With the widespread application of uncertain knowledge graph (UKG) in open-world knowledge representation, the probabilistic relations they model have become increasingly complex. Sole reliance on embedding representations or simple pattern-matching methods is now insufficient to meet the demand for high-quality reasoning results. Therefore, the introduction of high-confidence, interpretable rule mining mechanisms is essential for enhancing both the reasoning capability and knowledge interpretability of UKGs. To this end, an uncertain knowledge graph completion model named UBERT-RM (uncertain knowledge graph BERT-rule mining) is proposed, which integrates BERT-based semantic modeling with structural path rule mining. UBERT-RM constructs a unified end-to-end framework that organically combines subgraph modeling, path generation, and confidence prediction. Contextual semantic representations of triples are extracted using BERT, while the path generation module employs a Transformer decoder structure to autoregressively generate high-confidence relation paths. In the rule parsing module, a dynamic confidence threshold mechanism is introduced to filter and parse nodes, ensuring that the final reasoning paths remain semantic coherence while achieving high reliability and interpretability. For confidence prediction, the generated rules and target triples are jointly encoded, followed by deep semantic interaction and information aggregation through a multi-head self-attention mechanism. A multilayer perceptron is then employed to perform regression modeling of the tail-entity confidence, thereby completing a closed-loop reasoning process from path mining to confidence prediction. Experimental results on the CN15k and NL27k datasets demonstrate that the proposed UBERT-RM achieves state-of-the-art performance on the link prediction task.