# Shrinking Regression Coefficients Fits a Bayesian regression model with a shrinkage prior on the coefficient. The model is [REMOVE_ME]$$ %y_i \sim N(x_i \beta, \sigma^2) \\ %1 / \sigma^2 \sim Gamma(df/2, ss/2) \\ %g_1(\beta) \sim N(b1, v1) \\ %g_2(\beta) \sim N(b2, v2) \\ %\dots%y[i] ~ N(x[i,] * beta, sigma^2)%1 / sigma^2 ~ Gamma(df/2, ss/2)%group(beta, 1) ~ N(b1, v1)%group(beta, 2) ~ N(b2, v2)%... [REMOVE_ME_2]$$ In this notation, c("$group(beta, k) ~\n$", "$ N(bk, vk)$") indicates that the subset of coefficients in group k are a priori independent draws from the specified normal distribution. In addition, each subset-level prior may include a hyperprior, in which case the subset-level prior parameters will be updated as part of the MCMC. The hyperprior has the form of independent priors on the mean and precision parameters: [REMOVE_ME]$$ %b_i ~ N(prior.mean, prior.variance) \\ %1 / v_i ~ Chisq(df, guess.at.sd). \\ %%bi ~ N(prior.mean, prior.variance)1.0 / vi ~ Chisq(df, guess.at.sd). [REMOVE_ME_2]$$ 1.1 ## Description Fits a Bayesian regression model with a shrinkage prior on the coefficient. The model is $$ %y_i \sim N(x_i \beta, \sigma^2) \\ %1 / \sigma^2 \sim Gamma(df/2, ss/2) \\ %g_1(\beta) \sim N(b1, v1) \\ %g_2(\beta) \sim N(b2, v2) \\ %\dots%y[i] ~ N(x[i,] * beta, sigma^2)%1 / sigma^2 ~ Gamma(df/2, ss/2)%group(beta, 1) ~ N(b1, v1)%group(beta, 2) ~ N(b2, v2)%... $$ In this notation, c("$group(beta, k) ~\n$", "$ N(bk, vk)$") indicates that the subset of coefficients in group k are a priori independent draws from the specified normal distribution. In addition, each subset-level prior may include a hyperprior, in which case the subset-level prior parameters will be updated as part of the MCMC. The hyperprior has the form of independent priors on the mean and precision parameters: $$ %b_i ~ N(prior.mean, prior.variance) \\ %1 / v_i ~ Chisq(df, guess.at.sd). \\ %%bi ~ N(prior.mean, prior.variance)1.0 / vi ~ Chisq(df, guess.at.sd). $$ ```r ShrinkageRegression(response, predictors, coefficient.groups, residual.precision.prior = NULL, suf = NULL, niter, ping = niter / 10, seed = NULL) CoefficientGroup(indices, mean.hyperprior = NULL, sd.hyperprior = NULL, prior = NULL) ``` ## Arguments - `response`: The numeric vector of responses. - `predictors`: The matrix of predictors, including an intercept term, if desired. - `coefficient.groups`: A list of objects of type `CoefficientGroup`, defining the pattern in which the coefficients should be shrunk together. Each coefficient must belong to exactly one `CoefficientGroup`. - `residual.precision.prior`: An object of type `SdPrior` describing the prior distribution of the residual standard deviation. - `suf`: An object of class `RegressionSuf` containing the sufficient statistics for the regression model. If this is `NULL` then it will be computed from `response` and `predictors`. If it is supplied then `response` and `predictors` are not used and can be left missing. - `niter`: The desired number of MCMC iterations. - `ping`: The frequency with which to print status updates. - `seed`: The integer-valued seed (or `NULL`) to use for the C++ random number generator. - `indices`: A vector of integers giving the positions of the regression coefficients that should be viewed as exchangeable. - `mean.hyperprior`: A `NormalPrior` object describing the hyperprior distribution for the average coefficient. - `sd.hyperprior`: An `SdPrior` object describing the hyperprior distribution for the standard deviation of the coefficients. - `prior`: An object of type `NormalPrior` giving the initial value of the distribution describing the collection of coefficients in this group. If either hyperprior is `NULL` then the corresponding prior parameter will not be updated. If both hyperpriors are non-`NULL` then this parameter can be left unspecified. ## Returns `ShrinkageRegression` returns a list containing MCMC draws from the posterior distribution of model parameters. Each of the following is a matrix, with rows corresponding to MCMC draws, and columsn to distinct parameters. * coefficients: regression coefficients. * residual.sd: the residual standard deviation from the regression model. * group.means: The posterior distribution of the mean of each coefficient group. If no mean hyperprior was assigned to a particular group, then the value here will be a constant (the values supplied by the `prior` argument to `CoefficientGroup` for that group). * group.sds: The posterior distribution of the standard deviation of each coefficient group. If no sd.hyperprior was assigned to a particular group, then the value here will be a constant (the values supplied by the `prior` argument to `CoefficientGroup` for that group). `CoefficientGroup` is a configuration utility used to define which coefficients should be shrunk together. It returns an object (list) formatted in the manner expected by `ShrinkageRegression`. ## Author(s) Steven L. Scott ## Examples ```r b0 <- -1 b1 <- rnorm(20, 3, .2) b2 <- rnorm(30, -4, 7) nobs <- 10000 beta <- c(b0, b1, b2) X <- cbind(1, matrix(rnorm(nobs * (length(beta) - 1)), nrow = nobs, ncol = length(beta) - 1)) y.hat <- X %*% beta y <- rnorm(nobs, y.hat, .5) groups <- list(intercept = CoefficientGroup(1, prior = NormalPrior(0, 100)), first = CoefficientGroup(2:21, mean.hyperprior = NormalPrior(0, 100), sd.hyperprior = SdPrior(.2, 1)), second = CoefficientGroup(22:51, mean.hyperprior = NormalPrior(0, 100), sd.hyperprior = SdPrior(7, 1))) model <- ShrinkageRegression(y, X, groups, residual.precision.prior = SdPrior(.5, 1), niter = 1000) ```

shrinkage.regression function