Opinion mining and demographic attribute inference have many applications in social science. In this paper, we propose models to infer daily joint probabilities of multiple latent attributes from Twitter data, such as political sentiment and demographic attributes. Since it is costly and time-consuming to annotate data for traditional supervised classification, we instead propose scalable Learning from Label Proportions (LLP) models for demographic and opinion inference using U.S. Census, national and state political polls, and Cook partisan voting index as population level data. In LLP classification settings, the training data is divided into a set of unlabeled bags, where only the label distribution in of each bag is known, removing the requirement of instance-level annotations. Our proposed LLP model, Weighted Label Regularization (WLR), provides a scalable generalization of prior work on label regularization to support weights for samples inside bags, which is applicable in this setting where bags are arranged hierarchically (e.g., county-level bags are nested inside of state-level bags). We apply our model to Twitter data collected in the year leading up to the 2016 U.S. presidential election, producing estimates of the relationships among political sentiment and demographics over time and place. We find that our approach closely tracks traditional polling data stratified by demographic category, resulting in error reductions of 28-44% over baseline approaches. We also provide descriptive evaluations showing how the model may be used to estimate interactions among many variables and to identify linguistic temporal variation, capabilities which are typically not feasible using traditional polling methods.