Untitled1.ipynb

Credible interval change with sample size and contrasting outcomes.ipynb

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      "metadata": {
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      "cell_type": "code",
      "source": "%matplotlib inline\nimport pymc3 as pm\nimport numpy as np\nimport seaborn as sns",
      "execution_count": 2,
      "outputs": []
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "First, let's run an analysis of 100 binomial samples, with zero positive outcomes:"
    },
    {
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      "cell_type": "code",
      "source": "n1 = 100\nx1 = 0",
      "execution_count": 3,
      "outputs": []
    },
    {
      "metadata": {
        "trusted": true,
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      },
      "cell_type": "code",
      "source": "with pm.Model() as first_dataset:\n    θ = pm.Beta('θ', 1, 1)\n    x = pm.Binomial('x', n=n1, p=θ, observed=x1)\n    \n    trace1 = pm.sample(2000)",
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "text": "Applied logodds-transform to θ and added transformed θ_logodds_ to model.\nAssigned NUTS to θ_logodds_\n100%|██████████| 2000/2000 [00:00<00:00, 2721.90it/s]\n",
          "name": "stderr"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "The parameter estimate is 0.012, with a credible interval of length 0.031."
    },
    {
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      "cell_type": "code",
      "source": "pm.summary(trace1)",
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "text": "\nθ:\n\n  Mean             SD               MC Error         95% HPD interval\n  -------------------------------------------------------------------\n  \n  0.012            0.025            0.001            [0.000, 0.031]\n\n  Posterior quantiles:\n  2.5            25             50             75             97.5\n  |--------------|==============|==============|--------------|\n  \n  0.000          0.003          0.008          0.014          0.037\n\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "Now, let's add another 100 samples, but this time with 10 positive outcomes:"
    },
    {
      "metadata": {
        "trusted": true,
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      "cell_type": "code",
      "source": "n2 = 100\nx2 = 10\n\nwith pm.Model() as combined_dataset:\n    θ = pm.Beta('θ', 1, 1)\n    x = pm.Binomial('x', n=n1+n2, p=θ, observed=x1+x2)\n    \n    trace2 = pm.sample(2000)",
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "stream",
          "text": "Applied logodds-transform to θ and added transformed θ_logodds_ to model.\nAssigned NUTS to θ_logodds_\n100%|██████████| 2000/2000 [00:00<00:00, 3415.85it/s]\n",
          "name": "stderr"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true,
        "collapsed": false
      },
      "cell_type": "code",
      "source": "pm.summary(trace2)",
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "stream",
          "text": "\nθ:\n\n  Mean             SD               MC Error         95% HPD interval\n  -------------------------------------------------------------------\n  \n  0.056            0.030            0.002            [0.025, 0.087]\n\n  Posterior quantiles:\n  2.5            25             50             75             97.5\n  |--------------|==============|==============|--------------|\n  \n  0.027          0.042          0.053          0.064          0.091\n\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "Notice that the credible interval is twice as large, even with a doubling of the sample size!"
    }
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