{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Regression with Outliers\n", "\n", "In the standard __Gaussian process regression__ setting it is assumed that the observations are __Normally distributed__ about the latent function. In the package this can applied using either the GP or GPE functions with which *exact Gaussian process* models.\n", "\n", "One of the drawbacks of exact GP regression is that by assuming Normal noise the GP is __not robust to outliers__. In this setting, it is more appropriate to assume that the distribution of the noise is heavy tailed. For example, with a __Student-t distribution__,\n", "$$\n", "\\mathbf{y} \\ | \\ \\mathbf{f},\\nu,\\sigma \\sim \\prod_{i=1}^n \\frac{\\Gamma(\\nu+1)/2}{\\Gamma(\\nu/2)\\sqrt{\\nu\\pi}\\sigma}\\left(1+\\frac{(y_i-f_i)^2}{\\nu\\sigma^2}\\right)^{-(\\nu+1)/2}\n", "$$\n", "\n", "Moving away from the Gaussian likelihood function (i.e. Normally distributed noise) and using the Student-t likelihood means that we can no longer analytically calculate the GP marginal likelihood. We can take a Bayesian perspective and sample from the joint distribution of the latent function and model parameters." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "