genModel.ipynb

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    "# EBM as A Generative Model {#sec-gen}\n",
    "\n",
    "We can use EBM to generate synthetic biomarker data if we know:\n",
    "\n",
    "- The order ($S$) in which different biomarkers get affected by the disease. \n",
    "- Parameters (i.e., mean and standard deviation) of biomarkers' distribution when they are affected ($\\theta$) and not affected ($\\phi$) by the disease. \n",
    "- Stages ($k_j$) that each participant is in. \n",
    "\n",
    "Data we can generate looks like this:\n",
    "\n",
    "![Sample Data](img/sample_data.png){#fig-sample-data .lightbox}\n",
    "\n",
    "This data is from a single participant. \n",
    "\n",
    "As we mentioned above, to generate this data, we need to know:\n",
    "\n",
    "- $S$, i.e., the order of biomarkers. In the above example, $S$ is HIP-FCI, PCC-FCI, HIP-GMI, FUS-GMI, FUS-FCI.\n",
    "- $\\mathcal N(\\theta_{\\mu}, \\theta_{\\sigma})$ and $\\mathcal N(\\phi_{\\mu}, \\phi_{\\sigma})$ for each of the five biomarkers, which are known but not shown directly here in the dataset. \n",
    "- $k_j$, which is 2 in the above example. \n",
    "\n",
    "We explain how this data is constructed in the following, column by column. \n",
    "\n",
    "First, the `participant` id is $67$. The `biomarker` indicates each of the five biomarkers examined and measured. The `measurement` is the biomarkers' measurement. `k_j` is the participant's stage. If this stage is above 0, it means `Diseased = True`. `S_n` indicates the $n$-th rank in the order. If `k_j < S_n`, it means the participant's stage hasn't reached that biomarker's rank; therefore, this biomarker is `not affected`. If `k_j >= S_n`, then this biomarker is `affected`. \n",
    "\n",
    "If a biomarker is `affected`, then its measurement comes from $\\mathcal N(\\theta_{\\mu}, \\theta_{\\sigma})$ of that biomarker; if `not_affected`, $\\mathcal N(\\phi_{\\mu}, \\phi_{\\sigma})$.\n",
    "\n",
    "## Generative Process\n",
    "\n",
    "The generative process of biomarker measurements can be described as:\n",
    "\n",
    "$$\n",
    "X_{nj} \\mid S, k_j, \\theta_n, \\phi_n \\sim I(z_j = 1) \\bigg[ I(S(n) \\leq k_j) \\, p(X_{nj} \\mid \\theta_{n}) + I(S(n) > k_j) \\, p(X_{nj} \\mid \\phi_{n}) \\bigg] \\\\\n",
    "+ \\left(1 - I(z_j = 1) \\right) p(X_{nj} \\mid \\phi_{n})\n",
    "$$ {#eq-gen}\n",
    "\n",
    "\n",
    "<!-- \\begin{align*}\n",
    "X_{nj} \\mid S, k_j, \\theta_n, \\phi_n &\\sim I(z_j == 1) \\bigg[ I(S(n) \\leq k_j) \\, p(X_{nj} \\mid \\theta_{n}) + I(S(n) > k_j) \\, p(X_{nj} \\mid \\phi_{n}) \\bigg] \\\\\n",
    "&\\quad + \\left(1 - I(z_j == 1) \\right) p(X_{nj} \\mid \\phi_{n})\n",
    "\\end{align*} -->\n",
    "\n",
    "This model says that given that we know $S, k_j, \\theta_n, \\text{and } \\phi_n$, we can draw the biomarker measurement from a distribution. \n",
    "\n",
    "$S \\sim \\mathrm{UniformPermutation}(\\cdot)$\n",
    "\n",
    "$S$ follows a distribution of uniform permutation. That means the ordering of biomarkers is random. \n",
    "\n",
    "$k_j \\sim \\mathrm{DiscreteUniform}(N)$\n",
    "\n",
    "$k_j$ follows a discrete uniform distribution, which means a participant is equally likely to fall in a progression stage (e.g., from $0$ to $5$, where $0$ indicate this participant is healthy.)"
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    "## Graphical Explanation\n",
    "\n",
    "In the following, we explain the generative model in three different scenarios using [graphical models](https://en.wikipedia.org/wiki/Graphical_model): (1) All participants are healthy; (2) Both healthy and diseased participants, but all biomarkers are affected among diseased people; (3) Both healthy and diseased participants, but we do not whether biomarkers are affected or not among patients.\n",
    "\n",
    "### Scenario 1\n",
    "\n",
    "If all participants are healthy:\n",
    "\n",
    "$$\n",
    "X_{nj} \\sim p(X_{nj} \\mid \\phi_{n})\n",
    "$$ {#eq-gen-s1}\n",
    "\n",
    "Where\n",
    "\n",
    "$X_{nj}$ indicates the measurement of biomarker $n$ in participant $j$.\n",
    "\n",
    "$\\phi_{n}$ represents $\\mathcal N(\\phi_{\\mu}, \\phi_{\\sigma})$ for biomarker $n$.\n",
    "\n",
    "The graphical model would look like:\n",
    "\n",
    "![Graphical Model of Scenario 1](img/g1.png){#fig-g1 .lightbox}\n",
    "\n",
    "### Scenario 2\n",
    "\n",
    "If we have oth diseased and healthy participants, and all biomarkers are affected among diseased participants.\n",
    "\n",
    "$$\n",
    "X_{nj} \\sim I(z_j == 1) p(X_{nj} \\mid \\theta_n) + (1-I(z_j == 1))p(X_{nj} \\mid \\phi_n)\n",
    "$$ {#eq-gen-s2}\n",
    "\n",
    "Where:\n",
    "\n",
    "$z_j = 1$ indicates this participant is diseased and $z_j = 1$ represents a healthy participant. \n",
    "\n",
    "$I(True) = 1$ and $I(False) = 0$.\n",
    "\n",
    "$\\theta_{n}$ represents $\\mathcal N(\\theta_{\\mu}, \\theta_{\\sigma})$ for biomarker $n$.\n",
    "\n",
    "The graphical model would look like:\n",
    "\n",
    "![Graphical Model of Scenario 2](img/g2.png){#fig-g2 .lightbox}\n",
    "\n",
    "### Scenario 3\n",
    "\n",
    "If we have both healthy and diseased participants, but we do not whether biomarkers are affected or not among patients, see @eq-gen.\n",
    "\n",
    "This is the model in usual cases. \n",
    "\n",
    "The graphical model looks like:\n",
    "\n",
    "![Graphical Model of Scenario 3](img/g3.png){#fig-g3 .lightbox}"
   ]
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