Description
Overview: Experiments with Bayesian Linear Regression
Your goals in this assignment are to explore the role of regularization in linear regression and to
investigate two methods for model selection (evidence maximization and cross validation). In all
your experiments you should report the performance in terms of the mean square error
MSE = 1
N
X
i
(φ(xi)
T w − ti)
2
where the number of examples in the corresponding dataset is N.
Data
Data for this assignment is provided in a zip file pp2data.zip on Canvas.
We have 4 datasets and each dataset comes in 4 files with the training set in train-name.csv the
corresponding labels (regression values) in trainR-name.csv and similarly for test set. We have
two real datasets crime and wine and two artificial datasets artsmall and artlarge. Note that
the train/test splits are fixed and we will not change them in the assignment (in order to save work
and run time).
For the artificial data you can compare the MSE results to the MSE of the hidden true functions
generating the data that give 0.533 (artsmall), and 0.557 (artlarge).
Task 1: Regularization
In this part we use regularized linear regression, i.e., given a dataset, the solution vector w is given
by equation (3.28) of Bishop’s text.
For each of the 4 datasets plot the training set MSE and the test set MSE as a function of the
regularization parameter λ (use integer values in the range 0 to 150). For each dataset it is useful
to put both curves on the same plot. In addition, compare these to the MSE of the true functions
given above.
In your report provide the results/plots and discuss them: Why can’t the training set MSE be used
to select λ? How does λ affect error on the test set? Does this differ for different datasets? How
do you explain these variations?
Note: The experiments in this task tell us which value of λ is best in every case in hindsight. That
is, we need to see the test data and its labels in order to choose λ. This is clearly not a realistic
setting and it does not give reliable error estimates. The next two tasks investigate methods for
choosing λ automatically without using the test set.
Task 2: Model Selection using Cross Validation
In this part we use 10 fold cross validation on the training set to pick the value of λ in the same
range as above, then retrain on the entire train set and evaluate on the test set. To avoid confusion,
the procedure for doing this is explained at the end of the assignment.
Implement this scheme, apply it to the 4 datasets and report the values of λ selected, associated
MSE and the run time. How do the results compare to the best test-set results from part 1 both
in terms of the choice of λ and test set MSE?
Task 3: Bayesian Model Selection
In this part we consider the formulation of Bayesian linear regression with the simple prior w ∼
N (0,
1
α
I). Recall that the evidence function (and evidence approximation) gives a method to pick
the parameters α and β. Referring to Bishop’s book, the solution is given in equations (3.91),
(3.92), (3.95), where mN and SN are given in (3.53) and (3.54). As discussed in class these yield
an iterative algorithm for selecting α and β using the training set. We can then calculate the MSE
on the test set using the MAP (mN ) for prediction.
This scheme is pretty stable and converges in a reasonable number of iterations. You can initialize
α, β to random values in the range [1, 10] and stop the algorithm when the difference in α, β values
is < 0.0001.
Implement this scheme, apply it to the 4 datasets and report the values of α, β, the effective
λ = α/β, the associated MSE and the run time. How do the results compare to the best test-set
results from part 1 both in terms of the choice of λ and test set MSE?
Task 4: Discussion of Results
Tabulate together the values obtained in tasks 1-3 and use this to discuss the following questions.
How do the two model selection methods compare in terms of effective λ, test set MSE and run
time? Do the results suggest conditions where one method is preferable to the other?
Submission
Please submit two separate items via Canvas:
(1) A zip file pp2.zip with all your work and the report. The zip file should include: (1a) Please
write a report on the experiments, their results, and your conclusions as requested above. Prepare
a PDF file with this report. (1b) Your code for the assignment, including a README file that
explains how to run it. When run your code should produce all the results and plots as requested
above. Your code should assume that the data files will have names as specified above and will
reside in sub-directory pp2data/ of the directory where the code is executed. We will read your
code as part of the grading – please make sure the code is well structured and easy to follow (i.e.,
document it as needed). This portion can be a single file or multiple files.
(2) One PDF “printout” of all contents in 1a,1b: call this YourName-pp2-everything.pdf. One
PDF file which includes the report, a printout of the code and the README file. We will use
this file as a primary point for reading your submission and providing feedback so please include
anything pertinent here.
Grading
Your assignment will be graded based on (1) the clarity of the code, (2) its correctness, (3) the
presentation and discussion of the results, (4) our ability to test the code.
Addendum: 10 Fold Cross Validation for Parameter Selection (with
a fixed train/test split)
We have already used cross validation for estimating accuracy in project 1. Cross validation can
also be used for parameter selection if we make sure to use the train set only.
To select parameter a of algorithm A(a) over an enumerated range a ∈ V1, . . . , VK using dataset D
we do the following:
1. Split the data D into 10 disjoint portions.
2. For each value of a in V1, . . . , VK:
(a) For each i in 1 . . . 10
i. Train A(a) on all portions but i and test on i recording the error on portion i
(b) Record the average performance of a on the 10 folds.
3. Pick the value of a with the best average performance.
Now, in the above, D only includes the training set and the parameter is chosen without knowledge
of the test data. We then retrain on the entire train set D using the chosen value and evaluate the
result on the test set.
