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fi_lartillot_simulation_gpu_nsamp.ipynb
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{
"cell_type": "markdown",
"metadata": {
"id": "view-in-github",
"colab_type": "text"
},
"source": [
"<a href=\"https://colab.research.google.com/gist/avivajpeyi/76f1d691061f027a2d2fde87b291208e/fi_lartillot_simulation_gpu_nsamp.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
]
},
{
"cell_type": "markdown",
"source": [
"# FI Lartillot Simulation"
],
"metadata": {
"id": "cBK95jimnZtf"
}
},
{
"cell_type": "code",
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"from tqdm.auto import tqdm, trange\n",
"import matplotlib.pyplot as plt\n",
"import jax\n",
"import jax.numpy as jnp\n",
"from jax import random\n",
"from jax.scipy.stats import norm\n",
"from jax.scipy.special import logsumexp\n",
"from jax.numpy import prod, sum, sin, nansum, nanprod, power, log, exp, sqrt\n",
"from jax import jit\n",
"import os\n",
"import jax\n",
"from jax import config\n",
"\n",
"config.update(\"jax_enable_x64\", True)\n",
"\n",
"OUTDIR = 'out_simulation'\n",
"if not os.path.exists(OUTDIR):\n",
" os.makedirs(OUTDIR)\n",
"\n",
"PI = jnp.pi\n",
"\n",
"\n",
"@jit\n",
"def ln_fi_post(θstar, θ, r):\n",
" n, d = θ.shape\n",
" assert θstar.shape == (1,d)\n",
" integrand = sin(r * (θstar-θ) ) / (θstar-θ)\n",
" sum_prod_integrand = nansum(nanprod(integrand, axis=1))\n",
" return log(sum_prod_integrand / (n * power(PI, d)))\n",
"\n",
"\n",
"@jit\n",
"def ln_fi_z(θstar, θ, d=1, v=1, r=100):\n",
" return ln_likelihood(θstar, v=v) + ln_prior(θstar) - ln_fi_post(θstar, θ, r=r)\n",
"\n",
"\n",
"@jit\n",
"def get_err(θstar, θ, d=1, v=1, r=100):\n",
" return ln_fi_z(θstar, θ, d=d, v=v, r=r) - ln_true_z(d=d, v=v)\n",
"\n",
"\n",
"@jit\n",
"def ln_true_z(d, v=1):\n",
" return log(power(2.*PI * (1+v), -d/2.))\n",
"\n",
"@jit\n",
"def ln_true_post(θ, v=1):\n",
" return sum(norm.logpdf(θ, loc=0, scale=jnp.sqrt(v/(v+1))))\n",
"\n",
"@jit\n",
"def ln_likelihood(θ, v=1):\n",
" n, d = θ.shape\n",
" return log(\n",
" power(2.*PI*v, -d/2) * nanprod(exp(-power(θ,2) / (2 * v)))\n",
" )\n",
"\n",
"@jit\n",
"def ln_prior(θ):\n",
" return log(\n",
" prod(norm.pdf(θ, loc=0, scale=1))\n",
" )\n",
"\n",
"def sample_θ(d:int=1, v:float=1, nsamples:int=int(1e6)):\n",
" return np.random.normal(loc=0, scale=np.sqrt(v/(v+1)), size=(nsamples,d))\n",
"\n",
"def error_vs_ns(d=1,v=1, r=100, num_runs=10):\n",
" ns = np.linspace(500, int(2e7), num_runs, dtype=int)\n",
" err = np.zeros(num_runs)\n",
" θstar = jnp.zeros((1,d))\n",
" for i in trange(num_runs):\n",
" err[i] = get_err(\n",
" θstar=θstar,\n",
" θ = sample_θ(d=d, v=v, nsamples=ns[i]),\n",
" d=d, v=v, r=r\n",
" )\n",
"\n",
" return ns, err\n",
"\n",
"\n",
"def test_lnl_lnpri_functions(d, v):\n",
" θstar = jnp.zeros((1,d))\n",
"\n",
" lnl = ln_likelihood(θstar, v=v)\n",
" lnpri = ln_prior(θstar)\n",
" lnpost = ln_true_post(θstar, v=v)\n",
" lnz = lnl+lnpri-lnpost\n",
" truelnz = ln_true_z(d, v=v)\n",
" print(f\"lnl={lnl}\")\n",
" print(f\"lnpri={lnpri}\")\n",
" print(f\"lnpost={lnpost}\")\n",
" print(f'lnz={lnz}')\n",
" assert lnz == truelnz, ('LnZ using analytical ln-posterior doenst matches the analytical ln_z!!')\n",
"\n",
"def test_posterior_samples(d,v, n =int(1e5)):\n",
" θ = sample_θ(d, v, nsamples=n)\n",
" assert θ.shape == (n,d), f\"θ.shape ={θ.shape} !={(n,d)}\"\n",
" expected_std = np.sqrt(v / (v + 1))\n",
" obs_std = np.std(θ, axis=0)\n",
" assert obs_std.shape == (d,), f\"obs_std.shape={obs_std.shape}\"\n",
" check = jnp.all(jnp.isclose(expected_std, obs_std, atol=0.005))\n",
" assert check, f\"expected_std ! = std(θ) => {expected_std:.3f}, {obs_std}\"\n",
" θ1 = sample_θ(d=2, v=1, nsamples=4)\n",
" θ2 = sample_θ(d=2, v=1, nsamples=4)\n",
" assert np.sum(θ1-θ2) != 0, 'samples not unique!'\n",
"\n",
"\n",
"d, v = 20, 2\n",
"test_posterior_samples(d, v)\n",
"test_lnl_lnpri_functions(d,v)\n",
"\n",
"\n"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "3Oz2AL0wQZtI",
"outputId": "f476c14c-f390-4ded-8b38-8eddefd134f2"
},
"execution_count": 114,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"lnl=-25.31024246969291\n",
"lnpri=-18.37877066409345\n",
"lnpost=-14.324119583011809\n",
"lnz=-29.364893550774553\n"
]
}
]
},
{
"cell_type": "code",
"source": [
"\n",
"def run_sims(ndims=[1, 5]):\n",
" data = {}\n",
" v= 1\n",
" for d in tqdm(ndims):\n",
" data['ns'], data[f'd{d}'] = error_vs_ns(d=d, v=v, num_runs=100)\n",
" df = pd.DataFrame(data)\n",
" df=df.set_index('ns')\n",
"\n",
" df.to_csv('errors.csv')\n",
" plot_results(df)\n",
" return df\n",
"\n",
"\n",
"def plot_results(df):\n",
" for col in df.columns:\n",
" plt.figure(figsize=(5, 1.5))\n",
" plt.axhline(0, color='k')\n",
" plt.scatter(df.index, df[col], label=col)\n",
" plt.xlabel('N Samples')\n",
" plt.ylabel(\"LnZ Error\")\n",
" plt.legend()\n",
"\n",
"df = run_sims()\n"
],
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\n"
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"<Figure size 500x150 with 1 Axes>"
],
"image/png": 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\n"
},
"metadata": {}
}
]
},
{
"cell_type": "code",
"source": [
"def hist_ln_fis_at_r(d, v, num_expts, num_samp):\n",
" lnzs = np.zeros(num_expts)\n",
" θstar = jnp.zeros((1,d))\n",
" for i in trange(num_expts):\n",
" lnzs[i]= ln_fi_z(\n",
" θ = sample_θ(d, v, nsamples=num_samp),\n",
" θstar=θstar,\n",
" d=d, v=v, r=100\n",
" )\n",
" return lnzs, ln_true_z(d=d,v=v)\n",
"\n",
"def plot_hist(lnzs, trulnz, label, color='C0'):\n",
" plt.hist(lnzs, label=label, alpha=0.3, color=color)\n",
" plt.axvline(trulnz, color='k')\n",
" plt.xlabel(\"LnZ\")\n",
"\n",
"# lnzs, trulz = hist_ln_fis_at_r(d=1,v=1, num_expts=100, num_samp=int(1e5))\n",
"# plot_hist(lnzs, trulz)\n"
],
"metadata": {
"id": "I7mhzvHi_Q3b"
},
"execution_count": 126,
"outputs": []
},
{
"cell_type": "code",
"source": [
"lnzs_1e5, trulz = hist_ln_fis_at_r(d=5,v=0.01, num_expts=100, num_samp=int(1e5))\n",
"lnzs_1e6, trulz = hist_ln_fis_at_r(d=5,v=0.01, num_expts=100, num_samp=int(1e6))"
],
"metadata": {
"id": "9GHRMcnSHRu-"
},
"execution_count": 129,
"outputs": []
},
{
"cell_type": "code",
"source": [
"plot_hist(lnzs_1e5, trulz, label=\"n=1e5\", color='C0')\n",
"plot_hist(lnzs_1e6, trulz, label=\"n=1e6\", color='C1')\n",
"plt.title(f\"d=5, v=0.01, nRep=100\")\n",
"plt.legend()"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 490
},
"id": "SaygbXigKAwX",
"outputId": "241a1074-3cc6-4183-8e3f-2e0e099f80fc"
},
"execution_count": 127,
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7d9f9d7ad360>"
]
},
"metadata": {},
"execution_count": 127
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
],
"image/png": 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\n"
},
"metadata": {}
}
]
},
{
"cell_type": "code",
"source": [
"lnzs_1e5, trulz = hist_ln_fis_at_r(d=100,v=0.01, num_expts=100, num_samp=int(1e5))\n",
"lnzs_1e6, trulz = hist_ln_fis_at_r(d=100,v=0.01, num_expts=100, num_samp=int(1e6))\n",
"plot_hist(lnzs_1e5, trulz, label=\"n=1e5\", color='C0')\n",
"plot_hist(lnzs_1e6, trulz, label=\"n=1e6\", color='C1')\n",
"plt.title(f\"d=100, v=0.01, nRep=100\")\n",
"plt.legend()"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 554,
"referenced_widgets": [
"eff9cb2f15214097a69cc882ba43099e",
"94984e3ca4324576a726ea114a8d147a",
"e65b080bacda497683b56e8420cf5297",
"da08eef4abde421295f75f6202918651",
"a68a4a5b229e4a8295a1ee8c7c75eb59",
"56ee66210f0a47d2ac8de0db730ac53d",
"8dfbac71230c4a7dbe16464258a468ad",
"64173a3e3e8443d1948d52f5e5fdfa14",
"d0bfbc1fd67e433fbe5bd8a9439fcee9",
"cd593a1538a1423085def8a6e3589208",
"0eec09a27e6a4939bdc3440201398ab7",
"183d021c53f94aeaa4f396c9b510bb70",
"70211a1050574dfbbf308277973a2924",
"503c4d1aa8d84c08b33901a073dbbce3",
"96580848d54740cf949c5194d34297a4",
"da8de259d7a84c24b602598f17b00284",
"2b23a31e61e44ea5bcd400a5c0d88a0d",
"22922b1a7d58436093ca6cd5e52348aa",
"a675ca82be0344a88195083046d36058",
"815ec21bf0aa433bb5dbeed91dd7022f",
"155a42a8e2244189b86299ed825e947f",
"4fd1de8c7ac24088986a5de137fb488d"
]
},
"id": "48gzcYtVKWzV",
"outputId": "8eee13b5-d484-4a78-baf7-6d5865a35bcd"
},
"execution_count": 129,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
" 0%| | 0/100 [00:00<?, ?it/s]"
],
"application/vnd.jupyter.widget-view+json": {
"version_major": 2,
"version_minor": 0,
"model_id": "eff9cb2f15214097a69cc882ba43099e"
}
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
" 0%| | 0/100 [00:00<?, ?it/s]"
],
"application/vnd.jupyter.widget-view+json": {
"version_major": 2,
"version_minor": 0,
"model_id": "183d021c53f94aeaa4f396c9b510bb70"
}
},
"metadata": {}
},
{
"output_type": "execute_result",
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7d9fa06a3460>"
]
},
"metadata": {},
"execution_count": 129
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
],
"image/png": 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\n"
},
"metadata": {}
}
]
},
{
"cell_type": "markdown",
"source": [
"## Renate's version"
],
"metadata": {
"id": "vQRXLbGrEGba"
}
},
{
"cell_type": "code",
"source": [
"%load_ext rpy2.ipython"
],
"metadata": {
"id": "Chs6j3bMEHyk"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [
"%%R\n",
"#multivariate case\n",
"d = 5 # dimension\n",
"v = 2 # variance\n",
"# true evidence\n",
"thetastar = rep(0,d)\n",
"Z = (2*pi*(v+1))^(-d/2)\n",
"Z2 = 1/sqrt(2*pi*v)^d* exp(-sum(thetastar^2)/2/v) *1/sqrt(2*pi)^d*exp(-sum(thetastar^2)/2) / prod(dnorm(thetastar,mean=0,sd=sqrt(v/(v+1))))\n",
"\n",
"# fix R and look at varying MCMC sample size n:\n",
"n = seq(10000,1000000, by=50000 )\n",
"kmax = length(n)\n",
"R=100\n",
"phat = rep(0,kmax)\n",
"ptrue = prod(dnorm(thetastar,mean=0,sd=sqrt(v/(v+1))))\n",
"Zhat = rep(0,kmax)\n",
"for (k in 1:kmax){\n",
" theta = thetastar = matrix(0,nrow=n[k],ncol=d)\n",
" for (j in 1:d){\n",
" theta[,j] = rnorm(n[k],mean=0,sd=sqrt(v/(v+1)))\n",
" }\n",
" tmp = sin(R*(thetastar-theta))/(thetastar-theta)\n",
" phat[k] = 1/n[k]/pi^d*sum(apply(tmp,1,prod))\n",
"\n",
" likelihood = 1/sqrt(2*pi*v)^d*exp(-sum(thetastar[1,]^2)/2/v)\n",
" prior = 1/sqrt(2*pi)^d*exp(-sum(thetastar[1,]^2)/2)\n",
"\n",
" Zhat[k] = likelihood*prior/phat[k]\n",
"}\n",
"par(mfrow=c(2,1))\n",
"plot(n,Zhat)\n",
"abline(a=Z,b=0)\n",
"plot(n,phat)\n",
"abline(a=ptrue,0)\n",
"print(c(R,Z-Zhat[k],mean(Z-Zhat),sd(Zhat),ptrue-phat[k],mean(ptrue-phat),sd(phat)))\n"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 532
},
"id": "OVLTamGiEIzP",
"outputId": "4dbdf1bf-badb-40cd-bf18-ba0c94ea7df1"
},
"execution_count": null,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"[1] 1.000000e+02 6.280733e-04 6.117632e-04 1.287898e-04 -8.665389e-01\n",
"[6] -1.236168e-01 6.325309e-01\n"
]
},
{
"output_type": "display_data",
"data": {
"image/png": "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\n"
},
"metadata": {}
}
]
},
{
"cell_type": "code",
"source": [
"%%R\n",
"plot(n,log(Zhat)-log(Z))"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 497
},
"id": "HjhFHyDuHoJj",
"outputId": "739766ea-b04e-4f2b-9c7a-f1cd12e5ab99"
},
"execution_count": null,
"outputs": [
{
"output_type": "display_data",
"data": {
"image/png": 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\n"
},
"metadata": {}
}
]
},
{
"cell_type": "code",
"source": [],
"metadata": {
"id": "rctKGgq4HqnN"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [],
"metadata": {
"id": "MgvtMn-1KN6N"
},
"execution_count": null,
"outputs": []
}
]
}
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