Toosi et al. We define a t likelihood for the response variable, y, and suitable vague priors on all the model parameters: normal for α and β, half-normal for σ and gamma for ν. 17), \[ This density places the majority of the prior mass for values \(\nu < 50\), in which The scale mixture distribution of normal parameterization of the Student t distribution is useful for computational reasons. y ~ student_t(nu, mu, sigma); #> In file included from file199a4ffb80c1.cpp:8: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/src/stan/model/model_header.hpp:4: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/stan/math.hpp:4: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/stan/math/rev/mat.hpp:4: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/stan/math/rev/core.hpp:14: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/stan/math/rev/core/matrix_vari.hpp:4: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/stan/math/rev/mat/fun/Eigen_NumTraits.hpp:4: #> In file included from /Users/jrnold/Library/R/3.5/library/StanHeaders/include/stan/math/prim/mat/fun/Eigen.hpp:4: #> In file included from /Users/jrnold/Library/R/3.5/library/RcppEigen/include/Eigen/Dense:1: #> In file included from /Users/jrnold/Library/R/3.5/library/RcppEigen/include/Eigen/Core:531: #> /Users/jrnold/Library/R/3.5/library/RcppEigen/include/Eigen/src/Core/util/ReenableStupidWarnings.h:10:30: warning: pragma diagnostic pop could not pop, no matching push [-Wunknown-pragmas]. As can be seen, the function also plots the inferred linear regression and reports some handy posterior statistics on the parameters alpha (intercept), beta (slope) and y_pred (predicted values). See, #> http://mc-stan.org/misc/warnings.html#bfmi-low, #> Warning: Examine the pairs() plot to diagnose sampling problems. This probability distribution has a parameter ν, known as the degrees of freedom, which dictates how close to normality the distribution is: large values of ν (roughly ν > 30) result in a distribution that is very similar to the normal distribution, whereas low small values of ν produce a distribution with heavier tails (that is, a larger spread around the mean) than the normal distribution. Estimate some examples with known outliers and compare to using a normal In fact, let’s compare it with the line inferred from the clean data by our model, and with the line estimated by the conventional linear model (lm). When plotting the results of linear regression graphically, the explanatory variable is normally plotted on the x-axis, and the response variable on the y-axis. Traditional Bayesian quantile regression relies on the Asymmetric Laplace distribution (ALD) mainly because of its satisfactory empirical and theoretical performances. We can reparameterize the model to make \(\sigma\) and \(\nu\) less correlated by multiplying the scale by the degrees of freedom. \], \[ Consider the linear regression model with normal errors, \[ y_i \sim \dnorm\left(\ X \beta, \sigma_i^2 \right) . \[ We will also calculate the column medians of y.pred, which serve as posterior point estimates of the predicted response for the values in x.pred (such estimates should lie on the estimated regression line, as this represents the predicted mean response). // Uninformative priors on all parameters y_i \sim \dt\left(\nu, \mu_i, \sigma \sqrt{\frac{\nu - 2}{\nu}} \right) \[ sensitive to outliers. \] Gelman and Hill 2007, 125; Liu 2005) with approximately 99.8% of the probability within three standard deviations. Quite publication-ready. \] This tutorial illustrates how to interpret the more advanced output and to set different prior specifications in performing Bayesian regression analyses in JASP (JASP Team, 2020). Abstract. Although linear regression models are fundamental tools in statistical science, the estimation results can be sensitive to outliers. See help('pareto-k-diagnostic') for details. Robust Bayesian modelling for Covid-19 data in Italy Written by Robbayes-C19: Paolo Girardi, Luca Greco, Valentina Mameli, Monica Musio, Walter Racugno, Erlis Ruli and Laura Ventura on 02 June 2020.. On 21 February 2020, the first person-to-person transmission of SARS-CoV-2 – the virus responsible for Covid-19 – was reported in Italy. # As we are not going to build credible or prediction intervals yet, # we will not use M, P, x_cred and x_pred, # Define a sequence of x values for the credible intervals, # Define x values whose response is to be predicted, # HPD intervals of mean response (shadowed area), # Predicted responses and prediction intervals, highest posterior density (HPD) intervals. Suppose \(X \sim \dt(\nu, \mu, \sigma)\), then be robust with respect to the prior specification. \], \[ Bayesian robust regression uses distributions with wider tails than the normal … \pi_i &= \int_{-\infty}^{\eta_i} \mathsf{StudentT}(x | \nu, 0, (\nu - 2)/ \nu) dx \\ That is, the response variable follows a normal distribution with mean equal to the regression line, and some standard deviation σ. Thus, a linear regression with Laplace errors is analogous to a median regression. Historically, robust models were mostly developed on a case-by-case basis; examples include robust linear regression, robust mixture models, and bursty topic models. We can take a look at the MCMC traces and the posterior distributions for alpha, beta (the intercept and slope of the regression line), sigma and nu (the spread and degrees of freedom of the t-distribution). \[ This means that outliers will have less of an affect on the log-posterior of models using these distributions. Thus, we need a model that is able to recognise the linear relationship present in the data, while accounting the outliers as infrequent, atypical observations. 2013, Ch. Let’s pitch this Bayesian model against the standard linear model fitting provided in R (lm function) on some simulated data. \end{aligned} y_i \sim \dt\left(\nu, \mu_i, \sigma \right) The most commonly used Bayesian model for robust regression is a linear regression with independent Student-\(t\) errors (Geweke 1993; A. Gelman, Carlin, et al. Importantly, our sampling algorithm incorporates robust data models that … a Gamma distribution, \Var(X) = \frac{\nu}{\nu - 2} \sigma^2. The credible and prediction intervals reflect the distributions of mu_cred and y_pred, respectively. 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