EDITOR’S PROOF
Inferring Ideological Ambiguity from Survey Data 375277
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322depend upon country-level hyper-parameters. Letk= 1 ,...,Kdenote a country in
which the survey is taken. The mean ideological positionsμjkare assumedapriori
to follow truncated normal distributions so thatμjk|δ∼N( 0 ,ημ) 1[
μjk∈(c 1 −δ,cM− 1 +δ)]
, (8)
ln(δ)∼N(ci+ 1 −ci,vδ). (9)We s e tημ=100 resulting in a vague but proper prior distribution. The hyper-
parameterδisaprioriset to have a log-normal distribution with mean equal to
the distance between any two cut-off points. We setvδ=1, resulting in identifiable
and yet highly flexible model: under this specification, we have 0.44 prior probabil-
ity that the most extreme party is two units (one-fifth of the scale) away from the
smallest or largest cut-off point. For the remaining parameters, we setσjk^2 |bk∼Inv-Gamma(
a,(a− 1 )bk)
, (10)
bk∼Gamma(, ), (11)
ψik∼U( 1
2 ,^2)
, (12)
τik∼N( 0 , 1 ). (13)In (10), the shape and scale of the inverse gamma distribution is fixed so that the
E(σjk^2 )=bk. Settinga=4 yields a priori variance ofbk^2 /2. Lettingbeasmall
number (e.g., 0.1), yields a prior onbkwith large variance; thus, the priors end up
having a negligible effect on the estimates. The hierarchical priors induce adaptive
shrinkage: the estimates of ideological ambiguity in a countrykare shrunken to-
wards the common meanbk. The statistical advantages of the hierarchical shrinkage
are well-documented in the literature (Gelman et al. 2003 ).
Further, under priors in (12), each respondent can expand or shrink the perceptual
space at most by a factor of two. Notice that asψi→0, the distribution ofzij
collapses to a degenerate distribution with the mass atτi. This implies that, forψi
near zero, a respondent would place all parties on the same point. Similarly, ifψiis
very large, a respondent places all parties on the opposite extremes of the scale.
Since both of these alternatives are not common, we constraintψi’s to the specified
interval.
Relative to the scale of cut-off points, the prior distribution ofτiin (13)also
allows sizable idiosyncratic location shifts. Lastly, the selection model parameters
α 0 andα 1 are assumed to follow normal distributions with 0 mean and variance
of 100 (a higher variance reduces the speed of convergence without affecting the
results).4 Parameter Estimation
The model is estimated using Markov Chain Monte Carlo (MCMC) methods using
Gibbs sampling approach (Gelfand and Smith 1990 ). LetNkandJkdenote the num-
ber of respondents and number of parties in countrykrespectively. LetNjkdenote