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@petermchale
Last active December 7, 2023 19:18
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Using Unsupervised Learning to infer SNPs whose allelic expression is "hyper-variable" across samples: https://nbviewer.jupyter.org/gist/petermchale/c5e21f746fda65d8b379de88f224ac41
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Statement of problem and approach\n",
"\n",
"We have counts of reads containing ALT and REF alleles at a number of sites in the genome for a number of individuals (samples). We wish to discover sites at which those read counts are \"hyper-variable\" across individuals. \n",
"\n",
"How do we do this? The \"descriptive\" approach is to simply measure the observed dispersion of read counts. This is adequate under certain circumstances, but can fail under others, e.g. when sample size is small. \n",
"\n",
"The \"generative\" approach is to build a model of how the data were generated, and then fit it to the data to infer the \"dispersion parameter\", or use the model to build a statistical test that is then applied to the data to discover sites that lie in the tail of some observed distribution. We will adopt the \"generative\" approach. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Generative process\n",
"\n",
"Let $K$ be a random variable representing the number of ALT reads at a site of interest. \n",
"Assume \n",
"\n",
"\\begin{equation} \n",
"K \\sim \\mbox{Binomial}(n, Q) \n",
"\\end{equation} \n",
"\n",
"where $n$ is a deterministic variable representing the total number of read at the site of interest, and $Q$ is a random variable representing the probability that a random read at the site contains the ALT allele. \n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"number_samples = 10000 # large number is best to illustrate the approach\n",
"mean_depth_of_coverage = 50 # I generate synthetic read counts, \n",
" # but in practice we would use the experimentally observed read counts\n",
"\n",
"import numpy as np \n",
"\n",
"def generate_data(q):\n",
" # q = probability that a random read is ALT (one for each sample)\n",
" # n = total number of reads (one for each sample)\n",
" # k = number of ALT reads (one for each sample)\n",
" n = np.random.poisson(lam=mean_depth_of_coverage, size=len(q)) \n",
" k = np.random.binomial(n=n, p=q) \n",
" \n",
" n = n.astype(np.float32)\n",
" k = k.astype(np.float32)\n",
" \n",
" return n, k\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Truth #1: Distribution of $Q$ is a \"spike\""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's assume that $Q$ is deterministic. What is the expected distribution of ALT read counts over the samples? \n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"def histogram_probability(q): \n",
" plt.hist(q, bins=100)\n",
" plt.xlim([0, 1])\n",
" plt.xlabel('probability of random read being ALT at given site')\n",
" plt.ylabel('# samples')\n",
"\n",
"x_max = 2*mean_depth_of_coverage\n",
"\n",
"def histogram_read_count(k): \n",
" left_edge_of_first_bin = k.min() - 0.5\n",
" right_edge_of_last_bin = k.max() + 0.5\n",
" fig_, ax_ = plt.subplots(1, 1)\n",
" plt.hist(k, np.arange(left_edge_of_first_bin, right_edge_of_last_bin + 1, 1), label='empirical distribution')\n",
" plt.xlim([0, x_max])\n",
" plt.xlabel('number of ALT reads at given site')\n",
" plt.ylabel('number of samples')\n",
" return fig_, ax_\n",
"\n",
"import matplotlib.pyplot as plt \n",
"%matplotlib inline \n",
"\n",
"def truth_1(): \n",
" q = 0.25*np.ones(number_samples)\n",
" n, k = generate_data(q)\n",
" return n, k, q \n",
"\n",
"def plot_truth_1():\n",
" n, k, q = truth_1()\n",
" histogram_probability(q) \n",
" fig, ax = histogram_read_count(k)\n",
"\n",
" # A binomial distribution, where the n-parameter is Poisson distributed, and the q-parameter is fixed, is in fact a Poisson distribution: \n",
" # https://math.stackexchange.com/questions/308067/binomial-random-variable-with-number-of-trials-being-a-poisson-random-variable\n",
" # https://math.stackexchange.com/questions/1410540/what-is-the-distribution-of-a-binomial-variable-where-the-number-of-trials-is-it?rq=1\n",
" lmbda = q[0]*mean_depth_of_coverage \n",
"\n",
" from scipy.stats import poisson\n",
" x = np.arange(x_max) \n",
" ax.vlines(x, 0, poisson.pmf(x, lmbda)*number_samples, colors='r', lw=5, alpha=0.5, label='exact distribution')\n",
" _ = ax.legend()\n",
" \n",
"plot_truth_1()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Truth #2: Distribution of $Q$ is flat"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now suppose that $Q$ is a random variable whose value is equally likely to occur anywhere within the interval $[0, 1]$. You'd expect that the addition of this randomness to the process would make the distribution of read counts more dispersed; you'd be right: \n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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DuAw4OiJ+W/q8iJgfEVMiYsqYjTdtXIBmZvYWLUkaktYnJYwfR8TlefPzkibkxycAq1oRm5mZ9a0VZ08JOAt4JCK+U/PQAmB2Xp4NXNXs2MzMrH9jW1DmnsDhwAOSluVtXwJOBi6WdATwNHBIC2IzM7N+ND1pRMTtgPp4eFozYzEzs8HxFeFmZlbMScPMzIo5aZiZWTEnDTMzK+akYWZmxZw0zMysmJOGmZkVc9IwM7NiThpmZlbMScPMzIo5aZiZWTEnDTMzK+akYWZmxZw0zMysmJOGmZkVc9IwM7NiThpmZlbMScPMzIo5aZiZWTEnDTMzK+akYWZmxZw0zMysmJOGmZkVq1zSkLSvpMckLZd0QqvjMTOzHpVKGpLGAKcDHwcmA7MkTW5tVGZm1q1SSQPYDVgeEU9ExGvAT4HpLY7JzMyyqiWNbYBnatZX5G1mZlYBY1sdQC+qsy3etIM0D5iXV1/99TcOeLDhUbWH8cALrQ6iIlwXPVwXPVwXPXYa6hOrljRWANvVrG8LPFe7Q0TMB+YDSFoSEVOaF151uS56uC56uC56uC56SFoy1OdWrXtqMTBJ0vaSNgBmAgtaHJOZmWWVamlExOuS/hG4ARgDnB0RD7U4LDMzyyqVNAAi4jrgusLd5zcyljbjuujhuujhuujhuugx5LpQRAy8l5mZGdUb0zAzswpri6Qx0NQikjaUdFF+/G5JXc2PsjkK6uJYSQ9Lul/SzZLe1Yo4m6F0yhlJMySFpI49c6akLiQdmr8bD0n6SbNjbJaC38hESbdIujf/TvZrRZyNJulsSask1b0sQcmpuZ7ul7Rr0QtHRKX/kQbEfwXsAGwA3AdM7rXPZ4Az8/JM4KJWx93CuvgIsHFe/vRorou83zuB24BFwJRWx93C78Uk4F5gs7y+VavjbmFdzAc+nZcnA0+1Ou4G1cWHgF2BB/t4fD/gZ6Tr43YH7i553XZoaZRMLTIdOC8vXwpMk1TvQsF2N2BdRMQtEbEmry4iXevSiUqnnPkX4JvAK80MrslK6uLvgdMj4kWAiFjV5BibpaQuAtgkL29Kr2vBOkVE3Ab8v352mQ6cH8kiYJykCQO9bjskjZKpRd7YJyJeB1YDWzQluuYa7DQrR5COJDrRgHUh6f3AdhFxTTMDa4GS78V7gPdIukPSIkn7Ni265iqpi5OAT0laQTpT83PNCa1yhjRtU+VOua1jwKlFCvfpBMXvU9KngCnAhxsaUev0WxeS1gNOAeY0K6AWKvlejCV1UU0ltT5/IWnniHipwbE1W0ldzALOjYhvS9oD+FGui3WND69ShvR3sx1aGgNOLVK7j6SxpCZnf82ydlVSF0jaB/gycGBEvNqk2JptoLp4J7AzsFDSU6Q+2wUdOhhe+hu5KiL+GBFPAo+RkkinKamLI4CLASLiLmAj0rxUo03R35Pe2iFplEwtsgCYnZdnAD+PPNLTYQasi9wl8++khNGp/dYwQF1ExOqIGB8RXRHRRRrfOTAihjznToWV/EauJJ0kgaTxpO6qJ5oaZXOU1MXTwDQASX9KShq/aWqU1bAA+Nt8FtXuwOqIWDnQkyrfPRV9TC0i6WvAkohYAJxFamIuJ7UwZrYu4sYprItvAe8ALsnnAjwdEQe2LOgGKayLUaGwLm4APibpYWAt8IWI+O/WRd0YhXXxz8APJB1D6o6Z04kHmZIuJHVHjs/jNycC6wNExJmk8Zz9gOXAGmBu0et2YF2ZmVmDtEP3lJmZVYSThpmZFXPSMDOzYk4aZmZWzEnDzMyKOWm0CUkLB3NhmqQ5kk7r47E78/9d3TNgSpoi6dS8PFXSB0co7i3zzMP3SvqLkXjNfso6V9KMRpYxXLV13mv7VEmDmu5E0g8lTR7B2A7OswG/t2bbW+KVdKakZXnG3D/k5WWSDi4sZ+98XcBgYhsj6ReDec5IyHXyhbz8ydq6Ga0qf53GaCJpTESsbXQ5EfGWhJAveuu+8G0q8DJw5wgUNw14NCJm97dTs977cORJMFWV6SYi4sgRfslZwO2k65xO6qfcowAkvRu4NCJ2GWQ5ewMvkC64LJK/Gw096Oij3CtqVj8JrAMebXYcVeKWRhPko7VHJZ2X562/VNLG+bGnJH1V0u3AIZJ2yRPK3S/pCkmb1bzUpyTdKelBSbvl5++Wt92b/9+pZv/tJF2vdG+BE2vieblOjFMlXaN0L5KjgGPy0eNfSHpS0vp5v01yzOv3ev67lO7f0X0fj4mSdiHNMLtffq239XpO7/f+95IWS7pP0mU1dXSu0rz/d0p6ors1ka9kPS0f8V4LbFXz2tNynTygdF+BDWvK/FdJd0laImlXSTdI+pWko/r47B6R9H1gaa7Tj+XnL5V0iaR35H2/muN/UNL8nGSQ9IH8nu4CPtvX9wTYJH/mD+ej+fXy8/sq743Wp6SXJX09l7NI0tZ5+455fbGkr9X77PN+7wD2JE2xMSIXx0qarp5W5o2StpK0I3Ak8IX8nfhgr+dslb8/SyV9X9KzksZJGivppbzPZZI+VvOcC3JZYyV9R9Iv8/fwyPz4Pvk1L8+/hfP7iPeYXPf3SbogbztS0neVWsn7AafkuLskTcrfnXsk3SbpPSNRb5XX6jnfR8M/oIt05emeef1s4H/m5aeA42r2vR/4cF7+GvDdvLwQ+EH0zJP/YF7eBBibl/cBLsvLc4CVpNl+3wY8SL6fBPByTVzdrzMVuCYvn9QdX14/BzgoL88Dvl3nPV4NzM7LfwdcWRPHaX3US+/3vkXN8v8GPpeXzwUuIR3kTCZNfQ3pyO8m0pW/fwK8RJpGZiPS7J3vyfudDxxdU2b3vRROyfX9TmBLYFUfn906YPe8Pp50f4635/Xjga/m5c1rnvcj4BN1PtNvUef+Brn+XyHdB2JMfl8zBihvYc1nGjXlfRP4Sl6+BpiVl4/q/uzrlP8p4Ky8fCewa+/vSJ3nvBtY1s/3fjN6LiA+CvhGzWd7dB/POZN0tTrAAfl9jSP1iryUtx9SE2v3Z70h6b46J+TtG5LuHzKR9Lt4EZiQ63Zx9+fZq+yVwAZ5eVz+/0h6foMXkH8Hef0WYMe8vCdwY7P/trTin1sazfNMRNyRly8A9qp57CIASZuSvqy35u3nkRJEtwvhjXnyN5E0jjQ54yVK/c6nAO+r2f+miPjviPgDcHmvMgfjh/RMMTCXlER62wPovhvcjwZR1kU1yztL+oWkB4DDePN7uTIi1kXEw8DWeduHgAsjYm1EPAf8PG/fCXgyIv4zr/eux+4pRh4g3XjmdxHxG+CVXKe9/TrS/QYgTXw4GbhD0jLSnGfdd0f8SD6yfoDUBfO+Op/pj/qpi19Gug/EWtJnvdcA5dV6jZQgAO4h/bGH9Llckpf7u1vfLNK9J8j/z+pn31ITgRtzfRzLmz/PvuzVHUekKe1/V2efa4GPKrV29yfNNfcq8DFgbq6nu0nJpntSxkURsTLX7TJ66qfWQ8AFkg4D/thfkPl7sjtwWS7vdNKBS8fzmEbz9J6vpXb998N4jX8BbomIg5W6lhYWllksIu7IzfEPA2Miou7tI4dYVu17P5d0JHefpDmko+9utbP11k7pXK+cgW7A1f1a63q97jrq/yZqYxQpGb/pj6qkjYDvk478n5F0EukoWH3EWE+9z6tueXX8MfIhL2luqeLftqQtSEluZ0lBOhoPSceVvkYfTgf+NSKuU5p5uc9b8taGM9AOEbFG0h3AR4G/pucgRsBnIuLmN71gKrv2c+6rfv6SdCuB6cBXJO08QJwvxODHc9qeWxrNM1Fp7n7oGXB8k4hYDbyonrOMDgdurdnlrwEk7UWakXI1qaXxbH58Tq+X/KikzZXGEg4C7qDM70hdNrXOJx391mtlQOrS6O4LP4w676/AO4GV+QjysIL9bwNmKp1ZM4E8iytpoLJLaaAW3lqPw7EI2LP7tSVtnPuyN8qPv5DHB2YARLpfxer8mUH/72s3pdlZ1yN91rf3U95g4v2rvNzXWMUM0h3c3hVpVuDtgCcZesu026bAs3lsp/ZEiHrfr263A4cCKN27u6/9fkoaf9kD+I+87QbgM0q3R0DSTuo1jtYXSWOAbSPi58AXSN2VG/fa7Y24I90BcaXyGWOS1pP05yVltTsnjeZ5BJgt6X5gc+CMPvabDXwr77cLaVyj24tKp8ueSfrBQOq//j/5yGtMr9e6ndQdsow01lE6LfjVwMF5wK87gf2Y1Ed9YR/P+SdS18D9pD/Sny8sq9b/InUr3ETZGSpXAI+TupnOICeGiHiF1I12Se4aWUeqs2HL3VhzgAvze10EvDcnhx/kWK4k9Zt3mwucrjQQ/od+Xv4u4GTS+NOTwBV9lTeIkI8GjpX0S1Kf/uo6+8wi1WWty4C/ycs7SVpR8++QwrJPyq97K/B8zfargEPzAHnvM/lOBPaXtJTU+nme+i3x60ln5l0fEd1dSf9O+j4sy921Z1De4hoL/CTX8VLS+EvvrrELgS91D4STkvBRku4jdW0dUFhWW/Mst02Qv2DXRER/zd1KUzpjaXpEHN7qWKyc0hlof4iIkDSTNChe717qlZC7+V6PNMX5XqRB6E68cVbb8piGDUjS/wU+Tjrl0NrLB4DTchfRS6Qz26qsi9SqGkMah/iH1oZjvbmlYWZmxTymYWZmxZw0zMysmJOGmZkVc9IwM7NiThpmZlbMScPMzIr9f74JGGQn4p3zAAAAAElFTkSuQmCC\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"def truth_2(): \n",
" q = np.random.uniform(low=0.0, high=1.0, size=number_samples)\n",
" n, k = generate_data(q)\n",
" return n, k, q \n",
"\n",
"def plot_truth_2():\n",
" n, k, q = truth_2()\n",
" histogram_probability(q)\n",
" fig, ax = histogram_read_count(k)\n",
"\n",
"plot_truth_2()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Truth #3: $Q$ is beta-distributed"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now suppose that \n",
"\n",
"\\begin{equation} \n",
"Q \\sim \\mbox{Beta}(a, b) \n",
"\\end{equation} \n",
"\n",
"For suitable values of the parameters $a$ and $b$, the distribution of $Q$, and that of the ALT read counts, is \"half-way\" between the two cases considered above: \n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"from scipy import stats\n",
"\n",
"def truth_3(a, b): \n",
" q = stats.beta.rvs(a, b, size=number_samples)\n",
" n, k = generate_data(q)\n",
" return n, k, q\n",
"\n",
"def plot_truth_3():\n",
" n, k, q = truth_3(a=10, b=5)\n",
" histogram_probability(q) \n",
" fig, ax = histogram_read_count(k)\n",
"\n",
"plot_truth_3()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Inferring the distribution of $Q$ by maximum-likelihood estimation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So far, we have assumed $Q$ and then computed $K$. In practice, we are given $K$, and we must infer the distribution of $Q$, in spite of the fact that it is hidden from us. We can do this by fitting a model to the data. A simple approach is to assume that $Q$ is beta distributed and that $K$ is binomially distributed, because then (see [here](https://en.wikipedia.org/wiki/Beta-binomial_distribution#As_a_compound_distribution)):,\n",
"\n",
"\\begin{equation} \n",
"p(k | n,a,b) = {n\\choose k}\\frac{\\mathrm{B}(k+a,n-k+b)} {\\mathrm{B}(a,b)},\n",
"\\end{equation}\n",
"\n",
"where $B$ is the beta function. This allows us to analytically compute the likelihood of the data under this \\\"beta-binomial\\\" model. Maximizing this likelihood, by varying the parameters of the model, $a$ and $b$, generates a model that fits the data best. The best-fit values of $a$ and $b$ define the inferred distribution of $Q$. "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"import tensorflow as tf \n",
"sess = tf.InteractiveSession() # posterior sampling (see later in notebook) is buggy when using eager execution "
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"WARNING:tensorflow:From /anaconda2/envs/tf_tfp/lib/python3.5/site-packages/tensorflow/python/framework/op_def_library.py:263: colocate_with (from tensorflow.python.framework.ops) is deprecated and will be removed in a future version.\n",
"Instructions for updating:\n",
"Colocations handled automatically by placer.\n"
]
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 720x360 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 720x360 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 720x360 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"from scipy.special import binom\n",
"\n",
"# the contraint on a and b is: a>0, b>0\n",
"def compute_ab(a_unconstrained, b_unconstrained):\n",
" return tf.exp(a_unconstrained), tf.exp(b_unconstrained) \n",
" \n",
"def log_beta(x, y):\n",
" return tf.lgamma(x) + tf.lgamma(y) - tf.lgamma(x + y) \n",
"\n",
"def compute_loss(n, k, a_unconstrained, b_unconstrained):\n",
" term1 = tf.log(binom(n, k))\n",
" a, b = compute_ab(a_unconstrained, b_unconstrained) \n",
" term2 = log_beta(a + k, b + n - k)\n",
" term3 = log_beta(a, b)\n",
" return -tf.reduce_mean(term1 + term2 - term3)\n",
"\n",
"def train(n, k): \n",
" a_unconstrained = tf.Variable(0, dtype=tf.float32)\n",
" b_unconstrained = tf.Variable(0, dtype=tf.float32)\n",
"\n",
" optimizer = tf.train.GradientDescentOptimizer(learning_rate=0.25)\n",
"\n",
" a_list, b_list, loss_list = [], [], []\n",
" epochs = np.arange(100)\n",
"\n",
" loss = compute_loss(n, k, a_unconstrained, b_unconstrained)\n",
" training_step = optimizer.minimize(loss)\n",
" sess.run(tf.global_variables_initializer())\n",
" \n",
" for epoch in epochs: \n",
" a, b = compute_ab(a_unconstrained, b_unconstrained)\n",
" _, a_, b_, loss_ = sess.run([training_step, a, b, loss])\n",
" a_list.append(a_)\n",
" b_list.append(b_)\n",
" loss_list.append(loss_)\n",
" \n",
" return epochs, a_list, b_list, loss_list\n",
" \n",
"def plot_training(a_true, b_true, epochs, a_learn, b_learn, loss):\n",
" fig = plt.figure()\n",
" fig.set_size_inches(10, 5)\n",
" plt.plot([a_true] * len(epochs), 'r--',\n",
" [b_true] * len(epochs), 'b--')\n",
" plt.plot(epochs, a_learn, 'r',\n",
" epochs, b_learn, 'b')\n",
" plt.legend(['a_true', 'b_true', 'a_learn', 'b_learn'])\n",
" plt.xlabel('epochs')\n",
" \n",
" fig = plt.figure()\n",
" fig.set_size_inches(10, 5)\n",
" plt.plot(epochs, loss)\n",
" plt.xlabel('epochs')\n",
" plt.ylabel('loss')\n",
" \n",
" fig = plt.figure()\n",
" fig.set_size_inches(10, 5)\n",
" x = np.linspace(0, 1)\n",
" plt.plot(x, stats.beta.pdf(x, a_true, b_true), label='true pdf of Q')\n",
" plt.plot(x, stats.beta.pdf(x, a_learn[-1], b_learn[-1]), label='learned pdf of Q')\n",
" plt.xlabel('probability of random read being ALT at given site')\n",
" plt.legend()\n",
"\n",
"def learn_truth_3():\n",
" a, b = 10, 5\n",
" n, k, q = truth_3(a, b)\n",
" plot_training(a, b, *train(n, k))\n",
"\n",
"learn_truth_3()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Inferring the distribution of $a$ and $b$: Bayesian inference"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The Bayesian approach to inferring model parameters is to (i) assume a prior distribution of the parameters and (ii) update that distribution in light of evidence. In our case: \n",
"\n",
"\\begin{equation} \n",
"p(a, b | n, k) \\propto p(n, k | a, b) \\, p(a, b). \n",
"\\end{equation}\n",
"\n",
"where the constant of proportionality does not depend upon $a$ and $b$. That first factor on the right-hand side is called the likelihood of the data under the model and can be written \n",
"\n",
"\\begin{equation} \n",
"p(n, k | a, b) \\propto \\prod_i p(k_i | n_i, a, b),\n",
"\\end{equation}\n",
"\n",
"where, again, the constant of proportionality is independent of $a$ and $b$. Thus the posterior distribution of the model's parameters is given by: \n",
"\n",
"\\begin{equation} \n",
"p(a, b | n, k) \\propto \\prod_i p(k_i | n_i, a, b) \\, p(a, b). \n",
"\\end{equation}\n",
"\n",
"We have an analytic formula for $p(k_i | n_i, a, b)$ (see earlier in the notebook):\n",
"\n",
"\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"def unnormalized_likelihood_log_prob(n, k, parameters): \n",
" a, b = parameters[0], parameters[1]\n",
" term1 = tf.log(binom(n, k))\n",
" term2 = log_beta(a + k, b + n - k)\n",
" term3 = log_beta(a, b)\n",
"# return 0. # debugging: making the likelihood return 0 makes the posterior coincide with the prior\n",
" return tf.reduce_sum(term1 + term2 - term3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We are free to choose any functional form for the second factor, $p(a, b)$, which is called the \"prior\". \n",
"A sensible assumption for this problem, prior to accumulating evidence, is that the probability of choosing an ALT read does not vary much among samples:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"WARNING: The TensorFlow contrib module will not be included in TensorFlow 2.0.\n",
"For more information, please see:\n",
" * https://github.com/tensorflow/community/blob/master/rfcs/20180907-contrib-sunset.md\n",
" * https://github.com/tensorflow/addons\n",
"If you depend on functionality not listed there, please file an issue.\n",
"\n"
]
},
{
"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"import tensorflow_probability as tfp \n",
"tfd = tfp.distributions\n",
"\n",
"def prior_uniform(low, high): \n",
" return tfd.Independent(tfd.Uniform(low=low, high=high), reinterpreted_batch_ndims=1)\n",
"\n",
"def prior_biological(): \n",
" return prior_uniform(low=[19, 19], high=[21, 21])\n",
"\n",
"def plot_prior(prior):\n",
" parameters = prior().sample(100)\n",
" parameters_ = sess.run(parameters)\n",
" plt.scatter(parameters_[:,0], parameters_[:,1], color='blue')\n",
" plt.xlabel('a')\n",
" plt.ylabel('b')\n",
" plt.gca().set_aspect('equal', adjustable='box')\n",
"\n",
" fig = plt.figure() \n",
" for parameter in parameters_: \n",
" a, b = parameter\n",
" x = np.linspace(0, 1, 1000)\n",
" plt.plot(x, stats.beta.pdf(x, a, b), color='blue')\n",
" plt.xlabel('probability of random read being ALT at given site')\n",
" plt.ylabel('probability density function')\n",
"\n",
"plot_prior(prior_biological)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One way to relax this assumption is to expand the \"support\" of the distribution of $a$ and $b$: "
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"def prior_unbiological(): \n",
" return prior_uniform(low=[1, 1], high=[50, 50])\n",
"\n",
"plot_prior(prior_unbiological)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Sampling from the posterior distribution of model parameters "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"When a probability distribution is known only up to a constant factor, as is the case for the posterior $p(a, b | n, k)$, one can sample from that probability distribution using Markov Chain Monte Carlo:\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"posterior mean of (a, b): (10.0082426071167, 5.026904582977295)\n",
"true value of (a, b): (10, 5)\n"
]
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 720x720 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 720x360 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"def prior_log_prob(prior, parameters): \n",
" return prior().log_prob(parameters)\n",
"\n",
"def unnormalized_posterior_log_prob(n, k, prior, parameters):\n",
" return unnormalized_likelihood_log_prob(n, k, parameters) + prior_log_prob(prior, parameters)\n",
"\n",
"def sample_from_posterior(a=10, b=5, prior=prior_unbiological, number_steps=1000, number_burnin_steps=100, step_size=0.1): \n",
" n, k, q = truth_3(a, b) \n",
" \n",
" import functools\n",
" unnormalized_posterior_log_prob_partial = functools.partial(unnormalized_posterior_log_prob, n, k, prior)\n",
" \n",
" inner_transition_kernel = tfp.mcmc.RandomWalkMetropolis(\n",
" new_state_fn=tfp.mcmc.random_walk_normal_fn(scale=step_size),\n",
" target_log_prob_fn=unnormalized_posterior_log_prob_partial)\n",
"\n",
" # transform the random variables, a and b, to a space in which they are NOT constrained to be positive:\n",
" tfb = tfp.bijectors \n",
" unconstraining_bijector = tfb.Exp() \n",
" \n",
" transition_kernel = tfp.mcmc.TransformedTransitionKernel(\n",
" inner_kernel=inner_transition_kernel,\n",
" bijector=unconstraining_bijector)\n",
"\n",
" parameters_initial = prior().mean() \n",
" \n",
" parameters, kernel_results = tfp.mcmc.sample_chain(\n",
" num_results=number_steps,\n",
" current_state=parameters_initial,\n",
" kernel=transition_kernel,\n",
" num_burnin_steps=number_burnin_steps)\n",
" \n",
" parameters_, kernel_results_ = sess.run([parameters, kernel_results])\n",
" \n",
" return parameters_, kernel_results_, a, b\n",
"\n",
"def plot_posterior(parameters_inferred, kernel_results, a_true, b_true):\n",
" parameters_inferred = np.array(parameters_inferred)\n",
" a_inferred, b_inferred = parameters_inferred[:,0], parameters_inferred[:,1]\n",
" \n",
" fig = plt.figure() \n",
" fig.set_size_inches(10, 10)\n",
"\n",
" import seaborn as sns\n",
" sns.kdeplot(a_inferred, b_inferred, shade=True)\n",
" \n",
" def normalize(value, values): \n",
" return value/float(len(values))\n",
" \n",
" steps = np.arange(len(parameters_inferred))\n",
" for step in steps:\n",
" plt.scatter(a_inferred[step], b_inferred[step], color=plt.cm.jet(normalize(step, steps))) \n",
" markersize = 250\n",
" plt.scatter(a_inferred[0], b_inferred[0], marker='s', s=markersize, facecolors='none', edgecolors='black', label='first sample from posterior post-burn-in')\n",
" plt.scatter(a_inferred[-1], b_inferred[-1], marker='o', s=markersize, facecolors='none', edgecolors='black', label='last sample from posterior post-burn-in')\n",
" \n",
" plt.scatter(a_true, b_true, marker='x', color='black', s=markersize, label='truth')\n",
" \n",
" plt.xlabel('a')\n",
" plt.ylabel('b')\n",
" plt.legend() \n",
" plt.gca().set_aspect('equal', adjustable='box')\n",
"\n",
" fig = plt.figure() \n",
" fig.set_size_inches(10, 5)\n",
" x = np.linspace(0, 1)\n",
" for step in steps:\n",
" plt.plot(x, stats.beta.pdf(x, a_inferred[step], b_inferred[step]), color=plt.cm.jet(normalize(step, steps)))\n",
" \n",
" plt.plot(x, stats.beta.pdf(x, a_true, b_true), 's', color='black', label='true pdf of Q')\n",
"\n",
" plt.xlabel('probability of random read being ALT at given site')\n",
" plt.ylabel('probability density function') \n",
" plt.legend() \n",
" \n",
" print('posterior mean of (a, b): ({}, {})'.format(np.mean(a_inferred), np.mean(b_inferred)))\n",
" print('true value of (a, b): ({}, {})'.format(a_true, b_true))\n",
"\n",
"plot_posterior(*sample_from_posterior(prior=prior_unbiological, number_steps=1000, number_burnin_steps=500))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice that the posterior mean is close to the truth; [this is expected when the prior is uniform](https://wiseodd.github.io/techblog/2017/01/01/mle-vs-map/). \n",
"\n",
"Notice also that there is not much variation in the distribution of $Q$ as one samples from its posterior distribution; this implies that the maximum-likelihood-inferred distribution of $Q$ has small \"error bars\". "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Interestingly, a more biologically plausible prior doesn't even admit the maximum-likelihood estimate: "
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"posterior mean of (a, b): (20.844560623168945, 19.06966209411621)\n",
"true value of (a, b): (10, 5)\n"
]
},
{
"data": {
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\n",
"text/plain": [
"<Figure size 720x720 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 720x360 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plot_posterior(*sample_from_posterior(prior=prior_biological, number_steps=1000, number_burnin_steps=500))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Next steps \n",
"\n",
"* The posterior distribution of $a$ and $b$ is concentrated around the truth (for the \"non-biological\" prior). Is this because the model coincides with the truth, in the sense that both are beta-binomial? One way to test this idea is to generate a truth that is not beta binomial and infer a posterior distribution using a beta-binomial model. \n",
"* Write code to compute $\\sigma$, the standard deviation of $p(q)$, using the formula at https://en.wikipedia.org/wiki/Beta_distribution\n",
"* Is the distribution of $\\sigma$ wider when (i) the number of counts is smaller and/or (ii) the number of samples is smaller?\n",
"* Write code to compute the distribution of $D = K - (n - K)$\n",
"* Generalize the code to include a tissue dimension, in addition to a sample dimension\n",
"* Generalize model to multiple SNPs in a gene. Concretely, generalize $n$, $K$ and $Q$ to vectors where $p(q)$ now includes inter-SNP correlations, modelled, e.g., by copulas (see Fig 2 of [this paper](https://arxiv.org/pdf/1609.00066.pdf) and [this notebook](https://github.com/tensorflow/probability/blob/master/tensorflow_probability/examples/jupyter_notebooks/Gaussian_Copula.ipynb))\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Technical note \n",
"\n",
"In the above, we were able to compute \n",
"\n",
"\\begin{equation} \n",
"p(k|n) = \\int_0^1 dq \\, p(k | n, q) \\,p(q)\n",
"\\end{equation}\n",
"\n",
"analytically for the special case where $p(q)$ is a beta distribution. Had we chosen a different functional form for $p(q)$, e.g. a mixture of betas to model the hypothesis that $p(q)$ is multi-modal, we might not have been able to compute the integral analytically, forcing us to compute it numerically. One way to do that is Monte Carlo integration: \n",
"\n",
"* generate $m$ samples from $p(q)$ \n",
"* for each $q_i$, compute $p(k | n, q_i)$ \n",
"* then approximate the integral as \n",
"\\begin{equation} \n",
"p(k|n) \\approx \\frac{1}{m} \\sum_{i=1}^m p(k | n, q_i) \n",
"\\end{equation} \n",
"\n",
"See p5 of [this paper](https://arxiv.org/abs/1711.10604) and [tfp.monte_carlo](https://www.tensorflow.org/probability/api_docs/python/tfp/monte_carlo). "
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@petermchale
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Excellent resource for Bayesian Optimization:
"Taking the Human Out of the Loop: A Review of Bayesian Optimization"
https://ieeexplore.ieee.org/document/7352306

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