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In this paper, we extend the free-energy principle to video quality assessment (VQA) by incorporating with the recent psychophysical study on human visual speed perception (HVSP). A novel video quality metric, namely the free-energy principle inspired video quality metric (FePVQ), is therefore developed and applied to perceptual video coding optimization. The free-energy principle suggests that the human visual system (HVS) can actively predict "orderly" information and avoid "disorderly" information for image perception. Basically, "orderly" is associated with the skeletons and edges of objects, and "disorderly" mostly concerns textures in images. Based on this principle, an image is separated into orderly and disorderly regions, and processed differently in image quality assessment. For videos, visual attention, or fixation, is associated with the objects with significant motion according to HVSP, resulting in a motion strength factor in the FePVQ so that the free-energy principle is extended into spatio-temporal domain for VQA. In addition, we investigate the application of the FePVQ in perceptual rate distortion optimization (RDO). For this purpose, the FePVQ is realized with low computational cost by using the relative total variation model and the block-wise motion vectors of video coding to simulate the free-energy principle and the HVSP, respectively. The experimental results indicate that the proposed FePVQ is highly consistent with the HVS perception. The linear correlation coefficient and Spearman's rank-order correlation coefficient are up to 0.8324 and 0.8281 on the LIVE video database. Better perceptual quality of encoded video sequences is achieved by FePVQ-motivated RDO in video coding.