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fix formatting of conclusions
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examples/generalized_linear_models/GLM-robust-with-outlier-detection.ipynb

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"PLot posterior joint distribution"
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"Plot posterior joint distribution"
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]
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},
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{
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"source": [
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"**Observe**:\n",
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"\n",
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"##### The posterior preditive fit for:\n",
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"The posterior preditive fit for:\n",
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"+ the **OLS model** is shown in **Green** and as expected, it doesn't appear to fit the majority of our datapoints very well, skewed by outliers\n",
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"+ the **Student-T model** is shown in **Orange** and does appear to fit the 'main axis' of datapoints quite well, ignoring outliers\n",
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"+ the **Hogg Signal vs Noise model** is shown in two parts:\n",
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" + **Blue** for inliers fits the 'main axis' of datapoints well, ignoring outliers\n",
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" + **Red** for outliers has a very large variance and has assigned 'outlier' points with more log likelihood than the Blue inlier model \n",
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" \n",
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" \n",
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"##### We see that the **Hogg Signal vs Noise model** also yields specific estimates of _which_ datapoints are outliers:\n",
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"We see that the **Hogg Signal vs Noise model** also yields specific estimates of _which_ datapoints are outliers:\n",
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"+ 17 'inlier' datapoints, in **Blue** and\n",
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"+ 3 'outlier' datapoints shown in **Red**.\n",
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"+ From a simple visual inspection, the classification seems fair, and agrees with Jake Vanderplas' findings.\n",
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"+ I've annotated these Red and the most outlying inliers to aid visual investigation\n",
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" \n",
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" \n",
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"##### Overall:\n",
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"Overall:\n",
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"+ the **Hogg Signal vs Noise model** behaves as promised, yielding a robust regression estimate and explicit labelling of inliers / outliers, but\n",
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"+ the **Hogg Signal vs Noise model** is quite complex, and whilst the regression seems robust, the traceplot shoes many divergences, and the model is potentially unstable\n",
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"+ if you simply want a robust regression without inlier / outlier labelling, the **Student-T model** may be a good compromise, offering a simple model, quick sampling, and a very similar estimate."

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