I'm conducting an experiment with the MNIST digit data - handwritten digits 0-9, each example composed of 28x28 bitmap of pixels.

Imagine a collection of examples is drawn at random without class labels, and submitted to a third party (or "oracle") for classification, but instead of receiving example-wise labels, the oracle only tells you what proportion of the selected examples belong to each class. So maybe in a given bag of examples you know that 15% are "2", 45% are "9" and 40% are "3", but not which is which. To be clear, every example in the bag has a single, discrete "correct" answer. You can draw as many random bags as you'd like for submission, however degenerate cases aren't allowed (e.g. a bag of 1 example).

I feel confident that I could build a convnet to learn summary vectors from the input examples, but what I'd really like is the ability to "play back" the convnet in order to, say, see what it would project for 100% "2".

Can this be done? Is there a better approach?

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You are referring to the prior class distribution of the categories in a collection. The prior distribution is implicitly given, by means of the training set provided to the neural network.

The play-back option is called inverse mapping, en is computationally intractable for a regular feed-forward neural network. Instead, the convolutional deep-learning neural networks do learn prototypical patterns. I recommend you to train deep learning neural networks (LeCunn-style) and study their prototypical vectors, after successful learning has been achieved.

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