  Our opinions, which things we like or dislike, depend on the opinions of
those around us. Nowadays, we are influenced by the opinions of online
strangers, expressed in comments and ratings on online platforms. Here, we
perform novel "academic A/B testing" experiments with over 2,500 participants
to measure the extent of that influence. In our experiments, the participants
watch and evaluate videos on mirror proxies of YouTube and Vimeo. We control
the comments and ratings that are shown underneath each of these videos. Our
study shows that from 5$\%$ up to 40$\%$ of subjects adopt the majority opinion
of strangers expressed in the comments. Using Bayes' theorem, we derive a
flexible and interpretable family of models of social influence, in which each
individual forms posterior opinions stochastically following a logit model. The
variants of our mixture model that maximize Akaike information criterion
represent two sub-populations, i.e., non-influenceable and influenceable
individuals. The prior opinions of the non-influenceable individuals are
strongly correlated with the external opinions and have low standard error,
whereas the prior opinions of influenceable individuals have high standard
error and become correlated with the external opinions due to social influence.
Our findings suggest that opinions are random variables updated via Bayes' rule
whose standard deviation is correlated with opinion influenceability. Based on
these findings, we discuss how to hinder opinion manipulation and
misinformation diffusion in the online realm.
