1. Bayesian Hierarchical Model: Many Theta's in Multiple Conditions
Lecture was not given on Monday 2/6/2017 because of UW closure due to snow. A lecture that is largely the same as the present one will be given as lec06-2.p548.w17.pptm on Wednesday 2/8/2017.
Bayesian Hierarchical Model: Many Theta's in Multiple Conditions
P548: Bayesian Stats with Psych Applications Instructor: John Miyamoto 02/06/2017: Lecture 06-1
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2. Figure 8.2: Bayesian Model for Flips of a Single Coinq X N
q ~ beta(1, 1)
X ~ binomial(q, N)
Kruschke, Fig. 8.2, p. 196
LW, Fig. 2.1, p. 18
Psych 548:, Miyamoto, Win ‘16

3. Figure 9.7: J coins sampled from the same mintk w a b qs
yij
s = 1, ..., J
i = 1, ..., n
Amu and Bmu are not represented as random variables in the diagram on the right because they are treated as constants in this model. They are not sampled from a hyperhyperdistribution.
Skappa & Rkappa are not represented as random variables in the diagram on the right because they are treated as constants in this model. Specifically, Skappa = pow(10,2)/pow(10,2) = 1, Rkappa = 10/pow(10,2).
Note that kappa is a random variable that is sampled from a gamma( Skappa, Rkappa ) distribution.
Kruschke Fig. 9.7
UW Psych 548, Miyamoto, Spr '12

4. Figure 9.13 (ed.2): J coins sampled from C mintsw k A B
c = 1, 2, 3 wc kc
Amu and Bmu are not represented as random variables in the diagram on the right because they are treated as constants in this model. They are not sampled from a hyperhyperdistribution.
Skappa & Rkappa are not represented as random variables in the diagram on the right because they are treated as constants in this model. Specifically, Skappa = pow(10,2)/pow(10,2) = 1, Rkappa = 10/pow(10,2).
Note that kappa is a random variable that is sampled from a gamma( Skappa, Rkappa ) distribution. ac bc
s = 1, ..., J qs
i = 1, ..., n
yij
Kruschke Fig p. 252
UW Psych 548, Miyamoto, Spr '12

5. Simplified Version, LW Notation. Figure 9. 13 (ed
Simplified Version, LW Notation. Figure 9.13 (ed.2): J coins sampled from N mints w k w k A B
c = 1, 2, 3
c = 1, 2, 3 wc kc wc kc
Amu and Bmu are not represented as random variables in the diagram on the right because they are treated as constants in this model. They are not sampled from a hyperhyperdistribution.
Skappa & Rkappa are not represented as random variables in the diagram on the right because they are treated as constants in this model. Specifically, Skappa = pow(10,2)/pow(10,2) = 1, Rkappa = 10/pow(10,2).
Note that kappa is a random variable that is sampled from a gamma( Skappa, Rkappa ) distribution.
s = 1, ..., J ac bc qs
s = 1, ..., J qs
i = 1, ..., n
yij
i = 1, ..., n
yij
UW Psych 548, Miyamoto, Spr '12

6. Correspondence Between Model Syntax & Graphical Model
# Hyperpriors for omega0 and kappa0, the parameters of
# the hyperdistribution from which omega[cc] and kappa[cc]
# are sampled.
omegaO ~ dbeta( 1.0, 1.0 )
kappaO <- kappaMinusTwoO + 2
kappaMinusTwoO ~ dgamma( 0.01, 0.01 )
# Compute the A0 and B0 parameters of the hyper beta distribution
A0 <- omegaO*(kappaO - 2) + 1
B0 <- (1 - omegaO)*(kappaO - 2) + 1
# Hyperprior for omega[cc] and kappa[cc],
# the condition-specific omega and kappa
for ( mm in 1:Ncat ) {
omega[mm] ~ dbeta( A0, B0 )
  kappaMinusTwo[mm] ~ dgamma( 0.01, 0.01 )
  kappa[mm] <- kappaMinusTwo[mm] + 2 }
# Prior for theta[j]
for ( j in 1:Nsubj ) {
aSubj[j] <- omega[p[j]]*(kappa[p[j]]-2)+1
bSubj[j] <- (1-omega[p[j]])*(kappa[p[j]]-2)+1
  theta[j] ~ dbeta( aSubj[j], bSubj[j] )
# Likelihood function
for ( i in 1:Nsubj ) {
z[i] ~ dbin( theta[i], N[i] )
} #close bracket for the model syntax w k A B
c = 1, 2, 3 wc kc
Amu and Bmu are not represented as random variables in the diagram on the right because they are treated as constants in this model. They are not sampled from a hyperhyperdistribution.
Skappa & Rkappa are not represented as random variables in the diagram on the right because they are treated as constants in this model. Specifically, Skappa = pow(10,2)/pow(10,2) = 1, Rkappa = 10/pow(10,2).
Note that kappa is a random variable that is sampled from a gamma( Skappa, Rkappa ) distribution. ac bc
s = 1, ..., J qs
i = 1, ..., n
yij
UW Psych 548, Miyamoto, Spr '12

7. Set Up for Instructor
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Classroom Support Services (CSS), 35 Kane Hall,
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Start laptop & projector before connecting them together
If necessary, reboot the laptop
Psych 548, Miyamoto, Aut ‘16