Brain-like variational inference

Inference in both brains and machines can be formalized by optimizing a shared objective: maximizing the evidence lower bound (ELBO) in machine learning, or minimizing variational free energy $\left(F\right)$ in neuroscience ( $\text{ELBO}=-F$ ). While this equivalence suggests a unifying framework, it leaves open how inference is implemented in neural systems. Here, we show that online natural gradient descent on $F$ , under Poisson assumptions, leads to a recurrent spiking neural network that performs variational inference via membrane potential dynamics. The resulting model-the iterative Poisson variational autoencoder ( $\text{i}P$ -VAE)- replaces the encoder network with local updates derived from natural gradient descent on $F$ . Theoretically, $\text{i}P$ -VAE yields a number of desirable features such as emergent normalization via lateral competition, and hardware-efficient integer spike count representations. Empirically, $\text{i}P$ -VAE outperforms both standard VAEs and Gaussian-based predictive coding models in sparsity, reconstruction, and biological plausibility. $\text{i}P$ -VAE also exhibits strong generalization to out-of-distribution inputs, exceeding hybrid iterative-amortized VAEs. These results demonstrate how deriving inference algorithms from first principles can yield concrete architectures that are simultaneously biologically plausible and empirically effective.