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We present the Deep Convolutional Gaussian Mixture Model (DCGMM), a new probabilistic approach for image modeling capable of density estimation, sampling and tractable inference. DCGMM instances exhibit a CNN-like layered structure, in which the principal building blocks are convolutional Gaussian Mixture (cGMM) layers. A key innovation w.r.t. related models like sum-product networks (SPNs) and probabilistic circuits (PCs) is that each cGMM layer optimizes an independent loss function and therefore has an independent probabilistic interpretation. This modular approach permits intervening transformation layers to harness the full spectrum of (potentially non-invertible) mappings available to CNNs, e.g., max-pooling or half-convolutions. DCGMM sampling and inference are realized by a deep chain of hierarchical priors, where a sample generated by a given cGMM layer defines the parameters of sampling in the next-lower cGMM layer. For sampling through non-invertible transformation layers, we introduce a new gradient-based sharpening technique that exploits redundancy (overlap) in, e.g., half-convolutions. DCGMMs can be trained end-to-end by SGD from random initial conditions, much like CNNs. We show that DCGMMs compare favorably to several recent PC and SPN models in terms of inference, classification and sampling, the latter particularly for challenging datasets such as SVHN. We provide a public TF2 implementation.