To efficiently process visual data at scale, researchers have proposed two techniques for lowering the computational overhead associated with the underlying deep learning models. The first approach consists of leveraging a specialized, lightweight model to directly answer the query. The second approach focuses on filtering irrelevant frames using a lightweight model and processing the filtered frames using a heavyweight model. These techniques suffer from two limitations. With the first approach, the specialized model is unable to provide accurate results for hard-to-detect events. With the second approach, the system is unable to accelerate queries focusing on frequently occurring events as the filter is unable to eliminate a significant fraction of frames in the video. In this paper, we present THIA, a video analytics system for tackling these limitations. The design of THIA is centered around three techniques. First, instead of using a cascade of models, it uses a single object detection model with multiple exit points for short-circuiting the inference. This early inference technique allows it to support a range of throughput-accuracy tradeoffs. Second, it adopts a fine-grained approach to planning and processes different chunks of the video using different exit points to meet the user's requirements. Lastly, it uses a lightweight technique for directly estimating the exit point for a chunk to lower the optimization time. We empirically show that these techniques enable THIA to outperform two state-of-the-art video analytics systems by up to 6.5X, while providing accurate results even on queries focusing on hard-to-detect events.