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krivonogov/Monte Carlo integration, Gibbs sampling.ipynb

Created October 5, 2016 13:50
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Monte Carlo integration, Gibbs sampling.ipynb
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"%matplotlib inline\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import seaborn as sns"
]
},
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"cell_type": "markdown",
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"source": [
"# Monte Carlo Integration"
]
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"### Strong Law of Large Numbers\n",
"Let $X_1$, $X_2$, $\\ldots$ be a sequence of **independent** and **identically distributed** random variables, each having a finite mean $\\mu = \\mathrm E(X_i)$.\n",
"\n",
"Then with probability $1$,\n",
"$$ \\frac{X_1 + X_2 + \\cdots + X_N}{N} \\to \\mu \\text{ as } N \\to \\infty $$"
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"source": [
"### Monte-Carlo Integration"
]
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"To calculate integral\n",
"$$ I = \\int g(x) p(x) \\mathrm dx $$\n",
"for some probability distribution $p$, we can sample $N$ independant values $X_i$ from $p$ and then calculate an approximation\n",
"$$ \\hat I = \\frac{1}{N} \\sum_{i=1}^N g(X_i)$$\n",
"\n",
"By the Strong Law of Large Numbers for random variables $g(X_i)$\n",
"$$ \\hat I \\to I \\text{ as } N \\to \\infty$$"
]
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"source": [
"### Example"
]
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"Area of the unit disk $x^2 + y^2 \\le 1$ is equal to $\\pi$, so we can approximate $\\pi$ if we calculate numerically\n",
"$$ \\pi = \\int\\limits_{x^2 + y^2 \\le 1} \\mathrm dx \\mathrm dy$$"
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"Let us choose\n",
"$$ g(x) = \\left\\{\\begin{aligned} &1, \\text{ if } x^2 + y^2 \\le 1 \\\\ &0, \\text{ otherwise} \\end{aligned}\\right.$$\n",
"and also a random variable uniformly distributed in unit square $-1 \\le x, y \\le 1$."
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"Then\n",
"$$\\int g(x, y) p(x, y) \\mathrm dx \\mathrm dy = \\frac{1}{4} \\int\\limits_{-1 \\le x, y \\le 1} g(x, y) \\mathrm dx \\mathrm dy = \\frac{1}{4} \\int\\limits_{x^2 + y^2 \\le 1} \\mathrm dx \\mathrm dy = \\frac{\\pi}{4},$$\n",
"so we can approximate it by Monte-Carlo integration."
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"source": [
"def g(x, y):\n",
" if x ** 2 + y ** 2 <= 1:\n",
" return 1.\n",
" return 0."
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"def monte_carlo_integration_over_uniform(g_, N):\n",
" sample_x = np.random.uniform(-1, 1, N)\n",
" sample_y = np.random.uniform(-1, 1, N)\n",
" return sum(map(g_, sample_x, sample_y)) / N"
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"pi_numeric = 4 * monte_carlo_integration_over_uniform(g, 1000000)\n",
"pi_numeric"
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"source": [
"### Example"
]
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"source": [
"Let us calculate the variance of $Beta(\\alpha, \\beta)$ distributed random variable $X$"
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"The real solution is to calculate integral\n",
"$$\\mathrm{Var}(X) = \\displaystyle\\frac{\\Gamma (\\alpha + \\beta)}{\\Gamma (\\alpha) \\Gamma (\\beta)}\\int_0^1 (x - \\mu)^2 x^{\\alpha - 1} (1 - x)^{\\beta - 1} \\mathrm{d}x,$$\n",
"where $\\mu$ is a mean and is equal to $\\displaystyle\\frac{\\alpha}{\\alpha + \\beta}$"
]
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"Or simply to take it from https://en.wikipedia.org/wiki/Beta_distribution"
]
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"source": [
"The answer is $\\mathrm{Var}(X) = \\displaystyle\\frac{\\alpha \\beta}{(\\alpha + \\beta)^2 (\\alpha + \\beta + 1)}$"
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"alpha = 3.\n",
"beta = 5.\n",
"mean = alpha / (alpha + beta)\n",
"true_variance = alpha * beta / ((alpha + beta) ** 2 * (alpha + beta + 1))\n",
"true_variance"
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"But we can also calculate it numerically using Monte-Carlo approximation:"
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"def g(x):\n",
" return (x - mean) ** 2"
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"def monte_carlo_integration_over_beta(g_, N):\n",
" sample = np.random.beta(alpha, beta, N)\n",
" # sum(map(g_, sample)) / N\n",
" return np.mean(np.vectorize(g_)(sample))"
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"num_variance = monte_carlo_integration_over_beta(g, 10000)\n",
"num_variance"
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"source": [
"We can trace approximation error depending on sample size $N$"
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"source": [
"n_range = range(1000, 200000, 1000)\n",
"num_variances = np.asarray([monte_carlo_integration_over_beta(g, N) for N in n_range])\n",
"error = np.abs(num_variances - true_variance)\n",
"moving_average_error = np.convolve(error, np.ones(10)/10, 'same')\n",
"sns.set_style(\"darkgrid\")\n",
"plt.plot(n_range, moving_average_error, c='b')"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"Monte-Carlo works fine when we can sample *independant* values from our distribution, but what can we do when it is impossible? For example, in case of posterior distribution without close algebraic form? We need something else."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Markov chains"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<img src=\"https://upload.wikimedia.org/wikipedia/commons/thumb/9/95/Finance_Markov_chain_example_state_space.svg/400px-Finance_Markov_chain_example_state_space.svg.png\">"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"transition_probability = np.array([[0.9, 0.075, 0.025], [0.15, 0.8, 0.05], [0.25, 0.25, 0.5]]).T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us take some initial random distributions of market state:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
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"source": [
"initial_distribution = np.array([1, 0, 0]).reshape(3, 1)\n",
"np.linalg.matrix_power(transition_probability, 100).dot(initial_distribution)"
]
},
{
"cell_type": "code",
"execution_count": 12,
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"source": [
"initial_distribution = np.array([0, 1, 0]).reshape(3, 1)\n",
"np.linalg.matrix_power(transition_probability, 100).dot(initial_distribution)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
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"source": [
"initial_distribution = np.array([0.01, 0.53, 0.46]).reshape(3, 1)\n",
"np.linalg.matrix_power(transition_probability, 100).dot(initial_distribution)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"No matter an initial state distribution we end up in **stationary distribution**, which is a normalized eigenvector of transition matrix:\n",
"$$ \\mathrm P \\pi = \\pi $$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
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"execution_count": 14,
"metadata": {},
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],
"source": [
"eigen_values, eigen_vectors = np.linalg.eig(transition_probability)\n",
"eigen_vector_of_one = eigen_vectors[:, eigen_values > 1.0 - 1e-8] # eigen_values == 1.0\n",
"stationary_distribution = eigen_vector_of_one / eigen_vector_of_one.sum()\n",
"stationary_distribution"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
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]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"np.allclose(transition_probability.dot(stationary_distribution), stationary_distribution)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Monte Carlo integration on the Markov Chain and Ergodic theorem"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"If we can create a Markov chain with stationary distribution equals to a distribution $p$ (from which we can't sample directly independant values), then the draws in our chain (once it converged to stationary state) are like draws from $p$. But they are still *slightly* dependant (every value depends on previous one), so we can't use Monte Carlo integration directly."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Luckily, there is the **Ergodic theorem** (analog of Strong Law of Large Numbers for Markov chains)\n",
"\n",
"Let $X_1$, $X_2$, $\\ldots$, $X_N$ be values from an *\"ergodic\"* Markov chain with stationary distribution $\\pi$, and $\\mathrm E_\\pi (g(X)) < \\infty$.\n",
"\n",
"Then with probability $1$,\n",
"$$ \\frac{1}{N} \\sum_{i=1}^N g(X_i) \\to \\int g(x) \\pi(x) \\mathrm dx \\text{ as } N \\to \\infty$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"So once we have Markov chain with stationary distribution equals to $p$, we can calculate an integral over $p$ using Ergodic theorem"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# MCMC algorithms"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Gibbs Sampling"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Gibbs sampling is an algorithm which helps us to draw a sample from some multivariate distribution using full conditional distributions of every variable.\n",
"\n",
"Let $p(x, y)$ be some distribution with full conditional distributions $p(x \\mid y)$ and $p(y \\mid x)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. Pick some initial value $(X_0, Y_0)$\n",
"2. Draw $X_1$ from $p(x \\mid y = Y_0)$\n",
"3. Draw $Y_1$ from $p(y \\mid x = X_1)$\n",
"4. Repeat steps 2 and 3 to obtain following values $(X_2, Y_2)$, $\\ldots$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since we draw $(X_i, Y_i)$ using only $(X_{i-1}, Y_{i-1})$, the resulting sequence is a Markov chain. Its properties depend on distribution $p$ (or more precisely on full conditionals), but for many families of distributions it is easy to show that chain irreducible, reccurent and aperiodic when for every set $\\mathcal X \\times \\mathcal Y$ with positive volume and for every point $(X, Y)$\n",
"$$\\int_{\\mathcal X} p(x \\mid y=Y) \\mathrm dx > 0 \\text{ and } \\int_{\\mathcal Y} p(y \\mid x=X) \\mathrm dy > 0,$$\n",
"i.e. from every point we can possibly go to any set with positive volume (if can stay in the neighborhood we have aperiodicity, if we can go to any other set we have irreducibility, if we can go to other point and then return to initial point's neighborhood we have reccurency)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's take multivariate normal distribution $\\mathrm N \\left(\\pmatrix{0 \\\\ 0}, \\pmatrix{1 &\\rho \\\\ \\rho & 1} \\right)$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Full conditional distributions are\n",
"$$ p(x \\mid y) \\sim \\mathrm N(\\rho y, 1 - \\rho^2) $$\n",
"$$ p(y \\mid x) \\sim \\mathrm N(\\rho x, 1 - \\rho^2) $$"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"sample_size = 10000\n",
"burnin = 10\n",
"\n",
"rho = 0.7\n",
"sigma = 1 - rho ** 2"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"x = np.zeros(sample_size + burnin, np.float64)\n",
"y = np.zeros(sample_size + burnin, np.float64)\n",
"\n",
"x[0], y[0] = 7, 7\n",
"\n",
"for i in xrange(1, sample_size + burnin):\n",
" x[i] = np.random.normal(rho * y[i - 1], sigma)\n",
" y[i] = np.random.normal(rho * x[i], sigma)"
]
},
{
"cell_type": "code",
"execution_count": 18,
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