The ABC approach rejects or modifies samples to make the moments of the selected samples match the desired distribution. Unlike moment matching, this approach does not train the mode to automatically produce samples with the correct moments but rather it modifies the sample directly. A simple explanation is given below: 

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1. Variational AutoEncoders (_VAE_)
2. Generative Adversarial Networks (_GAN_)
3. Diffusion Inversion
4. Approximate Bayesian computation (_ABC_) framework

Generative Modeling Architectures

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Leyshon, T. (2021, March 22). The abcs of approximate bayesian computation. Medium. https://towardsdatascience.com/the-abcs-of-approximate-bayesian-computation-bfe11b8ca341. 

The ABCs of Approximate Bayesian Computation

One of the most fundamental architectures for generative modeling. It was proposed in 2013 by Diederik P. Kingma and Max Welling. 

Variational AutoEncoders

This generative model scheme is set apart from MCMC sampling and ancestral sampling because it is based on nonequilibrium thermodynamics. This approach assumes that the probabilistic distribution we will sample from has some structure. This structure is gradually broken down by some diffusion that increases the entropy in the distribution thus making it unstructured. 


To create the generative model, we then train a model that reverses the process and gradually returns structure to our distribution. Similarly to MCMC, we apply many iterations to produce our sample; however, in this case our model is iterating in order to transform the entropy-filled distribution to our target distribution.   


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