts

2
About BUGS
  • Bayesian analysis Using Gibbs Sampling
  • Developed by UK Medical Research Council and the
    Imperial College of Science, Technology and
    Medicine, London.
  • http//www.mrc-bsu.cam.ac.uk/bugs

3
Why Use BUGS
  • Sophisticated implementation of Markov Chain
    Monte Carlo for any given model
  • No derivation required
  • Useful for double-checking hand coded results
  • Great for problems with no exact analytic solution

4
What does every BUGS file need?
  • Model
  • Specify the Likelihood and Prior distributions
  • Data
  • External data in either rectangular array or as R
    data type
  • Initial Values
  • Starting values for MCMC parameters

5
Model
  • a syntactical representation of the model, in
    which the distributional form of the data and
    parameters are specified.
  • assigns distributions
  • lt- assigns relations
  • for loops to assign i.i.d. data.

6
Distributions
  • Syntax can be found in User Manual, under
    Distributions
  • Parameterization may be different than in R
  • dnorm(µ,t), t is the precision (not the standard
    deviation).

7
Data
  • Place Data and Global variables in one list file
  • Vectors represented just as in R, c(., .)
  • Matrices require structure command

8
Initial Values
  • User requested initial values can be supplied for
    multiple chains
  • Put starting values for all variables in one list
    statement
  • BUGS can generate its own starting values

9
How to run
  • Place your Model, Data, and Initial Values in one
    file.
  • Open Specification from the model menu
  • Highlight model statement
  • Click Check Model
  • Highlight list in Data section
  • Click Load Data

10
How to run continued
  • Still in Model Specification
  • Click Compile
  • Choose how many chains to run
  • Highlight list in initial values section
  • Click load inits
  • Alternatively click gen inits

11
How to run (Part 3)
  • Open Samples from Inference menu
  • Enter all variables of interest (one at a time)
    into node box, and click set
  • Open Update from Model menu
  • Enter how many iterations wanted

12
Inference
  • Open Samples from Inference Menu
  • Enter in node box
  • Click History to view Trace plots
  • Click Density for density plots
  • Click Stats for summary statistics
  • Change value of beg for burnout

13
Example 1
  • Poisson-Gamma Model
  • Data
  • of Failures in power plant pumps
  • Model
  • of Failures follows Poisson(?i ti)
  • ?i failure rate follows Gamma(a, ß)
  • Prior for a is Exp(1)
  • Prior for ß is Gamma(.1, 1)

14
Example 1 Continued
  • Computational Issues
  • Gamma is natural conjugate for Poisson
    distribution
  • Posterior for ß follows Gamma
  • Non-standard Posterior for a

15
Model Step
  • model
  • for (i in 1 N)
  • thetai dgamma(alpha, beta)
  • lambdai lt- thetai ti
  • xi dpois(lambdai)
  • alpha dexp(1)
  • beta dgamma(0.1, 1.0)

16
Data Step and Initial Values
  • Data
  • list(t c(94.3, 15.7, 62.9, 126, 5.24, 31.4,
    1.05, 1.05, 2.1, 10.5), x c(5, 1, 5,14, 3, 19,
    1, 1, 4, 22), N 10)
  • Initial Values
  • list(alpha 1, beta 1)

17
Example 2
  • Data 30 young rats whose weights were measured
    weekly for five weeks.
  • Yij is the weight of the ith rat measured at age
    xj.
  • Assume a random effects linear growth curve model

18
Example 2 Model
  • model
  • for( i in 1 N )
  • for( j in 1 J )
  • mui , j lt- alphai betai (tj -
    tbar)
  • Yi , j dnorm(mui , j, sigma.y)
  • alphai dnorm(mu.alpha , sigma.alpha)
  • betai dnorm(mu.beta, sigma.beta)
  • sigma.y dgamma(0.001,0.001)
  • mu.alpha dunif( -1.0E9,1.0E9)
  • sigma.alpha dgamma(0.001,0.001)
  • mu.beta dunif(-1.0E9,1.0E9)
  • sigma.beta dgamma(0.001,0.001)

19
R2winBUGS
  • R package
  • Runs BUGS through R
  • Same computational advantages of BUGS with
    statistical and graphical capacities of R.

20
Make Model file
  • Model file required
  • Must contain BUGS syntax
  • Can either be written in advance or by R itself
    through the write.model() function

21
Initialize
  • Both data and Initial values stored as lists
  • Create param vector with names of parameters to
    be tracked

22
Run
  • bugs(datafile , initial.vals, parameters,
    modelfile, n.chains1, n.iter5000,
    n.burnin2000, n.thin1, bugs.directory"c/Progra
    m Files/WinBUGS14/", working.directoryNULL)
  • Extract results from sims.matrix

23
How BUGS works
  • BUGS determines the complete conditionals if
    possible by
  • Closed form distributions
  • Adaptive-Rejection Sampling
  • Slice-Sampling
  • Metropolis Algorithm

24
Recall Rejection Sampling
25
Adaptive Rejection Sampling(ARS)
  • ARS is a method for efficiently sampling from any
    univariate probability density function which is
    log-concave.
  • Useful in applications of Gibbs sampling, where
    full-conditional distributions are algebraically
    messy yet often log-concave

26
Idea
  • ARS works by constructing an envelope function of
    the log of the target density
  • Envelope used for rejection sampling
  • Whenever a point is rejected by ARS, the envelope
    is updated to correspond more closely to the true
    log density.
  • reduces the chance of rejecting subsequent points

27
Results
  • As the envelope function adapts to the shape of
    the target density, sampling becomes
    progressively more efficient as more points are
    sampled from the target density.
  • The sampled points will be independent from the
    exact target density.

28
References
  • W. R. Gilks, P. Wild (1992), "Adaptive Rejection
    Sampling for Gibbs Sampling," Applied Statistics,
    Vol. 41, Issue 2, 337-348.
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