[{"id":354,"title":"Unifying generative models and exact likelihood-free inference with conditional bijections","sha":"5c2f74aedbf3618ca180382e393c7617","user_id":null,"state":"published","body":"Recent work in density estimation uses a bijection $f : X \\to Z$ (e.g. an invertible flow or autoregressive model) and a tractable density $p(z)$  (e.g. [[1]](https://arxiv.org/abs/1410.8516) [[2]](http://www.dmi.usherb.ca/~larocheh/projects_nade.html) [[3]](https://arxiv.org/abs/1410.6460) [[4]](https://arxiv.org/abs/1505.05770)). \r\n\\begin{equation}\r\np(x) = p(f_\\phi(x)) \\left| \\det\\left ( \\frac{\\partial f_\\phi(x)}{\\partial x_T} \\right) \\right | \\;,\r\n\\end{equation}\r\nwhere $\\phi$ are the internal network parameters for the bijection $f_\\phi$. Learning proceeds via gradient ascent $\\nabla_\\phi \\sum_i \\log p(x_i)$ with data $x_i$ (i.e. maximum likelihood wrt. the internal parameters $\\phi$). Since $f$ is invertible, then this model can also be used as a generative model for $X$.\r\n\r\nThis can be generalized to the conditional density $p(x|\\theta)$ by utilizing a family of bijections $f_{\\theta} : X \\to Z$ parametrized by $\\theta$ (e.g. [[5]](https://arxiv.org/pdf/1609.03499.pdf) [[6]](https://arxiv.org/abs/1611.05209)). \r\n\\begin{equation}\r\np(x|\\theta) = p(f_{\\phi; \\theta}(x)) \\left| \\det \\left ( \\frac{\\partial f_{\\phi; \\theta}(x)}{\\partial x_T} \\right) \\right |\r\n\\end{equation}\r\nHere $\\theta$ and $x$ are input to the network (and its inverse) and $\\phi$ are internal network parameters. Again, learning proceeds via gradient ascent $\\nabla_\\phi \\sum_i \\log p(x_i|\\theta_i)$ with data $x_i,\\theta_i$.\r\n\r\nWe observe that not only can this model be used as a conditional generative model $p(x|\\theta)$, but it can also be used to perform asymptotically exact, amortized likelihood-free inference on $\\theta$.\r\n\r\nThis is particularly interesting when $\\theta$ is identified with the parameters of an intractable, non-differentiable computer simulation or the conditions of some real world data collection process.","created_at":"2016-12-08T11:48:03.802Z","updated_at":"2016-12-10T02:41:10.507Z","zenodo_id":198541,"doi":"https://doi.org/10.5281/zenodo.198541","subject":null,"tags":["machine learning","likelihood-free inference","density estimation"],"vote_count":1,"view_count":1,"score":1.1,"deleted":false,"muted":false,"tweeted":true,"attachment_file_name":null,"attachment_content_type":null,"attachment_file_size":null,"attachment_updated_at":null}]