Renyi elbo derivation

renyi.ipynb

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    "#### Renyi variational bound\n",
    "$$ \\mathcal{L} = \\frac{1}{1-\\alpha} \\log \\mathbb{E}_q \\left(\\frac{p}{q}\\right)^{1-\\alpha}. $$"
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   "source": [
    "Putting $f = \\log(p) - \\log(q)$, we have\n",
    "$$\n",
    " \\mathcal{L} = \\frac{1}{1 - \\alpha} \\log{\\mathbb{E}_q\\left(e^{(1-\\alpha)f}\\right)}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "\\nabla \\mathcal{L} = \\frac{1}{1 - \\alpha} \\frac{\\nabla \\mathbb{E}_q \\left(e^{(1-\\alpha)f}\\right)}{\\mathbb{E}_q\\left(e^{(1-\\alpha)f}\\right)}.\n",
    "$$"
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   "source": [
    "#### Reparameterizable\n",
    "$$\n",
    "\\nabla \\mathcal{L} = \\frac{\\mathbb{E}_q \\left(e^{(1-\\alpha)f}\\nabla f\\right)}{\\mathbb{E}_q\\left(e^{(1-\\alpha)f}\\right)} = \\mathbb{E}_q \\frac{e^{(1-\\alpha)f}}{\\mathbb{E}_q\\left(e^{(1-\\alpha)f}\\right)} \\nabla f = \\mathbb{E}_q \\left(w\\nabla f\\right),\n",
    "$$\n",
    "where $w$ is the weight with\n",
    "$$\\log w = (1- \\alpha)f - \\log \\mathbb{E}_q\\left(e^{(1-\\alpha)f}\\right).$$"
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   "source": [
    "#### Non-reparameterizable\n",
    "Though have not been discussed in literature, similar to Pyro's [SVI Part III](http://pyro.ai/examples/svi_part_iii.html), we have\n",
    "$$\n",
    "\\nabla \\mathcal{L} = \\mathbb{E}_q \\left(w\\nabla f\\right) + \\frac{1}{1-\\alpha} \\frac{\\mathbb{E}_q \\left(e^{(1-\\alpha)f} \\nabla \\log q\\right)}{\\mathbb{E}_q\\left(e^{(1-\\alpha)f}\\right)} = \\mathbb{E}_q (w\\nabla f) + \\mathbb{E}_q \\left(\\frac{w}{1 - \\alpha}\\nabla \\log q\\right) .\n",
    "$$"
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