Problem 2. Consider the problem of predicting the cutting force (y) in a laser assisted mechanical micromachining (LAMM) process. 48 experiments were performed at a lab by varying the nominal depth of cut (0:1), cutting speed (332), laser power (333), and laser location (1:4), and measured the cutting force. The data. is given in LAMM.csv, which is taken from the article: Singh, Joseph, and Melkote (2011, Int. J. Adv. Man. Tech., 53:221230). The cutting force is assumed to follow a Normal distribution with mean: 14$) 2 13033?1 exp {232332 * 5333396434} and constant variance 02. The engineers had a. computer model to predict the cutting forces, which is used for postulating the following prior distribution for the parameters: 130 N W ,31 ~ N(0.,891) 132 ~ N(0.0014, 1) 83 ~ -'V(0.0268, 1) ,34 ~ N(0.0034, 1) Based on some replicated measurements of the cutting force, the following prior was postu- lated for the error variance: 02 N Inverse Gamma(48, 23). Answer the following questions. 1. Obtain the posterior samples of the six parameters (.130, . . . 2.54102) using MCMC. Pro- vide their mean and 95% credible intervals. (use 100,000 samples with 10,000 burn-in in BUGS. 3000 if using pvmc.) 2. Find the predictive distribution of the cutting force at 11:1 2 10,032 = 50,333 = 5, and 334 = 200. Provide the mean and 95% credible intervals. 3. The engineers know that all the coefcients .50, . . . \"84 should be positive for the model to make physical sense. Postulate a new prior to incorporate this information and obtain the posterior samples as in part (1) Compare the mean and credible intervals with those obtained in part (1)