diff --git a/BDA 5.9.3.ipynb b/BDA 5.9.3.ipynb index abfbe93..b3050ee 100644 --- a/BDA 5.9.3.ipynb +++ b/BDA 5.9.3.ipynb @@ -128,16 +128,16 @@ }, { "cell_type": "code", - "execution_count": 350, + "execution_count": 355, "metadata": {}, "outputs": [], "source": [ - "answers=sm.sampling(data=dict({'means':p55['effect'],'se':p55['se']}),iter=00)" + "answers=sm.sampling(data=dict({'means':p55['effect'],'se':p55['se']}),iter=5000)" ] }, { "cell_type": "code", - "execution_count": 351, + "execution_count": 356, "metadata": {}, "outputs": [ { @@ -145,31 +145,31 @@ "output_type": "stream", "text": [ "Inference for Stan model: anon_model_1c0a010b4129370aa04f0b4b9f729b4d.\n", - "4 chains, each with iter=500; warmup=250; thin=1; \n", - "post-warmup draws per chain=250, total post-warmup draws=1000.\n", + "4 chains, each with iter=5000; warmup=2500; thin=1; \n", + "post-warmup draws per chain=2500, total post-warmup draws=10000.\n", "\n", " mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat\n", - "theta[0] 12.07 0.66 9.38 -3.89 5.93 11.21 17.23 34.12 200 1.01\n", - "theta[1] 7.54 0.39 6.89 -6.07 3.02 7.48 12.15 21.17 312 1.01\n", - "theta[2] 5.45 0.46 8.58 -12.34 0.2 6.07 10.92 20.96 342 1.0\n", - "theta[3] 7.16 0.37 7.21 -7.68 2.72 7.31 11.83 20.44 383 1.01\n", - "theta[4] 4.51 0.43 6.58 -10.0 0.23 5.07 9.18 15.99 235 1.02\n", - "theta[5] 5.53 0.41 7.27 -10.59 1.01 6.16 10.74 18.24 313 1.01\n", - "theta[6] 11.03 0.53 7.5 -2.83 5.64 11.03 15.77 27.15 199 1.01\n", - "theta[7] 8.3 0.39 7.93 -6.88 3.08 8.28 13.26 24.5 414 1.0\n", - "mu 7.66 0.41 5.62 -3.36 4.06 7.92 11.47 18.73 192 1.02\n", - "tau 7.74 0.71 6.11 0.4 3.91 6.29 9.9 22.49 75 1.05\n", - "results[0] 11.81 0.57 12.82 -9.18 3.71 10.54 17.23 43.38 513 1.01\n", - "results[1] 7.26 0.52 12.33 -16.66 0.94 7.04 13.57 33.52 563 1.01\n", - "results[2] 5.43 0.42 12.64 -22.88 -1.07 5.99 12.55 29.09 902 1.0\n", - "results[3] 7.01 0.41 11.92 -16.05 1.03 7.18 13.39 29.57 861 1.0\n", - "results[4] 4.62 0.46 11.55 -23.93 -1.48 5.45 11.65 26.17 638 1.0\n", - "results[5] 5.83 0.44 11.71 -20.83 0.05 6.61 11.82 28.91 724 1.0\n", - "results[6] 10.76 0.56 11.87 -11.69 3.67 10.23 17.43 36.29 443 1.0\n", - "results[7] 8.01 0.54 12.75 -16.27 1.62 7.82 14.62 31.63 561 1.0\n", - "lp__ -18.65 1.21 5.8 -27.82 -22.04 -19.13 -16.31 0.06 23 1.18\n", + "theta[0] 11.66 0.17 8.25 -2.16 6.27 10.51 16.07 30.71 2223 1.0\n", + "theta[1] 8.02 0.1 6.27 -4.44 4.18 7.82 11.76 21.02 3674 1.0\n", + "theta[2] 6.36 0.12 7.68 -10.59 2.17 6.78 11.15 20.84 3806 1.0\n", + "theta[3] 7.79 0.11 6.43 -5.29 4.02 7.62 11.79 20.79 3689 1.0\n", + "theta[4] 5.19 0.13 6.38 -8.66 1.46 5.72 9.44 16.84 2357 1.0\n", + "theta[5] 6.3 0.12 6.62 -7.68 2.38 6.54 10.57 18.98 3277 1.0\n", + "theta[6] 10.99 0.15 6.78 -0.97 6.41 10.4 14.99 25.86 2181 1.0\n", + "theta[7] 8.61 0.13 7.97 -7.29 4.13 8.21 12.91 25.66 3946 1.0\n", + "mu 8.2 0.11 5.3 -1.85 5.01 7.96 11.32 18.88 2354 1.0\n", + "tau 6.84 0.2 5.58 0.93 2.99 5.46 9.11 20.69 751 1.01\n", + "results[0] 11.6 0.18 12.19 -8.79 4.97 9.96 16.83 40.26 4692 1.0\n", + "results[1] 7.97 0.13 10.69 -14.26 2.73 7.74 13.36 29.84 6547 1.0\n", + "results[2] 6.4 0.15 11.84 -18.96 1.18 6.95 12.45 28.63 5970 1.0\n", + "results[3] 7.79 0.14 11.1 -14.73 2.66 7.66 13.29 30.0 6438 1.0\n", + "results[4] 5.24 0.16 11.09 -19.44 0.35 6.13 11.03 24.67 4830 1.0\n", + "results[5] 6.36 0.15 10.98 -17.1 1.36 6.66 11.93 28.06 5695 1.0\n", + "results[6] 10.93 0.16 11.19 -8.59 4.94 9.77 16.05 35.59 4800 1.0\n", + "results[7] 8.64 0.14 11.63 -13.75 2.97 8.25 14.03 33.19 6700 1.0\n", + "lp__ -17.68 0.41 5.52 -27.62 -21.48 -18.16 -14.18 -5.46 179 1.04\n", "\n", - "Samples were drawn using NUTS at Sat Apr 21 16:45:21 2018.\n", + "Samples were drawn using NUTS at Sat Apr 21 16:45:42 2018.\n", "For each parameter, n_eff is a crude measure of effective sample size,\n", "and Rhat is the potential scale reduction factor on split chains (at \n", "convergence, Rhat=1).\n" @@ -182,12 +182,12 @@ }, { "cell_type": "code", - "execution_count": 352, + "execution_count": 357, "metadata": {}, "outputs": [ { "data": { - "image/png": "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\n", + "image/png": "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\n", "text/plain": [ "
" ] @@ -204,7 +204,7 @@ }, { "cell_type": "code", - "execution_count": 353, + "execution_count": 358, "metadata": {}, "outputs": [], "source": [ @@ -217,7 +217,7 @@ }, { "cell_type": "code", - "execution_count": 354, + "execution_count": 359, "metadata": {}, "outputs": [ { @@ -227,14 +227,14 @@ "Chance is empirical probability that given school is best\n", " effect se Chance\n", "school \n", - "A 28 15 0.191\n", - "B 8 10 0.109\n", - "C -3 16 0.105\n", - "D 7 11 0.124\n", - "E -1 9 0.077\n", - "F 1 11 0.088\n", - "G 18 10 0.186\n", - "H 12 18 0.120\n", + "A 28 15 0.2047\n", + "B 8 10 0.1160\n", + "C -3 16 0.1010\n", + "D 7 11 0.1075\n", + "E -1 9 0.0766\n", + "F 1 11 0.0854\n", + "G 18 10 0.1761\n", + "H 12 18 0.1327\n", "Only thing that worries me is that Gelman has A best with prob=10%\n" ] } @@ -248,7 +248,7 @@ }, { "cell_type": "code", - "execution_count": 332, + "execution_count": 360, "metadata": {}, "outputs": [ { @@ -259,14 +259,14 @@ " as good as or better than corresponding column\n", " \n", " A B C D E F G H\n", - "A 1.00 0.61 0.64 0.60 0.68 0.66 0.53 0.58 \n", - "B 0.39 1.00 0.53 0.50 0.59 0.56 0.41 0.47 \n", - "C 0.36 0.47 1.00 0.47 0.55 0.52 0.37 0.44 \n", - "D 0.40 0.50 0.53 1.00 0.58 0.56 0.42 0.47 \n", - "E 0.32 0.41 0.45 0.42 1.00 0.47 0.33 0.38 \n", - "F 0.34 0.44 0.48 0.44 0.53 1.00 0.35 0.42 \n", - "G 0.47 0.59 0.63 0.58 0.67 0.65 1.00 0.57 \n", - "H 0.42 0.53 0.56 0.53 0.62 0.58 0.43 1.00 \n" + "A 1.00 0.59 0.62 0.60 0.66 0.64 0.52 0.58 \n", + "B 0.41 1.00 0.54 0.51 0.57 0.56 0.42 0.49 \n", + "C 0.38 0.46 1.00 0.47 0.54 0.51 0.39 0.45 \n", + "D 0.40 0.49 0.53 1.00 0.56 0.54 0.41 0.48 \n", + "E 0.34 0.43 0.46 0.44 1.00 0.48 0.35 0.41 \n", + "F 0.36 0.44 0.49 0.46 0.52 1.00 0.38 0.43 \n", + "G 0.48 0.58 0.61 0.59 0.65 0.62 1.00 0.57 \n", + "H 0.42 0.51 0.55 0.52 0.59 0.57 0.43 1.00 \n" ] } ], @@ -336,7 +336,7 @@ " }\n", "}\n", "'''\n", - "sm=pystan.StanModel(model_code=stan_code_2)" + "sm2=pystan.StanModel(model_code=stan_code_2)" ] }, { @@ -345,7 +345,7 @@ "metadata": {}, "outputs": [], "source": [ - "answers=sm.sampling(data=dict({'means':p55['effect'],'se':p55['se']}),iter=5000)\n" + "answers=sm2.sampling(data=dict({'means':p55['effect'],'se':p55['se']}),iter=5000)\n" ] }, { @@ -492,7 +492,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.6.5" + "version": "3.6.4" } }, "nbformat": 4, diff --git a/BDA 5.9.8.ipynb b/BDA 5.9.8.ipynb index d3ea631..c3ad5cd 100644 --- a/BDA 5.9.8.ipynb +++ b/BDA 5.9.8.ipynb @@ -6,6 +6,10 @@ "source": [ "# Discrete Mixture Models\n", "\n", + " This solution differs from the one published here:\n", + "http://www.stat.columbia.edu/~gelman/book/solutions3.pdf and is probably wrong\n", + "\n", + "\n", "Discrete mixture models: if $p_m(\\theta)$, for $m=1,\\ldots,M$ are conjugate prior densities for the sampling model $y|\\theta$, show that the class of finite mixture prior densities given by \n", "$$\n", "p(\\theta)=\\sum_{1}^{M} \\lambda_m p_m(\\theta)\n", @@ -46,70 +50,163 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In the special case under consideration, $p_1$ is normal with mean $-1$ and $\\sigma=.5$, $p_2$ is normal with mean $1$ and $\\sigma=.5$ and we can set $\\lambda_1=.1$ and $\\lambda_2=.9$. The $p_m(\\{y_{i}\\})$ can be calculated from the $t$ distribution. Drawing a sample of size $10$ from $p_1$ and getting a sample mean of $-.25$ and a sample variance of $1$ gives a $t$-statistics of $\\sqrt{10}(-.25+1)$ in the first case and $\\sqrt{10}(-.25-1)$ in the second. " + "\n", + " This is the part that was wrong -- I leave it here for historical purposes, but the correct part follows. \n", + "

\n", + "In the special case under consideration, $p_1$ is normal with mean $-1$ and $\\sigma=.5$, $p_2$ is normal with mean $1$ and $\\sigma=.5$ and we can set $\\lambda_1=.1$ and $\\lambda_2=.9$. The $p_m(\\{y_{i}\\})$ can be calculated from the $t$ distribution. Drawing a sample of size $10$ from $p_1$ and getting a sample mean of $-.25$ and a sample variance of $1$ gives a $t$-statistics of \n", + "$$\\frac{(\\overline{y}-\\mu)}{s/\\sqrt{N}}=\\frac{(-.25+1)}{1/\\sqrt{10}}=\\sqrt{10}(-.25+1)$$ in the first case and $\\sqrt{10}(-.25-1)$ in the second. \n", + "

\n", + " Now we continue with what is correct \n", + "\n", + "We need to properly interpret $p_m(\\{y_{i}\\})$ and for that we should remember where it comes from. We\n", + "rewrote\n", + "$$\n", + "p(\\{y_{i}\\}|\\theta)p_{m}(\\theta)=p_m(\\{y_{i}\\},\\theta)=p_{m}(\\{y_{i}\\})p_{m}(\\theta|\\{y_{i}\\})\n", + "$$\n", + "We're dealing here with normal distributions. On the left, the quadratic form that is the log-likelihood of the\n", + "relevant bivariate normal is\n", + "$$Q=\\frac{(\\overline{y}-\\theta)^2}{\\sigma^2}+\\frac{(\\theta-\\mu)^2}{\\tau^2}\n", + "$$\n", + "where, more specifically, $\\overline{y}=-.25$, $\\mu=\\pm 1$, $\\sigma^2=1/10$, and $\\tau^2=0.25=0.5^2$. \n", + "The first term comes from $p_{m}(\\overline{y}|\\theta)$ and the second from $p_{m}(\\theta)$.\n", + "\n", + "Pure algebra (by expanding, writing $Q$ as a quadratic in $\\theta$, and completing the square) gives us\n", + "the expression\n", + "$$\n", + "Q=\\frac{(\\theta-\\mu_{1})^2}{\\tau_1^2}+\\frac{(\\overline{y}-\\mu)^2}{\\sigma^2+\\tau^2}\n", + "$$\n", + "where \n", + "$$\n", + "\\mu_{1}=\\frac{\\mu/\\tau^2+\\overline{y}/\\sigma^2}{\\tau_1^2}\n", + "$$\n", + "and \n", + "$$\n", + "\\frac{1}{\\tau_1^2}=\\frac{1}{\\sigma^2}+\\frac{1}{\\tau^2}\n", + "$$\n", + "\n", + "In the context of the problem under discussion, the two terms tell us that\n", + "$$\n", + "\\theta\\sim N(\\mu_1,\\tau_1^2)\n", + "$$\n", + "where $\\tau_1^2=1/(1/.1+1/.25)=1/14$ and $\\mu_1=(-.25/.1\\pm 1/.25)/14$ giving $-.46$ and $.11$. \n", + "and $p_{m}(\\{y_{i}\\})=N(-.25,\\pm 1, .1+.25)$.\n", + "\n", + "\n" ] }, { "cell_type": "code", - "execution_count": 14, + "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "0.041664931082753924\n", - "0.0035119750957915393\n" + "0.30191827840729224 0.07235502834102417\n" ] } ], "source": [ "import numpy as np\n", "from scipy.stats import norm, t\n", - "t_1=np.sqrt(9)*.75\n", - "t_2=np.sqrt(9)*1.25\n", - "print(t.pdf(t_1,df=9))\n", - "print(t.pdf(t_2,df=9))" + "import matplotlib.pyplot as plt\n", + "p2y=norm.pdf(-.25,1,np.sqrt(.35))\n", + "p1y=norm.pdf(-.25,-1,np.sqrt(.35))\n", + "lambda1,lambda2=(.1,.9)\n", + "print(p1y,p2y)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, + "outputs": [], + "source": [ + "wt1=lambda1*p1y/(lambda1*p1y+lambda2*p2y)\n", + "wt2=lambda2*p2y/(lambda1*p1y+lambda2*p2y)" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, "outputs": [ { - "data": { - "text/plain": [ - "0.5655172413793104" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" + "name": "stdout", + "output_type": "stream", + "text": [ + "0.3167705292212528 0.6832294707787472\n" + ] } ], "source": [ - ".1*.041/(.1*.041+.9*.0035)" + "print(wt1, wt2)" ] }, { "cell_type": "code", - "execution_count": 17, + "execution_count": 20, + "metadata": {}, + "outputs": [], + "source": [ + "def posterior(prior_mean,prior_variance,sample_mean,pop_variance,n):\n", + " post_var=1/((1/prior_variance) + n/pop_variance)\n", + " post_mean=(prior_mean/prior_variance+sample_mean*n/pop_variance)/(1/post_var)\n", + " return post_mean, post_var" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "-0.4642857142857143 0.10714285714285714\n" + ] + } + ], + "source": [ + "post_mean1,post_var1=posterior(-1,.25,-.25,1,10)\n", + "post_mean2,post_var2=posterior(1,.25,-.25,1,10)\n", + "print(post_mean1,post_mean2)" + ] + }, + { + "cell_type": "code", + "execution_count": 22, "metadata": {}, "outputs": [ { "data": { + "image/png": "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\n", "text/plain": [ - "0.43448275862068964" + "

" ] }, - "execution_count": 17, "metadata": {}, - "output_type": "execute_result" + "output_type": "display_data" } ], "source": [ - ".9*.0035/(.1*.041+.9*.0035)" + "x=np.linspace(-3,3,1000)\n", + "y1=lambda1*norm.pdf(x,-1,.5)+lambda2*norm.pdf(x,1,.5)\n", + "y2=wt1*norm.pdf(x,post_mean1,np.sqrt(post_var1))+wt2*norm.pdf(x,post_mean2,np.sqrt(post_var2))\n", + "fig,ax=plt.subplots(1)\n", + "ax.plot(x,y1,color='red',label='prior')\n", + "ax.plot(x,y2,color='blue',label='posterior')\n", + "ax.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This now matches the solution published on the web!" ] }, { diff --git a/Useful Formulae.ipynb b/Useful Formulae.ipynb index 3fc65f4..1d0ddd3 100644 --- a/Useful Formulae.ipynb +++ b/Useful Formulae.ipynb @@ -52,66 +52,6 @@ " " ] }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "0.7403867575800461" - ] - }, - "execution_count": 26, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "post_sample(-.25,1,.25,-.25,1,10)+post_sample(-.25,-1,.25,-.25,1,10)" - ] - }, - { - "cell_type": "code", - "execution_count": 29, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "0.05095226579074726" - ] - }, - "execution_count": 29, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - ".1*post_sample(-.25,-1,.25,-.25,1,10)/(post_sample(-.25,1,.25,-.25,1,10)+post_sample(-.25,-1,.25,-.25,1,10))" - ] - }, - { - "cell_type": "code", - "execution_count": 30, - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "0.4414296078832747" - ] - }, - "execution_count": 30, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - ".9*post_sample(-.25,1,.25,-.25,1,10)/(post_sample(-.25,1,.25,-.25,1,10)+post_sample(-.25,-1,.25,-.25,1,10))" - ] - }, { "cell_type": "code", "execution_count": null,