Scepticism towards childhood vaccines and genetically modified food has grown despite scientific evidence of their safety. Beliefs about scientific issues are difficult to change because they are entrenched within many related moral concerns and beliefs about what others think. We propose a cognitive network model which estimates the relationships, dissonance, and randomness between all related beliefs to derive predictions of the circumstances under which beliefs change. Using a probabilistic nationally representative longitudinal study, we found support for our model's predictions: Randomness of the belief networks decreased over time, for many participants their estimated dissonance related positively to their self-reported dissonance, and individuals who had high estimated dissonance of their belief network were more likely to change their beliefs to reduce this dissonance. This study is the first to combine a unifying predictive model with an experimental intervention and sheds light on dynamics of dissonance reduction leading to belief change.