Setup for inference

Setup for inference.ipynb

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   "source": [
    "Let $E$ be the selected variables and $E'$ be the non-zero variances.\n",
    "\n"
   ]
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    "There is some RFX model ${\\cal M} = \\{\\theta=(\\beta_E, \\sigma^2_{E'})\\}$. \n",
    "\n",
    "1. Let's assume in ${\\cal M}$ you can construct\n",
    "$\\hat{\\beta}_j \\sim N(\\beta_j, \\sigma^2_j)$ on fresh data.\n",
    "\n",
    "2. Let's assume the event\n",
    "$$\n",
    "(\\hat{E}(X,Y), \\hat{E}'(X,Y)) = (E,E')\n",
    "$$\n",
    "can be expressed as \n",
    "$$\n",
    "F_{(E,E')}(T)\n",
    "$$\n",
    "where $(E,E')$ are your observed effects and variances.\n",
    "Further, assume that in model ${\\cal M}$ the pair $(T, \\hat{\\beta}_j)$ are asymptotically normal with covariance\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    " \\sigma^2_j & \\rho \\\\\n",
    " \\rho & \\Sigma_T\n",
    "\\end{pmatrix}\n",
    "$$"
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   "source": [
    "Then, we'll define\n",
    "$$\n",
    "N = T - \\rho / \\sigma^2_j \\cdot \\hat{\\beta}_j\n",
    "$$\n",
    "and\n",
    "$$\n",
    "\\pi = \\pi(\\beta_j; n) = F(n + \\rho / \\sigma^2 \\cdot \\beta_j)\n",
    "$$\n",
    "and $\\bar{\\pi}(\\beta_j) = \\pi(\\beta_j;N_{obs})$"
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   "source": [
    "To test $H_0:\\beta_j=\\gamma$ use density proportional to\n",
    "    $$\n",
    "t \\to \\phi_{(\\gamma, \\sigma^2_j)}(t) \\cdot \\bar{\\pi}(t)\n",
    "    $$"
   ]
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