Same-Sex Marriage Estimates from Vote Compass

At 10am AEDT tomorrow, Wednesday 15 November 2017, the Australian Bureau of Statistics will release the results of the Australian Marriage Law Postal Survey (AMLPS).

Ahead of tomorrow’s release of results, I report the results of using the ABC’s 2016 Vote Compass survey to estimate support for same-sex marriage in each of Australia’s 150 Commonwealth electoral divisions (CEDs). These estimates draw on joint work with Dr Shaun Ratcliff and Luke Mansillo, also based at the USSC at the University of Sydney.

Over one million respondents took the ABC/Vote Compass survey in 2016, administered on-line via the ABC’s web site. This is a truly massive sample relative to typical public opinion surveys, yet comes at the cost of pronounced unrepresentativeness and bias. Respondents were not randomly sampled, but self-selected to take the survey either by seeing links to the survey on the ABC’s web site, hearing or seeing references to the survey in ABC programming, or word of mouth. Australians who do not listen to ABC radio or watch ABC TV or do not visit the ABC website (including those who do not use the Internet) can be considered to have a very small probability of taking the survey. Self-selection among the ABC’s audience is also surely an issue. In short, despite the massive size of the VC data, it cannot be considered a random sample of the Australian electorate, not without weighting or other adjustments. We are attracted to the methodological challenge these data provide.

Caveats

As a predictive exercise, the analysis of the VC data provided here has some limitations:

VC was administered around the time of the 2016 Australian Federal election, almost 18 monthgs ago. Public opinion towards SSM may well have changed since then, especially given the “Yes” and “No” campaigns associated with the administration of AMLPS.
VC asks a slightly different question about SSM than that on AMLPS.
VC respondents were offered a “don’t know” response option, which was not available on AMLPS. Even if all “don’t know” responses were not in favor of SSM, this would not drive our estimates down. Accordingly, we consider these estimates as lower bounds on estimates of support for SSM.
Our methods of statistical adjustment rely on Census data from 2011 mapped to 2016 CEDs. The demographic composition of CEDs has surely changed since 2016.

Methods of adjustment

We consider several different approaches to adjusting the Vote Compass SSM data. Most of these use model-based approaches, similar to the “MRP” or “multivariate regression and post-stratification” approach of Andy Gelman and co-workers. Indeed, all of the results reported below rely on post-stratification to some extent. The differences among the different results turn on how the post-stratification is done, and the type of model used in the “MR” stage of “MRP”. In every instance we rely on the presence of Census data tabulated by CED.

Post-stratifying variables

The variables we rely on include:

gender
age, categorized into four bins (18-29, 30-44, 45-64, 65+)
religion, categorized into five bins Catholic, Conservative Christian, None, Other, Centrist Protestant. “Conservative Christian” denominations are defined here as Church of Christ, Jehovah’s Witness, Presbyterian, Seventh-day Adventist, Apostolic, Baptist, Pentecostal, Latter-day Saints, Reformed and Brethren.
income, by quintile
education (Some school, High school, Some tertiary, University)
1st preference vote in 2016 (Coalition, Greens, Labor, Other, Not voting)

With the exception of 1st preference vote share, these variables are available in both the VC survey and the Census. On the Census side, we use the ABS tablebuilder product to produce cell counts for the cross-classification of the variables listed above by CED; we refer to this data set of cells and cell counts as the Census post-stratifying frame.

First preference vote share is not available on the Census. We impute vote preference proportions onto each cell in the Census post-stratifying frame by fitting a model using the randomForests package in R for each CED. Predictions of vote proportions from this modeling exercise are adjusted such that aggregated over the cells for each CED, the predicted vote proportions match the actual vote proportions observed in each CED at the 2016 election.

Post-stratifying methods

The methods utilized here are:

raking, a conventional approach to post-stratification, using iterative proportional fitting methods to weight survey data such that there is (nearly exact) correspondence between the marginal distributions of selected demographic variables in the Census and the weighted survey data, by CED. Raking variables are those listed above and the raking procedure is implemented using the survey package in R.
GLM. (1) Fit logistic regressions fit to the VC data withing each CED. (2) A post-stratification frame — a data set of Census cells spanning the variables listed above — is then run through the model for each CED, then aggregated to produce estimated levels of support for SSM at the CED level.
NNET. As for GLM, except the fitted model in stage one is a neural net, fit using the nnet function in the MASS package iun R.
BART. As for GLM, except the fitted model in stage one is a Bayesian additive regression tree, fit with the bartMachine package in R.
GLMER. Instead of fitting a model specific to each CED, a mixed-effects logistic regression model is fit over the entire VC data set, with random effects and divisional-level predictors entering the model, using the glmer function in the lme4 package in R. The divisional-level predictors used are
- log of Greens vote share in the 2016 House of Representatives election
- log of Coalition vote share in the 2016 House of Representatives election
- log of proportion of households that identify as same-sex
- log of proportion of households in the top income quintile
- log of proportion of households in the 2nd income quantile
- log of proportion of persons reporting a conservative Christian denomination
- log of proportion of persons reporting no religion

The Bayesian additive regression tree models have the best fit in each CED, and so this approach tends to be our preferred method.

Final round of shrinkage for BART estimates

Since each CED is treated separately in the approaches listed above (except the mixed-effects GLMER method), we introduce some Bayesian shrinkage with a simple hierarchical model.

Let \(\theta_i\) be the unknown level of support for SSM in electoral division \(i = 1, \ldots, n\). The BART procedure provides \(y_i\), the BART estimate of support for SSM in CED \(i\) and \(\sigma^2_i\), the posterior variance of the corresponding BART estimate. We posit that \(y_i \sim N(\theta_i, \sigma^2_i)\) and absent any additional information we would estimate \(\theta_i\) with the BART estimate \(y_i\).

Divisional-level covariates not used in the post-stratification exercise can be used to specify a prior for \(\theta_i\), via the model \(\theta_i \sim N(z_i'\gamma, \omega^2)\). Priors on \(\gamma\) and \(\omega^2\) complete the model. Posterior inference for \(\theta_i\) thus combines both the information from VC and the Census post-stratifying frame reflected in \(y_i\) and \(\sigma^2_i\), along with additional divisional-level covariates \(z_i\). Here \(z_i\) comprises

log of population density of each CED
log of proportion of a CED’s households that identify as same-sex
fixed effects for each Australian state and territory

Smoothing splines are fit over the 1st two predictors, using the jagam function in mgcv package generates model matrices for the smooth terms, that we then utilize in the Bayesian model fitting package, JAGS.

Results

National and state results

We can produce a national level estimate by summing over all Census cells in the post-stratification frame, or equivalently, over the CEDs. These results are displayed, below.

Percent in support of SSM, by jurisdiction.
	Estimated	Low	Upper
QLD	52.7	52.2	53.2
NSW	55.0	54.6	55.4
SA	55.3	54.5	56.0
NT	56.1	53.3	58.9
TAS	56.2	54.9	57.4
WA	56.5	55.8	57.2
VIC	60.3	59.8	60.7
ACT	66.7	65.9	67.6
National	56.3	56.0	56.5

We again note that these estimates don’t allocate “don’t know” responses, and should be considered lower bounds on actual support levels for SSM.

Commonwealth electoral divisions

The following table presents results for all 150 CEDs, ranked from highest levels of estimated support for to lowest levels of the “BART + Bayes” estimates, which correspond to the Bayes estimates of \(\theta_i\) in the model described in the preceeding paragraphs, expressed as percentages. Lower and upper bounds of a 95% credible interval for the Bayes estimates are also reported.

Note that in 0 divisions we estimate that support for SSM is below 50%. Note once again that these estimates don’t allocate “don’t know” responses, and should be considered lower bounds on actual support levels for SSM.

Estimated support for SSM, by CED, from statistical adjustment of Vote Compass 2016
Rank	Division	State	BART + Bayes	lo	up	BART	GLMER	NNET	GLM	Raked	Naive
1	Melbourne	VIC	77.0	75.0	79.1	77.1	75.2	75.0	78.6	79.4	88.1
2	Sydney	NSW	76.9	74.7	79.2	76.7	75.6	78.1	77.8	80.2	88.0
3	Melbourne Ports	VIC	76.1	74.2	78.0	76.2	74.2	75.0	76.6	78.0	86.8
4	Grayndler	NSW	73.1	70.9	75.4	73.1	73.1	75.8	73.1	75.4	87.8
5	Wentworth	NSW	72.7	70.5	74.9	73.2	71.6	72.2	73.5	72.2	85.4
6	Batman	VIC	72.1	68.9	75.2	72.2	71.3	70.2	70.2	69.9	88.9
7	Higgins	VIC	69.5	67.8	71.4	69.5	68.6	70.1	69.7	71.3	83.2
8	Wills	VIC	69.4	66.3	72.4	69.6	68.2	68.7	68.5	71.3	88.5
9	Brisbane	QLD	68.5	66.6	70.5	68.7	67.4	66.5	69.7	69.8	81.1
10	Fenner	ACT	67.6	66.3	68.9	67.6	67.4	66.7	67.4	68.8	79.5
11	Gellibrand	VIC	67.3	64.4	70.1	66.4	68.6	64.4	66.5	65.4	86.1
12	Goldstein	VIC	66.9	65.0	68.8	67.5	65.3	67.9	68.1	67.9	81.2
13	Griffith	QLD	66.1	63.9	68.1	66.6	65.1	68.2	67.2	68.9	80.9
14	Canberra	ACT	65.9	64.8	67.0	65.9	65.9	67.6	65.7	66.3	77.7
15	Perth	WA	65.9	63.7	68.0	65.6	65.3	64.9	64.2	69.0	80.4
16	Newcastle	NSW	65.3	63.0	67.4	65.8	64.4	66.0	65.6	66.6	78.4
17	Kooyong	VIC	64.3	62.5	66.2	64.5	62.1	64.2	63.6	65.1	78.7
18	Isaacs	VIC	64.3	61.8	66.9	65.0	62.3	66.3	65.6	62.7	77.0
19	Ballarat	VIC	63.8	61.6	65.9	64.1	62.0	62.7	63.7	65.7	76.9
20	Kingsford Smith	NSW	63.4	60.8	65.9	63.8	63.8	62.2	63.0	63.9	80.0
21	Hotham	VIC	63.4	60.7	66.0	63.9	61.5	65.7	63.7	60.6	78.0
22	North Sydney	NSW	63.3	61.5	65.4	63.7	62.3	62.0	63.2	64.2	78.2
23	Jagajaga	VIC	63.1	61.1	65.2	63.2	62.8	67.2	62.4	63.8	79.2
24	Denison	TAS	62.7	60.2	65.4	62.2	63.1	61.2	61.4	64.8	79.8
25	Lalor	VIC	62.4	59.8	64.9	62.1	60.2	60.3	60.4	61.7	70.7
26	Dunkley	VIC	62.3	60.2	64.4	62.4	60.2	62.5	62.5	62.1	76.0
27	Ryan	QLD	62.3	60.6	63.9	62.7	61.4	63.2	62.8	64.1	74.9
28	Gorton	VIC	62.1	59.3	64.9	63.1	60.9	62.0	62.4	62.1	70.9
29	Adelaide	SA	61.9	59.9	63.8	61.8	63.3	62.8	61.9	63.7	79.0
30	Curtin	WA	61.9	59.7	63.8	62.1	61.3	65.3	61.3	62.5	76.3
31	Brand	WA	61.8	59.3	64.0	62.3	60.3	58.0	62.3	64.0	70.2
32	Warringah	NSW	61.8	59.9	63.9	62.3	61.7	63.8	61.5	65.4	79.3
33	Lilley	QLD	61.7	59.4	64.0	62.1	60.4	61.3	60.3	63.9	73.9
34	Solomon	NT	61.6	58.6	64.1	61.8	61.0	57.7	61.3	61.4	71.4
35	Corio	VIC	61.5	58.9	64.1	62.0	60.2	60.6	61.8	62.9	75.4
36	Fremantle	WA	61.2	58.8	63.5	61.3	63.3	58.1	60.1	62.1	78.5
37	Port Adelaide	SA	61.2	58.5	64.0	61.2	62.4	61.5	61.7	60.7	76.7
38	Cunningham	NSW	60.9	58.3	63.4	61.6	59.0	59.2	59.8	62.6	75.1
39	Franklin	TAS	60.5	58.3	62.6	61.0	60.2	60.0	60.3	62.1	72.2
40	McEwen	VIC	60.0	57.9	62.1	60.4	60.4	59.1	59.9	62.1	72.9
41	Kingston	SA	59.8	57.6	61.9	60.2	61.8	60.9	59.7	61.3	71.7
42	Bendigo	VIC	59.7	57.5	61.8	59.8	60.2	59.4	58.4	63.0	76.5
43	Maribyrnong	VIC	59.6	57.0	62.5	59.3	61.6	61.2	58.7	60.2	80.7
44	Swan	WA	59.4	56.7	62.0	58.5	58.8	59.0	58.3	62.2	75.3
45	Stirling	WA	58.9	56.8	61.1	58.7	58.6	56.9	58.1	59.0	75.8
46	Oxley	QLD	58.7	56.3	61.1	59.4	57.2	59.3	59.2	59.2	68.6
47	Moore	WA	58.7	56.5	60.8	59.4	56.8	58.2	58.7	60.0	69.7
48	Mackellar	NSW	58.4	56.2	60.7	58.9	55.5	58.9	58.0	59.3	72.8
49	Hunter	NSW	58.2	55.8	60.7	59.3	57.6	56.8	60.0	58.2	69.7
50	Makin	SA	58.2	56.1	60.0	58.4	56.7	57.2	58.0	57.2	68.6
51	Casey	VIC	58.1	55.8	60.2	58.3	57.7	58.1	57.6	61.1	73.3
52	Shortland	NSW	58.1	55.7	60.5	58.7	57.8	54.7	58.0	58.8	70.6
53	La Trobe	VIC	58.0	56.1	60.2	57.7	58.8	56.3	58.2	59.1	74.3
54	Scullin	VIC	57.9	54.8	61.0	58.5	59.5	56.1	58.0	53.6	73.9
55	Holt	VIC	57.7	54.9	60.2	57.7	56.7	62.0	57.4	53.9	67.5
56	Hindmarsh	SA	57.6	55.2	59.8	57.4	57.7	56.4	57.2	57.8	75.2
57	Chisholm	VIC	57.3	55.0	59.8	56.8	57.0	56.5	55.8	58.7	75.2
58	Pearce	WA	57.2	54.8	59.5	57.7	56.5	54.3	59.3	58.1	67.2
59	Richmond	NSW	56.9	54.4	59.6	57.0	57.6	60.4	57.0	58.2	74.7
60	Wakefield	SA	56.9	54.3	59.3	57.7	57.7	55.7	57.7	58.8	68.9
61	Eden-Monaro	NSW	56.7	54.7	59.0	57.3	56.5	57.3	58.6	56.1	70.8
62	Rankin	QLD	56.6	54.0	59.2	57.3	53.9	58.3	56.6	59.8	63.1
63	Flinders	VIC	56.6	54.1	59.0	56.2	57.2	56.3	55.7	57.8	74.6
64	McMillan	VIC	56.4	54.0	58.8	56.3	55.3	55.6	56.0	57.0	71.2
65	Moncrieff	QLD	56.3	53.5	59.5	56.0	54.3	58.6	55.2	58.1	69.0
66	Deakin	VIC	56.3	54.3	58.1	56.1	54.9	55.9	55.2	57.2	72.3
67	Boothby	SA	56.3	54.4	58.0	56.1	58.0	55.5	56.9	58.0	74.4
68	Macarthur	NSW	56.3	53.6	58.9	56.6	54.2	55.4	57.1	57.3	67.1
69	Burt	WA	56.2	53.7	58.7	56.0	55.1	56.7	55.7	56.1	64.3
70	Corangamite	VIC	56.2	54.2	58.5	56.0	57.3	52.0	55.0	59.6	75.7
71	Paterson	NSW	56.2	53.6	58.7	56.7	54.8	58.0	57.9	57.2	67.1
72	Whitlam	NSW	56.2	53.5	58.8	57.0	56.0	62.0	56.6	57.0	67.2
73	Reid	NSW	56.1	53.8	58.4	55.6	55.9	55.6	55.5	54.6	72.5
74	Dobell	NSW	56.0	53.6	58.3	55.8	55.8	53.5	55.9	56.8	71.5
75	Herbert	QLD	55.7	53.1	58.4	55.8	55.4	55.3	56.3	59.5	69.0
76	Robertson	NSW	55.7	53.2	57.9	55.5	55.1	57.6	54.6	57.4	71.2
77	McPherson	QLD	55.6	52.9	58.1	55.5	54.7	56.5	54.6	58.7	70.4
78	Dickson	QLD	55.5	53.6	57.2	55.6	55.1	52.8	55.6	56.9	66.2
79	Hasluck	WA	55.4	53.2	57.9	54.9	54.9	56.0	55.2	56.3	68.2
80	Cowan	WA	55.4	53.0	57.9	55.2	56.1	53.7	54.3	54.1	68.8
81	Sturt	SA	55.4	53.4	57.6	55.2	55.1	54.2	54.4	55.1	72.5
82	Aston	VIC	55.4	53.1	57.5	55.3	54.5	50.8	55.0	56.0	69.4
83	Barton	NSW	55.2	52.2	58.4	53.2	55.6	54.0	56.3	53.4	76.0
84	Bowman	QLD	54.5	52.4	56.8	54.8	53.5	55.5	54.9	56.4	64.9
85	Forrest	WA	54.5	51.8	56.8	54.8	52.6	52.8	54.4	54.6	66.5
86	Bradfield	NSW	54.3	52.6	56.2	54.4	52.8	52.9	53.7	53.7	69.2
87	Petrie	QLD	54.3	51.9	56.7	54.2	52.4	53.8	54.5	54.4	64.9
88	Leichhardt	QLD	54.2	51.4	57.0	54.6	54.3	56.3	52.4	57.8	67.2
89	Bass	TAS	54.0	51.5	56.3	53.8	55.1	55.0	52.9	56.1	69.3
90	Moreton	QLD	53.7	51.4	55.9	53.0	55.8	51.0	53.8	55.3	73.2
91	Menzies	VIC	53.7	51.4	55.9	53.4	52.2	52.5	53.6	52.8	69.5
92	Indi	VIC	53.6	51.3	55.9	53.5	54.9	51.8	52.4	54.4	71.3
93	Watson	NSW	53.6	49.9	57.2	52.7	50.4	51.1	51.3	44.1	66.7
94	Bonner	QLD	53.4	51.3	55.6	53.1	53.7	48.1	53.2	54.9	68.3
95	Lyons	TAS	53.3	50.4	56.3	53.1	53.7	52.6	53.0	54.4	67.7
96	Hughes	NSW	53.3	51.2	55.3	53.1	50.7	53.0	52.1	53.7	68.1
97	Cook	NSW	53.3	50.7	55.9	52.9	50.5	56.4	51.3	52.0	68.9
98	Chifley	NSW	53.3	49.8	56.6	53.2	48.4	52.8	55.7	50.6	59.7
99	Bennelong	NSW	53.2	50.9	55.2	52.9	49.5	55.2	54.6	52.0	68.5
100	Greenway	NSW	53.2	50.7	55.6	52.9	49.7	55.4	53.5	53.4	62.0
101	Forde	QLD	53.0	50.5	55.3	52.7	52.3	55.7	51.9	54.2	62.4
102	Blair	QLD	53.0	50.6	55.6	53.3	53.6	49.9	50.5	56.4	66.0
103	Wannon	VIC	53.0	50.1	55.6	53.3	51.9	56.0	54.7	52.4	70.8
104	Mayo	SA	52.9	51.1	54.6	52.7	54.2	52.0	52.5	54.1	70.0
105	Fadden	QLD	52.8	50.5	55.3	52.5	53.1	52.2	52.3	54.9	66.7
106	Macquarie	NSW	52.8	50.6	55.1	51.9	54.6	51.5	51.3	54.4	71.4
107	Lindsay	NSW	52.7	50.0	55.4	51.7	52.3	50.1	50.5	51.6	67.4
108	Calare	NSW	52.5	49.9	54.9	52.8	51.2	53.4	53.9	53.1	68.3
109	Gilmore	NSW	52.2	49.8	54.6	52.2	52.0	49.4	50.1	55.1	68.1
110	Page	NSW	52.0	49.2	54.8	51.6	51.8	51.1	52.3	53.8	69.6
111	Parramatta	NSW	51.8	49.2	54.7	51.1	48.7	49.4	49.6	50.4	64.5
112	Mitchell	NSW	51.7	49.4	53.9	51.6	49.0	51.5	51.5	53.1	63.9
113	Fisher	QLD	51.6	49.3	54.0	51.5	51.8	52.6	51.4	53.4	64.4
114	Gippsland	VIC	51.5	49.0	54.0	51.1	49.4	49.8	52.1	51.9	67.6
115	Lingiari	NT	51.4	46.9	55.4	50.8	53.4	52.0	47.9	51.5	65.9
116	Banks	NSW	51.3	48.4	54.1	50.3	48.2	49.3	50.1	49.5	66.5
117	Canning	WA	51.2	48.5	53.7	50.8	49.1	53.3	51.9	51.9	62.9
118	Calwell	VIC	51.2	47.8	54.5	49.6	54.1	52.5	48.5	47.4	67.2
119	Tangney	WA	50.9	48.9	53.0	50.7	52.0	49.3	50.7	52.0	68.0
120	Bruce	VIC	50.8	48.0	53.4	49.8	49.8	49.3	49.1	48.5	65.5
121	Berowra	NSW	50.8	48.9	52.7	50.5	49.5	49.1	50.8	50.5	66.6
122	Dawson	QLD	50.8	47.5	53.7	52.0	49.0	59.4	49.5	54.6	61.9
123	Fairfax	QLD	50.7	48.4	52.9	50.3	51.2	46.3	49.9	52.1	64.7
124	Longman	QLD	50.7	48.1	53.5	50.2	49.5	46.9	51.1	52.3	61.3
125	Werriwa	NSW	50.7	46.6	54.5	51.3	48.5	50.1	NA	45.5	64.1
126	Braddon	TAS	50.4	47.6	53.3	50.6	51.1	53.9	49.5	52.0	63.1
127	Durack	WA	50.2	46.5	53.7	50.7	51.7	49.3	50.9	57.0	66.8
128	McMahon	NSW	50.1	46.5	53.7	50.6	49.1	55.5	50.1	46.9	62.5
129	Lyne	NSW	50.1	47.5	52.7	50.4	47.7	48.3	52.0	51.8	65.1
130	Mallee	VIC	50.0	47.0	52.9	49.7	47.3	51.7	50.4	51.5	67.3
131	Fowler	NSW	50.0	45.4	54.6	52.6	50.0	51.8	51.4	48.8	65.5
132	Hume	NSW	49.9	47.2	52.4	49.5	51.1	51.6	48.5	51.0	66.8
133	O’Connor	WA	49.5	47.0	51.9	49.6	49.9	46.4	48.3	49.8	63.5
134	Murray	VIC	49.5	46.6	52.2	48.9	47.2	46.9	49.0	49.3	67.7
135	Farrer	NSW	49.0	46.3	51.7	49.9	48.9	50.6	50.5	49.3	67.9
136	Blaxland	NSW	49.0	44.8	53.1	48.9	46.4	49.6	50.4	45.9	59.2
137	Wide Bay	QLD	47.3	44.9	49.8	46.8	47.6	46.1	47.3	49.2	59.8
138	Riverina	NSW	46.8	43.8	49.7	46.7	45.8	46.7	46.6	48.7	66.1
139	Wright	QLD	46.8	44.1	49.4	46.3	47.3	42.1	44.4	48.9	61.0
140	Cowper	NSW	46.5	43.7	49.1	44.6	46.9	49.5	44.1	48.5	68.4
141	Capricornia	QLD	46.1	43.5	48.8	46.1	49.4	44.5	45.6	48.3	59.8
142	Parkes	NSW	45.2	42.1	48.5	45.3	45.1	47.9	46.2	44.2	60.7
143	Groom	QLD	44.9	42.4	47.3	44.3	42.8	41.2	44.2	46.4	57.5
144	Grey	SA	44.6	41.5	47.5	44.9	47.7	43.7	46.3	47.7	61.1
145	Kennedy	QLD	44.5	41.3	47.5	44.9	43.5	46.3	43.9	47.0	56.9
146	Hinkler	QLD	44.1	40.6	47.4	42.5	41.5	36.8	41.4	42.4	53.1
147	New England	NSW	44.0	40.9	46.6	42.8	43.8	41.8	41.7	43.8	62.3
148	Barker	SA	43.7	41.4	46.2	43.0	43.8	45.9	42.8	45.2	60.2
149	Flynn	QLD	42.7	39.8	45.7	42.7	44.5	49.0	40.2	46.1	56.3
150	Maranoa	QLD	37.2	34.4	40.1	36.5	39.3	36.7	35.1	36.0	53.6

We compare the Bayes estimates and naive, raw estimates in the following dotplot:

Comparison of Bayes estimates and naive estimates.