The rapid proliferation of the Internet of Things and the dramatic resurgence of artificial intelligence based application workloads have led to immense interest in performing inference on energy-constrained edge devices. Approximate computing (a design paradigm that trades off a small degradation in application quality for disproportionate energy savings) is a promising technique to enable energy-efficient inference at the edge. This article introduces the concept of an approximate edge inference system ( AxIS ) and proposes a systematic methodology to perform joint approximations between different subsystems in a deep neural network (DNN)-based edge inference system, leading to significant energy benefits compared to approximating individual subsystems in isolation. We use a smart camera system that executes various DNN-based image classification and object detection applications to illustrate how the sensor, memory, compute, and communication subsystems can all be approximated synergistically. We demonstrate our proposed methodology using two variants of a smart camera system: (a) Cam Edge , where the DNN is executed locally on the edge device, and (b) Cam Cloud , where the edge device sends the captured image to a remote cloud server that executes the DNN. We have prototyped such an approximate inference system using an Intel Stratix IV GX-based Terasic TR4-230 FPGA development board. Experimental results obtained using six large DNNs and four compact DNNs running image classification applications demonstrate significant energy savings (≈ 1.6× -4.7× for large DNNs and ≈ 1.5× -3.6× for small DNNs), for minimal (<1%) loss in application-level quality. Furthermore, results using four object detection DNNs exhibit energy savings of ≈ 1.5× -5.2× for similar quality loss. Compared to approximating a single subsystem in isolation, AxIS achieves 1.05× -3.25× gains in energy savings for image classification and 1.35× -4.2× gains for object detection on average, for minimal (<1%) application-level quality loss.