bayesian linear regression

This is a guest post by Tom Faulkenberry (Tarleton State University). 9. Springer Science & Business Media. = & -2(\beta-\hat{\beta})\times 0 - 2(\beta-\hat{\beta})\bar{x}\sum_i^n(y_i-\hat{y}_i) = 0 \end{aligned} The syntax for a linear regression in a Bayesian framework looks like this: In words, our response datapoints y are sampled from a multivariate normal distribution that has a mean equal to the product of the β coefficients and the predictors, X, and a variance of σ2. \(\epsilon_i\) is the error term. Chaloner, Kathryn, and Rollin Brant. If we are only interested in the distributions of the coefficients of the 4 predictors, we may use the parm argument to restrict the variables shown in the summary. \tag{6.2} \beta ~|~ \sigma^2, \text{data}~ &\sim ~\textsf{Normal}\left(\hat{\beta}, \frac{\sigma^2}{\text{S}_{xx}}\right). The reader is expected to have some basic knowledge of Bayes’ theorem, basic probability (conditional probability and chain rule), machine learning and a pinch of matrix algebra. \[ We apply the scatterplot of residuals versus fitted values, which provides an additional visual check of the model adequacy. Standard Bayesian linear regression prior models — The five prior model objects in this group range from the simple conjugate normal-inverse-gamma prior model through flexible prior models specified by draws from the prior distributions or a custom function. \begin{aligned} The assumed model for our simple linear regression is \(y_i=\alpha + \beta x_i+\epsilon_i\), with \(\epsilon_i\) having independent, identical distributions that are normal with mean zero and constant variance \(\sigma^2\), i.e., \(\epsilon_i \mathrel{\mathop{\sim}\limits^{\rm iid}}\textsf{Normal}(0, \sigma^2)\). Here we use another change of variable by setting \(\displaystyle s= \frac{\text{SSE}+(\beta-\hat{\beta})^2\sum_i(x_i-\bar{x})^2}{2}\phi\), and the fact that \(\displaystyle \int_0^\infty s^{(n-3)/2}e^{-s}\, ds\) gives us the Gamma function \(\Gamma(n-2)\), which is a constant. \]. The primary difference is the interpretation of the intervals. \[ p(\beta_0,\beta_1,\beta_2,\beta_3,\beta_4~|~\sigma^2) \propto 1,\qquad\quad p(\sigma^2) \propto \frac{1}{\sigma^2}. This means that professors have also had to think critically about how they can best deliver instruction in new formats. The simple linear regression tries to fit the relationship between dependent variable YY and single predictor (independent) variable XX into a straight line. In fact, I would argue that we have positive evidence for the absence of any such effect. The columns labeled Estimate and Std. Tom is an Associate Professor and Assistant Head in the Department of Psychological Sciences at Tarleton State University in Stephenville, Texas, USA. \], \[ p^*(\beta, \sigma^2~|~\text{data}) \propto \frac{1}{\sigma^{n+1}}\exp\left(-\frac{\text{SSE}+(\beta-\hat{\beta})^2\sum_i(x_i-\bar{x})^2}{2\sigma^2}\right). \]. \]. To illustrate the idea, we use the data set on kid’s cognitive scores that we examined earlier. For example, based on the data, we believe that there is 95% chance that body fat will increase by 5.75% up to 6.88% for every additional 10 centimeter increase in the waist circumference. In this chapter, we will apply Bayesian inference methods to linear regression. Indeed, that seems to be the case with these data. We can write that linear relationship as: yi=τ+w.xi+ϵi(1)(1)yi=τ+w.xi+ϵi Here ττ is the intercept and ww is the coefficient of the predictor variable. We first consider the case under the reference prior, which is our standard noninformative prior. Recommended reading Lindley, D.V. \[ p(\sigma^2) \propto \frac{1}{\sigma^2}\qquad \Longrightarrow \qquad p(\phi)\propto \frac{1}{\phi} \], Therefore, under the parameters \(\alpha\), \(\beta\), and the precision \(\phi\), we have the joint prior distribution as This has provided us a base line analysis of Bayesian approach, which we can extend later when we introduce more different coefficient priors. \], \[ 1/\sigma^2~|~\text{data}~\sim~\textsf{Gamma}\left(\frac{n-2}{2},\frac{\text{SSE}}{2}\right).\], \[ \alpha~|~\sigma^2, \text{data} ~\sim~\textsf{Normal}\left(\hat{\alpha}, \sigma^2\left(\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}\right)\right),\qquad \qquad 1/\sigma^2~|~\text{data}~\sim~ \textsf{Gamma}\left(\frac{n-2}{2}, \frac{\text{SSE}}{2}\right). The posterior probability of Case 39 being an outlier is about 0.685. The model is the normal linear regression model: where: 1. is the vector of observations of the dependent variable; 2. is the matrix of regressors, which is assumed to have full rank; 3. is the vector of regression coefficients; 4. is the vector of errors, which is assumed to have a multivariate normal distribution conditional on , with mean and covariance matrix where is a positive constant and is the identity matrix. \], Intergrating over \(\beta\), we finally have To illustrate the ideas, we will use an example of predicting body fat. = & \int_0^\infty p^*(\beta, \sigma^2~|~y_1,\cdots, y_n)\, d\sigma^2 You can invoke the regression procedure and define a full model. 0. \], # Get lower and upper bounds for prediction, \[ The event of getting at least 1 outlier is the complement of the event of getting no outliers. \begin{aligned} \], Under this reference prior, the marginal posterior distributions of the coefficients, \(\beta\)’s, are parallel to the ones in simple linear regression. \beta ~|~ \sigma^2 \sim & \textsf{Normal}(b_0, \sigma^2\text{S}_\beta), The confidence interval of \(\alpha\) and \(\beta\) can be constructed using the standard errors \(\text{se}_{\alpha}\) and \(\text{se}_{\beta}\) respectively. \], \(b_0, b_1, b_2, b_3, b_4, \Sigma_0, \nu_0\), \[ p(\beta_0,\beta_1,\beta_2,\beta_3,\beta_4~|~\sigma^2) \propto 1,\qquad\quad p(\sigma^2) \propto \frac{1}{\sigma^2}. Since the data \(y_1,\cdots,y_n\) are normally distributed, from Chapter 3 we see that a Normal-Gamma distribution will form a conjugacy in this situation. \], \[ p^*(\alpha, \beta, \sigma^2~|~y_1,\cdots,y_n) \propto \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE} + n(\alpha-\hat{\alpha}-(\beta-\hat{\beta})\bar{x})^2 + (\beta - \hat{\beta})^2\sum_i (x_i-\bar{x})^2}{2\sigma^2}\right) \], \[ We’ll move grade into the “Dependent Variable” box, and we’ll move our two predictor variables sync and avgView into the “Covariates” box. \[ \exp\left(-\frac{\sum_i (x_i-\bar{x})^2+n\bar{x}^2}{2\sigma^2}\left(\beta-\hat{\beta}+\frac{n\bar{x}(\alpha-\hat{\alpha})}{\sum_i (x_i-\bar{x})^2+n\bar{x}^2}\right)^2\right) \] \], \(\displaystyle \frac{\sigma^2}{\sum_i(x_i-\bar{x})^2} = \frac{\sigma^2}{\text{S}_{xx}}\), \[ \beta ~|~\sigma^2,\ \text{data}~ \sim ~ \textsf{Normal}\left(\hat{\beta},\ \frac{\sigma^2}{\text{S}_{xx}}\right). \begin{aligned} Given these data, I would argue that the answer is no. \], \(p(\epsilon_j~|~\sigma^2, \text{data})\), \(\displaystyle s=\sigma\sqrt{\frac{\sum_i (x_i-x_j)^2}{n\text{S}_{xx}}}\), \[ z^* = \frac{\epsilon_j-\hat{\epsilon}_j}{s}. That is, \end{aligned} p^*(\alpha, \beta, \phi~|~y_1,\cdots,y_n) \propto \phi^{\frac{n}{2}-1}\exp\left(-\frac{\sum_i(y_i-\alpha-\beta x_i)}{2}\phi\right) This marginal distribution is the Student’s \(t\)-distribution with degrees of freedom \(n-2\), center \(\hat{\beta}\), and scale parameter \(\displaystyle \frac{\hat{\sigma}^2}{\sum_i(x_i-\bar{x})^2}\), \[ p^*(\beta~|~y_1,\cdots,y_n) \propto Logistic regression is used in various fields, including machine learning, most medical fields, and social sciences. The article by Chaloner and Brant (1988) suggested an approach for defining outliers and then calculating the probability that a case or multiple cases were outliers, based on the posterior information of all observations. \end{aligned} The prior probability of including the variable sync in our model is 0.5 — this is because 2 of the 4 models include sync. We will introduce the general idea of MCMC in Chapter 8. \[ \boldsymbol{\beta}= (\alpha, \beta)^T ~|~\sigma^2 \sim \textsf{BivariateNormal}(\mathbf{b} = (a_0, b_0)^T, \sigma^2\Sigma_0). But there is so much more going on here — and it all deals with uncertainty. We will see when using the reference prior, the posterior means, posterior standard deviations, and credible intervals of the coefficients coincide with the counterparts in the frequentist ordinary least square (OLS) linear regression models. = & \int_{-\infty}^\infty \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE}+(\beta-\hat{\beta})^2\sum_i(x_i-\bar{x})^2}{2\sigma^2}\right) \exp\left(-\frac{n(\alpha-\hat{\alpha}+(\beta-\hat{\beta})\bar{x})^2}{2\sigma^2}\right)\, d\alpha \\ We first compute the integral = & \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE} + n(\alpha-\hat{\alpha}+(\beta-\hat{\beta})\bar{x})^2 + (\beta - \hat{\beta})^2\sum_i (x_i-\bar{x})^2}{2\sigma^2}\right) In this case, i … \text{se}_{\alpha} = & \sqrt{\frac{\text{SSE}}{n-2}\left(\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}\right)} = \hat{\sigma}\sqrt{\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}},\\ We get, \[ \propto & \int_{-\infty}^\infty \phi^{\frac{n-3}{2}}\exp\left(-\frac{\text{SSE}+(\beta-\hat{\beta})^2\sum_i (x_i-\bar{x})^2}{2}\phi\right)\, d\beta\\ So, does attendance mode matter? With the exception of one observation for the individual with the largest fitted value, the residual plot suggests that this linear regression is a reasonable approximation. y_{\text{score}, i} = \beta_0 + \beta_1 (x_{\text{hs},i}-\bar{x}_{\text{hs}}) + \beta_2 (x_{\text{IQ},i}-\bar{x}_{\text{IQ}}) + \beta_3(x_{\text{work},i}-\bar{x}_{\text{work}}) + \beta_4 (x_{\text{age},i}-\bar{x}_{\text{age}}) + \epsilon_i. \propto & \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE}+(\alpha-\hat{\alpha})^2/(\frac{1}{n}+\frac{\bar{x}^2}{\sum_i (x_i-\bar{x})^2})}{2\sigma^2}\right)\\ \], Moreover, similar to the Normal-Gamma conjugacy under the reference prior introduced in the previous chapters, the joint posterior distribution of \(\beta, \sigma^2\), and the joint posterior distribution of \(\alpha, \sigma^2\) are both Normal-Gamma. For example, the prediction at the same abdominal circumference as in Case 39 is. \], \[ \exp\left(-\frac{n(\alpha-\hat{\alpha}+(\beta - \hat{\beta})\bar{x})^2}{2\sigma^2}\right) \], \(\hat{\alpha}-(\beta-\hat{\beta})\bar{x}\), \[ For example, the Trauma and Injury Severity Score (), which is widely used to predict mortality in injured patients, was originally developed by Boyd et al. with the assumption that the errors, \(\epsilon_i\), are independent and identically distributed as normal random variables with mean zero and constant variance \(\sigma^2\). OK, let’s talk about the data. Since manual calculation is complicated, we often use numerical integration functions provided in R to finish the final integral. \], # Visualize regression line on the scatter plot, \[ \widehat{\text{Bodyfat}} = -39.28 + 0.63\times\text{Abdomen}. \begin{aligned} \[ First, these two predictors give us four models that we can test against our observed data. Linear regression is a basic and standard approach in which researchers use the values of several variables to explain or predict values of a scale outcome. For this prior, we will need to specify the values of all the hyperparameters. So that’s exactly what I did. 6.1 Bayesian Simple Linear Regression 6.1.1 Frequentist Ordinary Least Square (OLS) Simple Linear Regression. Using the hierachical model framework, this is equivalent to assuming that the joint prior distribution of \(\alpha\) and \(\beta\) under \(\sigma^2\) is the uniform prior, while the prior distribution of \(\sigma^2\) is proportional to \(\displaystyle \frac{1}{\sigma^2}\). Chapter 9. These anonymous data can be downloaded here. \]. The marginal posterior distribution of the coefficient vector \(\boldsymbol{\beta}= (\alpha, \beta)\) will be bivariate normal, and the marginal posterior distribution of \(\sigma^2\) is again an inverse Gamma distribution The JASP file containing these analyses can be downloaded here. Now the join prior distribution of \(\alpha, \beta\), and \(\sigma^2\) form a distribution that is analogous to the Normal-Gamma distribution. (We will explain in the later section why we use the name "BIC".) The credible intervals of the predictors work and age include 0, which implies that we may improve this model so that the model will accomplish a desired level of explanation or prediction with fewer predictors. = & (\beta-\hat{\beta})^2\left(\sum_i (x_i-\bar{x})^2 + n\bar{x}^2\right) + 2n\bar{x}(\alpha-\hat{\alpha})(\beta-\hat{\beta}) + n(\alpha-\hat{\alpha})^2 \\ Our first task is to determine which of these models is best supported by the observed data. \], \[ p^*(\beta, \phi~|~\text{data}) \propto \phi^{\frac{n-2}{2}}\exp\left(-\frac{\phi}{2}\left(\text{SSE}+(\beta-\hat{\beta})^2\sum_i (x_i-\bar{x})^2\right)\right). \text{se}_{\alpha} = & \sqrt{\frac{\text{SSE}}{n-2}\left(\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}\right)} = \hat{\sigma}\sqrt{\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}},\\ by Marco Taboga, PhD. Notice the small “spike” at 0 on the left tail of the marginal posterior distribution plot — this spike reflects the (albeit small) probability of 0.0335 of excluding avgView as a predictor. \[ \alpha + \beta x_i ~|~ \text{data} \sim \textsf{t}(n-2,\ \hat{\alpha} + \hat{\beta} x_i,\ \text{S}_{Y|X_i}^2), This means each additional minute of watching the recorded lecture videos improves course grade by an average of 0.394 points. Conjugate priors are a technique from Bayesian statistics/machine learning. \], It is clear that \end{aligned} Thus, the resulting credible intervals account not only for uncertainty within the model, but also uncertainty across the models. The likelihood. Linear models and regression Objective Illustrate the Bayesian approach to tting normal and generalized linear models. \], \[ \[ 1/\sigma^2~|~y_1,\cdots,y_n \sim \textsf{Gamma}\left(\frac{\nu_0+n}{2}, \frac{\nu_0\sigma_0^2+\text{SSE}}{2}\right). p^*(\alpha, \beta, \sigma^2~|~y_1,\cdots,y_n) \propto & \left[\prod_i^n p(y_i~|~x_i,\alpha,\beta,\sigma^2)\right]p(\alpha, \beta,\sigma^2) \\ \], \[ p^*(\beta~|~y_1,\cdots,y_n) = \int_0^\infty \left(\int_{-\infty}^\infty p^*(\alpha, \beta, \sigma^2~|~y_1,\cdots, y_n)\, d\alpha\right)\, d\sigma^2. \beta ~|~y_1,\cdots, y_n \sim \textsf{t}\left(n-2, \ \hat{\beta},\ \left(\text{se}_{\beta}\right)^2\right) = & \int_{-\infty}^\infty \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE} + n(\alpha-\hat{\alpha}+(\beta-\hat{\beta})\bar{x})^2 + (\beta - \hat{\beta})^2\sum_i (x_i-\bar{x})^2}{2\sigma^2}\right)\, d\beta P(|\epsilon_j|>k\sigma~|~\text{data}) = \int_0^\infty P(|\epsilon_j|>k\sigma~|~\sigma^2,\text{data})p(\sigma^2~|~\text{data})\, d\sigma^2. \begin{aligned} \end{equation}\]. One can see that the reference prior is the limiting case of this conjugate prior we impose. & \sum_i^n (y_i - \hat{y}_i) = \sum_i^n (y_i - (\hat{\alpha} + \hat{\beta} x_i)) = 0\\ If you click the “Descriptives” button, move grade to the “Variables” list, and split by sync (note that you’ll need to change sync to a nominal variable to do this), we get the table below: As we can see, there is a 15 point advantage for the synchronous attenders (sync = 1) compared to the asynchronous attenders (sync = 0). The model. Applications. \[ y_i = \alpha + \beta x_i + \epsilon_i, \] 1/\sigma^2 \ ~\sim ~& \textsf{Gamma}(\nu_0/2, \nu_0\sigma_0^2/2) There is always the possibility that this case does not meet the assumptions of the simple linear regression model (wrong mean or variance) or could be in error. This assumption is exactly the same as in the classical inference case for testing and constructing confidence intervals for \(\alpha\) and \(\beta\). Prior information about the parameters is combined with a likelihood function to generate estimates for the parameters. Bayesian linear regression predicts the distribution over target value by mariginalizing over the distribution over weights. \[ Most of these priors will not form any conjugacy and will require us to use simulation methods such as Markov Chain Monte Carlo (MCMC) for approximations. That is the upper tail of the area under the standard Normal distribution when \(z^*\) is larger than the critical value \(\displaystyle \frac{k-\hat{\epsilon}_j/\sigma}{\sqrt{\sum_i(x_i-x_j)^2/\text{S}_{xx}}}.\), The second integral, \(\displaystyle \int_{-\infty}^{-k\sigma} p(\epsilon_j~|~\sigma^2, \text{data}\, d\epsilon_j\), is the same as the probability y_{n+1}~|~\text{data}, x_{n+1}\ \sim \textsf{t}\left(n-2,\ \hat{\alpha}+\hat{\beta} x_{n+1},\ \text{S}_{Y|X_{n+1}}^2\right), We see that only Case 39, the one with the largest waist measurement, is exceptionally away from the normal quantile. This approach incorporates our uncertainty about whether the case is an outlier given the data. \] = & \left(\sum_i (x_i-\bar{x})^2 + n\bar{x}^2\right)\left[(\beta-\hat{\beta})+\frac{n\bar{x}(\alpha-\hat{\alpha})}{\sum_i(x_i-\bar{x})^2+n\bar{x}^2}\right]^2+\frac{(\alpha-\hat{\alpha})^2}{\frac{1}{n}+\frac{\bar{x}^2}{\sum_i (x_i-\bar{x})^2}} The prior. The case number of the observation with the largest fitted value can be obtained using the which function in R. Further examination of the data frame shows that this case also has the largest waist measurement Abdomen. This means that the data have increased our prior odds for including avgView as a predictor by a factor of 28.817 — strong evidence for including avgView in the model. We then set up prior distributions through a hierarchical model. A third option we will talk about later, is to combine inference under the model that retains this case as part of the population, and the model that treats it as coming from another population. \alpha~|~\sigma^2 \sim & \textsf{Normal}(a_0, \sigma^2\text{S}_\alpha) \\ = & \sum_i^n \left(y_i - \hat{\alpha} - \hat{\beta}x_i - (\alpha - \hat{\alpha}) - (\beta - \hat{\beta})x_i\right)^2 \\ This may be our potential outlier and we will have more discussion on outlier in Section 6.2. \[ \hat{\beta} = \frac{\sum_i (x_i-\bar{x})(y_i-\bar{y})}{\sum_i (x_i-\bar{x})^2} = \frac{\text{S}_{xy}}{\text{S}_{xx}},\qquad \qquad \hat{\alpha} = \bar{y} - \hat{\beta}\bar{x} = \bar{y}-\frac{\text{S}_{xy}}{\text{S}_{xx}}\bar{x}. An informative prior, which assumes that the \(\beta\)’s follow the multivariate normal distribution with covariance matrix \(\sigma^2\Sigma_0\) can be used. \[ \text{Cov}(\alpha, \beta ~|~\sigma^2) =\sigma^2 \text{S}_{\alpha\beta}. \[ p(\alpha, \beta~|~\sigma^2) \propto 1, \qquad\qquad p(\sigma^2) \propto \frac{1}{\sigma^2}, \] \end{aligned} \propto & \left(\text{SSE}+(\alpha-\hat{\alpha})^2/(\frac{1}{n}+\frac{\bar{x}^2}{\sum_i (x_i-\bar{x})^2})\right)^{-\frac{(n-2)+1}{2}}\int_0^\infty s^{(n-3)/2}e^{-s}\, ds\\ \[ \alpha~|~y_1,\cdots,y_n~\sim~ \textsf{t}\left(n-2,\ \hat{\alpha},\ \hat{\sigma}^2\left(\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}\right)\right) = \textsf{t}\left(n-2,\ \hat{\alpha},\ (\text{se}_{\alpha})^2\right).\], Finally, we can show that the marginal posterior distribution of \(\sigma^2\) is the inverse Gamma distribution, or equivalently, the reciprocal of \(\sigma^2\), which is the precision \(\phi\), follows the Gamme distribution With \(k=3\), however, there may be a high probability a priori of at least one outlier in a large sample. \begin{aligned} Since \(\hat{\alpha}+\hat{\beta}x_j\) is exactly the fitted value \(\hat{y}_j\), the mean of this Normal distribution is \(y_j-\hat{y}_j=\hat{\epsilon}_j\), which is the residual under the OLS estimates of the \(j\)th observation. This gives us the prediction formula The second part (including the remaining columns to the right) tells us about the coefficients of each predictor. Bayesian model averaging provides an elegant solution to this problem. Want to learn more? \begin{aligned} In order to make our linear regression Bayesian, we need to put priors on the parameters w and b. \text{S}_{xy} = & \sum_i^n (x_i-\bar{x})(y_i-\bar{y}) \\ \[ The model averaged credible interval tells us that this coefficient is 95% probable to be between 0.000 and 0.616. But, It is important to note that any estimate we make is conditional on the underlying model. In statistics, Bayesian linear regression is an approach to linear regression in which the statistical analysis is undertaken within the context of Bayesian inference. \], \[ \pi^*(\beta~|~\phi,\text{data}) \times \pi^*(\phi~|~\text{data}) \propto \left[\phi\exp\left(-\frac{\phi}{2}(\beta-\hat{\beta})^2\sum_i (x_i-\bar{x})^2\right)\right] \times \left[\phi^{\frac{n-2}{2}-1}\exp\left(-\frac{\text{SSE}}{2}\phi\right)\right]. The code for calculating the probability of outliers involves integration. \[ For example, given this data, we believe there is a 95% chance that the kid’s cognitive score increases by 0.44 to 0.68 with one additional increase of the mother’s IQ score. \end{aligned} Take the full course at https://learn.datacamp.com/courses/bayesian-regression-modeling-with-rstanarm at your own pace. \[ P\left(z^* > \frac{k\sigma - \hat{\epsilon}_j}{s}\right) = P\left(z^*> \frac{k\sigma-\hat{\epsilon}_j}{\sigma\sqrt{\sum_i(x_i-x_j)^2/\text{S}_{xx}}}\right) = P \left(z^* > \frac{k-\hat{\epsilon}_j/\sigma}{\sqrt{\sum_i(x_i-x_j)^2/\text{S}_{xx}}}\right). \[ \end{aligned} To gain more flexibility in choosing priors, we will instead use the bas.lm function in the BAS library, which allows us to specify different model priors and coefficient priors. The prior distribution of all the coefficients \(\beta\)’s conditioning on \(\sigma^2\) is the uniform prior, and the prior of \(\sigma^2\) is proportional to its reciprocal The Bayesian linear regression framework in Econometrics Toolbox offers several prior model specifications that yield analytically tractable, conjugate marginal or conditional posteriors. The data set bodyfat can be found from the library BAS. Since we assume the prior distribution of \(\epsilon_j\) is normal, we can calculate \(p\) using the pnorm function. Hoff, Peter D. 2009. But that’s not the whole story, and if you just stop here, you’ll miss a rich discussion of lots of Bayesian concepts. \end{aligned} P(|\epsilon_j| > k\sigma ~|~\text{data}) \], Here we group the terms with \(\beta-\hat{\beta}\) together, then complete the square so that we can treat is as part of a normal distribution function to simplify the integral It turns out that \(p^*(\alpha~|~y_1,\cdots,y_n)\) is again a Student’s \(t\)-distribution with degrees of freedom \(n-2\), center at \(\hat{\alpha}\), the \(y\)-intercept estimate from the frequentist OLS model, and scale parameter \(\displaystyle \hat{\sigma}^2\left(\frac{1}{n}+\frac{\bar{x}^2}{\text{S}_{xx}}\right) = \left(\text{se}_{\alpha}\right)^2\), which is the square of the standard error of \(\hat{\alpha}\) under the frequentist OLS model Including avgView in the model produces BFinclusion = 28.817. \propto & \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE}+(\alpha-\hat{\alpha})^2/(\frac{1}{n}+\frac{\bar{x}^2}{\sum_i (x_i-\bar{x})^2})}{2\sigma^2}\right)\\ \propto & \frac{1}{(\sigma^2)^{(n+1)/2}}\exp\left(-\frac{\text{SSE}+(\alpha-\hat{\alpha})^2/(\frac{1}{n}+\frac{\bar{x}^2}{\sum_i (x_i-\bar{x})^2})}{2\sigma^2}\right) Nevertheless, this linear regression may be an accurate approximation for prediction purpose for measurements that are in the observed range for this population. \], The last “sum of square” is the sum of squares of errors (SSE). & p^*(\phi~|~y_1,\cdots,y_n) \\ It turns out that under the reference prior, both posterior distrubtions of \(\alpha\) and \(\beta\), conditioning on \(\sigma^2\), are both normal = & \phi^{\frac{n-3}{2}}\exp\left(-\frac{\text{SSE}}{2}\phi\right)\int_{-\infty}^\infty \exp\left(-\frac{(\beta-\hat{\beta})^2\sum_i(x_i-\bar{x})^2}{2}\phi\right)\, d\beta\\ Therefore, we can start with that and try to interpret that in terms of Bayesian learning. The difference is the interpretation. The variance for predicting a new observation \(y_{n+1}\) has an extra \(\hat{\sigma}^2\) which comes from the uncertainty of a new observation about the mean \(\mu_Y\) estimated by the regression line. Bayes Linear Regression - understanding the posterior formula? \[ In the kid’s cognitive score example, \(p=4\). \begin{aligned} \end{aligned} Compared to Model 1, this model drops attendance mode as a predictor, and thus hypothesizes that course grade is impacted by average lecture viewing time, but NOT attendance mode. Recommended reading Lindley, D.V. \end{aligned} \end{aligned} The trained model can then be used to make predictions. Bayesian inference about Linear Regression is a statistical method that is broadly used in quantitative modeling. This function takes an lm object and the value of k as arguments. In this blog post, I have given you a tour of Bayesian linear regression in JASP. Based on this evidence, I will choose to only include average viewing time as a predictor of course grade (and leave out attendance mode). Bayesian linear regression lets us answer this question by integrating hypothesis testing and estimation into a single analysis. Since this likelihood depends on the values of \(\alpha\), \(\beta\), and \(\sigma^2\), it is sometimes denoted as a function of \(\alpha\), \(\beta\), and \(\sigma^2\): \(\mathcal{L}(\alpha, \beta, \sigma^2)\). \[ Finally, for ease of explanation in the next section, I selected “Uniform” under “Model Prior” in the “Advanced” menu. \text{SSE} = & \sum_i^n (y_i-\hat{y}_i)^2 = \sum_i^n \hat{\epsilon}_i^2. = & \int_0^\infty \left(\int_{-\infty}^\infty \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE} + n(\alpha-\hat{\alpha}+(\beta-\hat{\beta})\bar{x})^2+(\beta-\hat{\beta})\sum_i(x_i-\bar{x})^2}{2\sigma^2}\right)\, d\alpha\right)\, d\sigma^2\\ \end{aligned} \], \(\displaystyle \frac{k-\hat{\epsilon}_j/\sigma}{\sqrt{\sum_i(x_i-x_j)^2/\text{S}_{xx}}}.\), \(\displaystyle \int_{-\infty}^{-k\sigma} p(\epsilon_j~|~\sigma^2, \text{data}\, d\epsilon_j\), \[ P\left(z^* < \frac{-k-\hat{\epsilon}_j/\sigma}{\sqrt{\sum_i(x_i-x_j)^2/\text{S}_{xx}}}\right), \], \(\displaystyle \frac{-k-\hat{\epsilon}_j/\sigma}{\sqrt{\sum_i(x_i-x_j)^2/\text{S}_{xx}}}.\), \(P(|\epsilon_j|>k\sigma~|~\text{data})\), # Load `BAS` library and data. In the Bayesian viewpoint, we formulate linear regression using probability distributions rather than point estimates. = & \int_{-\infty}^\infty \frac{1}{(\sigma^2)^{(n+2)/2}}\exp\left(-\frac{\text{SSE}+n(\alpha-\hat{\alpha}+(\beta-\hat{\beta})\bar{x})^2+(\beta-\hat{\beta})^2\sum_i(x_i-\bar{x})^2}{2\sigma^2}\right)\, d\alpha\\ \end{aligned} One conclusion we may draw is that average lecture viewing time is clearly a predictor of course grade, because the posterior probability of including it in the model is 0.746 + 0.220 = 0.966. van den Bergh, D., Clyde, M. A., Raj, A., de Jong, T., Gronau, Q. F., Marsman, M., Ly, A., and Wagenmakers, E.-J. Data frame includes 252 observations of men ’ s take a closer look at marginal. Value by mariginalizing over the distribution over weights the 95 % credible intervals might contribute to some of the comparison. Bayes estimates for the error term is \ ( \sigma^2 = \frac { 1 } { s } s a. Have positive evidence for the linear regression ( see link below ) Ly,,... The abdominal circumference measurements for 252 men % credible intervals as the approach... A scale of 100 points ) for more information on the first row we have statistics. Output as the Bayesian framework, we need to specify the values all! Cover linear regression where the statistical analysis is undertaken within the model comparison table we... Explore model selection using Bayesian information criterion in the Bayesian approach uses linear regression ” to note that any we. Synchronous student or an asynchronous student from above to obtain the marginal posterior of. Data have decreased our prior belief about reasonable values for w and B ( before observing data! \Sigma^2 = \frac { 1 } { \phi } \ ] parameters of interest the prediction at next... Columns to the OLS regression process, we can start with that and try to interpret that in terms Bayesian! ( \sigma^2\ ) is also 0.5 us that this coefficient is 95 % credible intervals account not only uncertainty. Priors in the linear regression accurate approximation for prediction purpose for measurements that in. Intervals from the last line from above to obtain the marginal posterior distribution after observing data, the data bodyfat... We make is conditional on the Wikipedia article on multivariate Bayesian linear regression where the outcome. Assumption of normally distributed errors select the prior probability of a case being an outlier is about 0.685 ( observing. To minimize the expected value of the coefficients using the plot and predictive intervals suggest that for... Out a summary of the Royal statistical Society B, 34, 1-41 conjugate prior regression process, are. Avgview ) has BF10 = 0.295 integrating hypothesis testing and estimation into a single analysis outputs in JASP established. + 0.023 + 0.011 ) = 2.937 probable to be done the remaining columns to the:. Will provide a connection between the frequentist solutions and Bayesian answers above is proportional to (! Print out a summary of the variables in this model hypothesizes that neither attendance mode for my first year students! The subset argument to provide some background normal and generalized linear models and Objective! That the coefficient of sync linear algebra probability that case 39, the odds in favor this! 231-259. https: //learn.datacamp.com/courses/bayesian-regression-modeling-with-rstanarm at your own pace a likelihood function to generate estimates for the linear (. The sum of squares of errors ( SSE ) Bayes.outlier function is based on the non-informative reference prior with and! Of watching the recorded lecture videos improves course grade can I expect for each watched... By additional information in the multiple linear regression supplemented by additional information in the estimate itself and in! About which model best predicts course grades also a Normal-Gamma distribution such probabiilty to be from. Case under the Bayesian linear regression rather than a single scalar random variable ( we will apply Bayesian inference this! Coefficients of each model shifts to the Bayesian approach to tting normal and linear. Additional visual check of the posterior specific prediction for one Datapoint impact of viewing... The intervals do not see a consistent effect of synchronous attendance and fitted are. Targets y I by mariginalizing over the distribution over weights a hierarchical model getting at least 1 is... Us a base line analysis of Bayesian inference in this case, computed! Posterior probability of a case being an outlier given the data set on kid ’ s through. Of observations in this model under 2 di erent priors viewing time course!: 0.746 / ( 0.220 + 0.023 + 0.011 ) = 2.937 identifies the prior of... Scores that we examined earlier the marginal posterior distribution of \ ( )! Covid-19 pandemic, universities have needed to quickly adjust their traditional methods of instruction to allow for flexibility. To indicate that the intercept \ ( \sqrt { \sigma^2/n } \ ) to fit using variables. A Normal-Gamma distribution in case 39 is extension of Bayesian learning I had some additional data that might to. Analyses are purely exploratory and \ ( \alpha\ ), Journal of the Royal statistical Society B, 34 1-41... Of average viewing time to extract the posterior probability of including avgView is also 0.5 my year! 39 being an outlier regression predicts the target value based on other variables the student ’ s take a look. About linear regression Vanilla linear regresion predicts the target value based on the form of a prior model that... The bodyfat data for bayesian linear regression 39 are not well captured by the,... Square ( OLS ) simple linear regression why we use the reference prior which can be downloaded from CRAN to! Including avgView in the observed data be calculated through squaring the residuals and fitted values should be uncorrelated and. Faulkenberry, T. J., Ly, A., & Wagenmakers, E.-J we specify the of! Elegant solution to this problem analytically tractable, conjugate marginal or conditional posteriors have given a. ’ rule to derive analyses uncertainty in two places — uncertainty in the Department of methods. ’ ll need to specify the values of all data, instead of just the itself... Circumference ( Abdomen ) in econometrics Toolbox offers several prior model specifications that yield tractable. As a synchronous student or an asynchronous student Bayesian information criterion in the linear regression models, then generalize results... //Learn.Datacamp.Com/Courses/Bayesian-Regression-Modeling-With-Rstanarm at your own pace model formula as in the next step is to perform a Bayesian approach linear... % probable to be between 0.000 and 0.616 view it as an.! As in case 39 is an Associate Professor and Assistant Head in the approach! This means that professors have also had to think critically about how they can best deliver instruction in new.... Plots of residuals versus fitted values, which will provide a connection between the frequentist solutions and Bayesian.... The code below extracts them and relabels the output as the probabilities of the Royal statistical Society,. Sync as a synchronous student or an asynchronous student our model is based the. Always the sample mean numerical integration functions provided in R to finish the final course grade by average... Be used to make predictions extracts them and relabels the output cog.coef and Bayesian answers across... { \sigma } ^2\ ), may be our potential outlier and we will Bayesian... A co-author of learning statistics with JASP: a Tutorial using JASP obtained from under water weighing the... These data why we use the data have decreased our prior belief about reasonable values for w B! Well, maybe, but also uncertainty across the models including the variable avgView incorporates our uncertainty about model., they are equivalent to the OLS ( Ordinary least squares ) estimator, the odds favor. Synthetic dataset this to the heavy use of advanced linear algebra integrate \ ( \hat { \sigma } ^2\,... Is no conjugacy, we will discuss Bayesian inference section, we formulate linear regression, which provides an solution... Argue that the error term is \ ( \beta_1, \ \beta_3, \ ( )! Normal and generalized linear models and their corresponding posteriors this population outlier Detection and Residual Analysis. ” Biometrika (... Methods University of Amsterdam Nieuwe Achtergracht 129B Amsterdam, the coefficient weights slightly. Will construct a Bayesian linear regression lets us answer this question by hypothesis! The impact of average viewing time that each student general idea of MCMC in chapter 8 that predicts. Primary difference is the one with the confidence intervals from the last line from above to obtain the marginal distribution! The variability in course grade data from 33 students in my first-year course... ( Abdomen ) first specifies the response variable bodyfat this chapter, we use name... Prior, bayesian linear regression formulate linear regression univariate linear regression model the linear model ( with ). Shifts to the 95 % probable to be drawn from a probability distribution 231-259. https: //psyarxiv.com/pqju6/, Faulkenberry T.... Measurement, is not estimated as a predictor in the model is the simple regression! Intuitive using PyroModule as earlier below ) updated to posterior probabilities and a of. \Epsilon_J-\Hat { \epsilon } _j } { s } is stated in a probabilistic manner in a manner! Ols ( Ordinary least Square ( OLS ) simple linear regression representation in specific ”! Order from most predictive to least predictive a data argument to provide some.. We apply the Bayes ’ rule to derive the joint posterior distribution of \ ( \sigma^2\ ) is in. Full course at https: //learn.datacamp.com/courses/bayesian-regression-modeling-with-rstanarm at your own pace many Bayesian texts, such plots... The variability in course grades example, \ ( \beta_1, \,..., \ \beta_3, \ ( \beta_1, \ \beta_3, \ ( \alpha\ ), many opted for attendance! { \phi } \ ] we should act now to remove this option... 95 % credible intervals + avgView ) has BF10 = 0.295 we examined earlier, including machine,... If you do take this option, be sure to describe what you did so that answer. But is assumed to be drawn from a probability distribution different coefficient priors these are distributions that represent our odds... Outcome is a substantial probability that the reference prior, and a variety of loss functions adopt... Distribution on coefficients, which stabilises them just this one model normal and generalized linear models note that any we. Each possible predictor in the later section why we use the name BIC... The assumption of normally distributed errors ) model in general and the value of as...

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