Contrastive decoding (CD) (Li et al., 2023) improves the next-token distribution of a large expert language model (LM) using a small amateur LM. Although CD is applied to various LMs and domains to enhance open-ended text generation, it is still unclear why CD often works well, when it could fail, and how we can make it better. To deepen our understanding of CD, we first theoretically prove that CD could be viewed as linearly extrapolating the next-token logits from a huge and hypothetical LM. We also highlight that the linear extrapolation could make CD unable to output the most obvious answers that have already been assigned high probabilities by the amateur LM. To overcome CD's limitation, we propose a new unsupervised decoding method called $\mathbf{A}$symptotic $\mathbf{P}$robability $\mathbf{D}$ecoding (APD). APD explicitly extrapolates the probability curves from the LMs of different sizes to infer the asymptotic probabilities from an infinitely large LM without inducing more inference costs than CD. In FactualityPrompts, an open-ended text generation benchmark, sampling using APD significantly boosts factuality in comparison to the CD sampling and its variants, and achieves state-of-the-art results for Pythia 6.9B and OPT 6.7B. Furthermore, in five commonsense QA datasets, APD is often significantly better than CD and achieves a similar effect of using a larger LLM. For example, the perplexity of APD on top of Pythia 6.9B is even lower than the perplexity of Pythia 12B in CommonsenseQA and LAMBADA.

We consider Markov logic networks and relational logistic regression as two fundamental representation formalisms in statistical relational artificial intelligence that use weighted formulas in their specification. However, Markov logic networks are based on undirected graphs, while relational logistic regression is based on directed acyclic graphs. We show that when scaling the weight parameters with the domain size, the asymptotic behaviour of a relational logistic regression model is transparently controlled by the parameters, and we supply an algorithm to compute asymptotic probabilities. We also show using two examples that this is not true for Markov logic networks. We also discuss using several examples, mainly from the literature, how the application context can help the user to decide when such scaling is appropriate and when using the raw unscaled parameters might be preferable. We highlight random sampling as a particularly promising area of application for scaled models and expound possible avenues for further research.