coin_analogy.ipynb

coin_analogy.ipynb

{
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          "end_time": "2017-04-17T20:26:00.865721"
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      },
      "cell_type": "code",
      "source": "import numpy as np\nimport collections\nimport scipy.stats as ss\nnp.random.seed(0)",
      "execution_count": 1,
      "outputs": []
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## Example 1\nConsider we have a list of coin throwing result. e.g. ['H' 'T' 'H' 'H' 'H', ...].\n\nTwo models are proposed to model the mechanism.\n* Model 1: Assuming a fair coin\n* Model 2: Consider unfair coin with different probability of throwing a 'H' and 'T' \n\nThe reproducing likelihood and predicting likelihood of both models are calculated and compared."
    },
    {
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          "start_time": "2017-04-17T20:26:00.866952",
          "end_time": "2017-04-17T20:26:00.872952"
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      },
      "cell_type": "code",
      "source": "def gen_synthetic(p_head, n_train, n_test):\n    x_all = np.random.choice(['H', 'T'], n_train + n_test, p=[p_head, 1 - p_head])\n    xtrain = x_all[:n_train]\n    xtest = x_all[n_train:]\n    return xtrain, xtest\n\nxtrain, xtest = gen_synthetic(0.7, 75, 25)\nprint(xtrain[:5])\nprint(xtest[:5])",
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "text": "['H' 'T' 'H' 'H' 'H']\n['H' 'H' 'H' 'H' 'H']\n",
          "name": "stdout"
        }
      ]
    },
    {
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      },
      "cell_type": "code",
      "source": "def eval_model(samples, p_head_est):\n    probs = np.where(samples == 'H', p_head_est, 1 - p_head_est)\n    return ss.mstats.gmean(probs)\n\n# Model 1\np_model1 = 0.5\nlikelihood_trn = eval_model(xtrain, p_model1)\nlikelihood_tst = eval_model(xtest , p_model1)\nprint (\"Model 1 with p_head_est {:.4f} has avg reproducing likelihood {:.4f} and avg predicting likelihood {:.4f}\".format(p_model1, likelihood_trn, likelihood_tst))\n\n# Model 2\np_model2 = collections.Counter(xtrain)['H'] / len(xtrain)\nlikelihood_trn = eval_model(xtrain, p_model2)\nlikelihood_tst = eval_model(xtest , p_model2)\nprint (\"Model 2 with p_head_est {:.4f} has avg reproducing likelihood {:.4f} and avg predicting likelihood {:.4f}\".format(p_model2, likelihood_trn, likelihood_tst))\n",
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "text": "Model 1 with p_head_est 0.5000 has avg reproducing likelihood 0.5000 and avg predicting likelihood 0.5000\nModel 2 with p_head_est 0.7333 has avg reproducing likelihood 0.5599 and avg predicting likelihood 0.6495\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "## Example 2\nConsider we have a list of coin throwing pair result. \nEach round the coin is threw 2 times and recorded as a pair.\ne.g. [['H' 'H'], ['T' 'T'], ['H' 'H'], ['H' 'H'], ['H' 'T'], ...].\n\nThree models are proposed to model the mechanism.\n* Model 1: Assuming a fair coin\n* Model 2: Consider an unfair coin with probability $p_H$ of getting 'H', and believing the probability is independent to round.\n* Model 3: Consider an unfair coin with round-sensitive probability $p_{H1}$ and $p_{H2}$ of getting 'H' at the first and second round.\n\nThe reproducing likelihood and predicting likelihood of these three models are calculated and compared."
    },
    {
      "metadata": {
        "trusted": true,
        "collapsed": false,
        "ExecuteTime": {
          "start_time": "2017-04-17T20:26:01.115215",
          "end_time": "2017-04-17T20:26:01.378901"
        }
      },
      "cell_type": "code",
      "source": "def gen_synthetic(p_head, n_train, n_test):\n    x_all = np.random.choice(['H', 'T'], 2 * (n_train + n_test), p=[p_head, 1 - p_head])\n    x_all = x_all.reshape((-1, 2))\n    # x_all = [tuple(x) for x in x_all]\n    xtrain = x_all[:n_train]\n    xtest = x_all[n_train:]\n    return xtrain, xtest\n\nxtrain, xtest = gen_synthetic(0.7, 75, 30)\nprint(xtrain[:5])\nprint(xtest[:5])",
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": "[['H' 'H']\n ['T' 'T']\n ['H' 'H']\n ['H' 'H']\n ['H' 'T']]\n[['H' 'T']\n ['T' 'T']\n ['T' 'H']\n ['H' 'H']\n ['T' 'H']]\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
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        "ExecuteTime": {
          "start_time": "2017-04-17T20:26:01.381313",
          "end_time": "2017-04-17T20:26:01.663465"
        }
      },
      "cell_type": "code",
      "source": "def eval_model(samples, p_head_est_r1, p_head_est_r2=None):\n    if p_head_est_r2 is None:\n        p_head_est_r2 = p_head_est_r1\n    prob = 1.0\n    for idx, x in np.ndenumerate(samples):\n        # print (idx, x)\n        if idx[1] == 0:\n            prob *= p_head_est_r1 if x == 'H' else 1 - p_head_est_r1\n        else:\n            prob *= p_head_est_r2 if x == 'H' else 1 - p_head_est_r2\n    return pow(prob, 1 / len(samples))\n\n# Model 1\np_model1 = 0.5\nlikelihood_trn = eval_model(xtrain, p_model1)\nlikelihood_tst = eval_model(xtest,  p_model1)\nprint (\"Model 1 with p_head_est {:.4f} has avg reproducing likelihood {:.4f} and avg predicting likelihood {:.4f}\".format(p_model1, likelihood_trn, likelihood_tst))\n\n# Model 2\np_model2 = collections.Counter(xtrain.flatten())['H'] / (len(xtrain) * 2)\nlikelihood_trn = eval_model(xtrain, p_model2)\nlikelihood_tst = eval_model(xtest,  p_model2)\nprint (\"Model 2 with p_head_est {:.4f} has avg reproducing likelihood {:.4f} and avg predicting likelihood {:.4f}\".format(p_model2, likelihood_trn, likelihood_tst))\n\n# Model 3\np_model3_r1 = collections.Counter(xtrain[:, 0])['H'] / len(xtrain)\np_model3_r2 = collections.Counter(xtrain[:, 1])['H'] / len(xtrain)\nlikelihood_trn = eval_model(xtrain, p_model3_r1, p_model3_r2)\nlikelihood_tst = eval_model(xtest,  p_model3_r1, p_model3_r2)\nprint (\"Model 3 with p_head_est [{:.4f}, {:.4f}] has avg reproducing likelihood {:.4f} and avg predicting likelihood {:.4f}\".format(p_model3_r1, p_model3_r2, likelihood_trn, likelihood_tst))",
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "text": "Model 1 with p_head_est 0.5000 has avg reproducing likelihood 0.2500 and avg predicting likelihood 0.2500\nModel 2 with p_head_est 0.7200 has avg reproducing likelihood 0.3055 and avg predicting likelihood 0.2513\nModel 3 with p_head_est [0.7067, 0.7333] has avg reproducing likelihood 0.3057 and avg predicting likelihood 0.2494\n",
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        }
      ]
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