We propose a framework for Bayesian Likelihood-Free Inference (LFI) based on Generalized Bayesian Inference. To define the generalized posterior, we use Scoring Rules (SRs), which evaluate probabilistic models given an observation. In LFI, we can sample from the model but not evaluate the likelihood; for this reason, we employ SRs which admit unbiased empirical estimates. We use the Energy and the Kernel SRs, for which our posterior enjoys consistency in a well-specified setting and outlier robustness, but our general framework applies to other SRs. The straightforward way to perform posterior inference relies on pseudo-marginal Markov Chain Monte Carlo (MCMC). While this works satisfactorily for simple setups, it mixes poorly, which makes inference impossible when many observations are present. Hence, we employ stochastic-gradient (SG) MCMC methods, which are rejection-free and have thus no mixing issues. The targets of both sampling schemes only approximate our posterior, but the error vanishes as the number of model simulations at each MCMC step increases. In practice, SG-MCMC performs better than pseudo-marginal at a lower computational cost when both are applicable and scales to higher-dimensional setups. In our simulation studies, we compare with related approaches on standard benchmarks and a chaotic dynamical system from meteorology; for the latter, SG-MCMC allows us to infer the parameters of a neural network used to parametrize a part of the update equations of the dynamical system.

翻译：我们提出了一种基于广义贝叶斯推断的贝叶斯无似然推断（LFI）框架。为定义广义后验分布，我们采用评分规则（SRs）来评估给定观测下的概率模型。在LFI中，虽然可以从模型中采样，但无法计算似然函数，因此我们选用可得到无偏经验估计的评分规则。我们采用能量评分规则和核评分规则，在设定正确时后验分布具有一致性且对异常值具有鲁棒性，但我们的通用框架同样适用于其他评分规则。执行后验推断的直接方法是基于伪边际马尔可夫链蒙特卡洛（MCMC）方法。尽管该方法在简单设定下表现良好，但其混合效率较差，当观测数据较多时会导致推断不可行。为此，我们采用无拒绝采样且无混合问题的随机梯度（SG）MCMC方法。两种采样方案的目标分布均仅近似于我们的后验分布，但随着每个MCMC步长中模型模拟次数的增加，近似误差会逐渐消失。实践表明，在两种方法均适用的场景下，SG-MCMC以更低的计算成本获得优于伪边际方法的性能，并能扩展至更高维度的设定。在模拟研究中，我们使用标准基准测试及气象学中的混沌动力系统与相关方法进行对比；针对后者，SG-MCMC使我们能够推断出用于参数化动力系统更新方程部分的神经网络参数。