[alew1]

Any interpretation (including interpreting the inputs to the neural net as a "prompt") is "slapped on" in some sense—at some level, it's all just numbers being added, multiplied, and so on.

But I wouldn't call the probabilistic interpretation "after the fact." The entire training procedure that generated the LM weights (the pre-training as well as the RLHF post-training) is formulated based on the understanding that the LM predicts p(x_t | x_1, ..., x_{t-1}). For example, pretraining maximizes the log probability of the training data, and RLHF typically maximizes an objective that combines "expected reward [under the LLM's output probability distribution]" with "KL divergence between the pretraining distribution and the RLHF'd distribution" (a probabilistic quantity).