Causal Inference via Predictive Coding

Bayesian and causal inference are fundamental processes for intelligence. Bayesian inference models observations: what can be inferred about y if we observe a related variable x? Causal inference models interventions: if we directly change x, how will y change? Predictive coding is a neuroscience-inspired method for performing Bayesian inference on continuous state variables using local information only. In this work, we go beyond Bayesian inference, and show how a simple change in the inference process of predictive coding enables interventional and counterfactual inference in scenarios where the causal graph is known. We then extend our results, and show how predictive coding can be generalized to cases where this graph is unknown, and has to be inferred from data, hence performing causal discovery. What results is a novel and straightforward technique that allows us to perform end-to-end causal inference on predictive-coding-based structural causal models, and demonstrate its utility for potential applications in machine learning.

Tighter Variational Bounds are Not Necessarily Better. A Research Report on Implementation, Ablation Study, and Extensions

This report explains, implements and extends the works presented in "Tighter Variational Bounds are Not Necessarily Better" (T Rainforth et al., 2018). We provide theoretical and empirical evidence that increasing the number of importance samples $K$ in the importance weighted autoencoder (IWAE) (Burda et al., 2016) degrades the signal-to-noise ratio (SNR) of the gradient estimator in the inference network and thereby affecting the full learning process. In other words, even though increasing $K$ decreases the standard deviation of the gradients, it also reduces the magnitude of the true gradient faster, thereby increasing the relative variance of the gradient updates. Extensive experiments are performed to understand the importance of $K$. These experiments suggest that tighter variational bounds are beneficial for the generative network, whereas looser bounds are preferable for the inference network. With these insights, three methods are implemented and studied: the partially importance weighted autoencoder (PIWAE), the multiply importance weighted autoencoder (MIWAE) and the combination importance weighted autoencoder (CIWAE). Each of these three methods entails IWAE as a special case but employs the importance weights in different ways to ensure a higher SNR of the gradient estimators. In our research study and analysis, the efficacy of these algorithms is tested on multiple datasets such as MNIST and Omniglot. Finally, we demonstrate that the three presented IWAE variations are able to generate approximate posterior distributions that are much closer to the true posterior distribution than for the IWAE, while matching the performance of the IWAE generative network or potentially outperforming it in the case of PIWAE.