  Understanding correlation judgement is important to designing effective
visualizations of bivariate data. Prior work on correlation perception has not
considered how factors including prior beliefs and uncertainty representation
impact such judgements. The present work focuses on the impact of uncertainty
communication when judging bivariate visualizations. Specifically, we model how
users update their beliefs about variable relationships after seeing a
scatterplot with and without uncertainty representation. To model and evaluate
the belief updating, we present three studies. Study 1 focuses on a proposed
''Line + Cone'' visual elicitation method for capturing users' beliefs in an
accurate and intuitive fashion. The findings reveal that our proposed method of
belief solicitation reduces complexity and accurately captures the users'
uncertainty about a range of bivariate relationships. Study 2 leverages the
``Line + Cone'' elicitation method to measure belief updating on the
relationship between different sets of variables when seeing correlation
visualization with and without uncertainty representation. We compare changes
in users beliefs to the predictions of Bayesian cognitive models which provide
normative benchmarks for how users should update their prior beliefs about a
relationship in light of observed data. The findings from Study 2 revealed that
one of the visualization conditions with uncertainty communication led to users
being slightly more confident about their judgement compared to visualization
without uncertainty information. Study 3 builds on findings from Study 2 and
explores differences in belief update when the bivariate visualization is
congruent or incongruent with users' prior belief. Our results highlight the
effects of incorporating uncertainty representation, and the potential of
measuring belief updating on correlation judgement with Bayesian cognitive
models.
