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For efficient neural network inference, it is desirable to achieve state-of-the-art accuracy with the simplest networks requiring the least computation, memory, and power. Quantizing networks to lower precision is a powerful technique for simplifying networks. As each layer of a network may have different sensitivity to quantization, mixed precision quantization methods selectively tune the precision of individual layers to achieve a minimum drop in task performance (e.g., accuracy). To estimate the impact of layer precision choice on task performance, two methods are introduced: i) Entropy Approximation Guided Layer selection (EAGL) is fast and uses the entropy of the weight distribution, and ii) Accuracy-aware Layer Precision Selection (ALPS) is straightforward and relies on single epoch fine-tuning after layer precision reduction. Using EAGL and ALPS for layer precision selection, full-precision accuracy is recovered with a mix of 4-bit and 2-bit layers for ResNet-50, ResNet-101 and BERT-base transformer networks, demonstrating enhanced performance across the entire accuracy-throughput frontier. The techniques demonstrate better performance than existing techniques in several commensurate comparisons. Notably, this is accomplished with significantly lesser computational time required to reach a solution.