neural_networks/tensorflow_probability/fit_gaussian_tfp.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fitting a normal distribution with tensorflow probability"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we look at fitting a normal distribution to some data using Tensorflow Probability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import tensorflow as tf\n",
    "import tensorflow_probability as tfp\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.style.use(\"seaborn-whitegrid\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate some data from a normal distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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     },
     "metadata": {}
    }
   ],
   "source": [
    "n = 1000\n",
    "true_mu = 2.0\n",
    "true_std = 3.4\n",
    "x = np.random.normal(loc=true_mu, scale=true_std, size=n)\n",
    "plt.hist(x, 30)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Model setup\n",
    "Tensorflow probability allows us to fit a network where the final layer output is not a scalar value,\n",
    "but a probability distribution.\n",
    "The normal distribution is parameterised by the mean and standard deviation.\n",
    "As such the model needs to output two values.\n",
    "\n",
    "We are fitting just a static distribution to the data.\n",
    "This is equivalent to fitting a network where the inputs are all 0.\n",
    "In this situation we only use the network biases as the weights become irrelevant with inputs of 0.\n",
    "\n",
    "To build the above - the model has a dense layer with 2 output nodes.\n",
    "For the final output layer we utilise the normal distribution object from tensorflow probability.\n",
    "The `DistributionLambda` layer allows us to use a lambda function to split the dense layer outputs\n",
    "and connect them to the `loc` and `scale` (mean and standard deviation) inputs of the normal distribution.\n",
    "The scale uses a `softplus` function to ensure the standard deviation is always positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "model = tf.keras.Sequential(\n",
    "    [\n",
    "        tf.keras.layers.Dense(1 + 1),\n",
    "        tfp.layers.DistributionLambda(\n",
    "            lambda t: tfp.distributions.Normal(\n",
    "                loc=t[..., :1],\n",
    "                scale=1e-3 + tf.math.softplus(0.05 * t[..., 1:]),\n",
    "            )\n",
    "        ),\n",
    "    ]\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cost function is setup as the negative loglikelihood:\n",
    "Tensorflow probably objects have a `log_prob(y)` method which returns the log probablility of the sample `y`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def negloglik(y, distr):\n",
    "    return -distr.log_prob(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We fit the model as normal using gradient descent and the `Adam` optimiser.\n",
    "As mentioned before, the input to the network is a vector of zeros.\n",
    "The target variable is our random data we want to fit the normal distribution to."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "model.compile(optimizer=tf.optimizers.Adam(learning_rate=0.2), loss=negloglik)\n",
    "\n",
    "dummy_input = np.zeros(x.shape)[:, np.newaxis]\n",
    "history = model.fit(dummy_input, x, epochs=100, verbose=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The loglikelihood has converged over the training:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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     },
     "metadata": {}
    }
   ],
   "source": [
    "plt.plot(history.history[\"loss\"])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Results\n",
    "We can obtain the fitted parameters from the `model.weights` biases.\n",
    "They do not match the exact parameters of the true distribution.\n",
    "They converge to the maximum likelihood estimator for a normal distribution.\n",
    "In this case it corresponds to the sample mean and standard deviation.\n",
    "\n",
    "We can compare these different estimates by creating probability distribution objects\n",
    "and then plotting the probability distribution function.\n",
    "As we can see the estimates are reasonable approximations of the true distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {}
    }
   ],
   "source": [
    "p_y1 = tfp.distributions.Normal(\n",
    "    loc=model.weights[1][0], scale=tf.math.softplus(0.05 * model.weights[1][1])\n",
    ")\n",
    "p_y2 = tfp.distributions.Normal(loc=np.mean(x), scale=np.std(x))\n",
    "p_y3 = tfp.distributions.Normal(loc=true_mu, scale=true_std)\n",
    "\n",
    "x_t = np.linspace(-10, 10, 50)\n",
    "y_t1 = p_y1.prob(x_t)\n",
    "y_t2 = p_y2.prob(x_t)\n",
    "y_t3 = p_y3.prob(x_t)\n",
    "\n",
    "plt.hist(x, bins=30, density=True, label=\"y\")\n",
    "plt.plot(x_t, y_t1, label=\"y_est\")\n",
    "plt.plot(x_t, y_t2, label=\"y_samp\")\n",
    "plt.plot(x_t, y_t3, label=\"y_true\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6-final"
  },
  "orig_nbformat": 2,
  "kernelspec": {
   "name": "python37664bitdataanalysisenvvenv63b40a3c1a484f6297148098436bce1d",
   "display_name": "Python 3.7.6 64-bit ('dataAnalysisEnv': venv)"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}