You may work in groups on the assignment, but you must write up your own answers in your own words.
Submit your outcomes in one PDF and also submit your codes so that I can implement them on my laptop.
This assignment is due on 20 May. See the course profile for information on penalties for late submission.
- The Bernstein polynomial density (BPD) is given as;
f(x|θ) := X
k
j=1
θjbeta(x|j, k − j + 1) (1)
where beta(x|a, b) ∝ x
a−1
(1 − x)
b−11(x ∈ [0, 1]) denotes the PDF of Be(a, b) with a > 0 and b > and
θ ∈ ∆k−1, the k − 1 dimensional unit simplex, i.e., θ ∈ Rk
- and Pk
j=1 θj = 1. By construction, we
see that the specification (1) is a weighted average of the beta densities, {beta(x|j, k − j + 1)}
k
j=1. In
order to understand that (1) is a histogram smoothing, make a diagram of {beta(x|j, k − j + 1)}
k
j=1
on the unit support in a similar fashion of Figure 8 in Kim (2015, QE) for each k ∈ {5, 7, 10}.
1
- Replicate the figure on page 31 of the lecture slides on “Bayesian Methods – Computation.” In partic-
ular,
• Refer to the first two bullet points on page 29 regarding the true data generating process. Use
the CDF inversion method (pages 21 and 22) to simulate the data from the truncated normal
distribution.
• In addition, see page 30 for computational details, e.g., tuning parameters and number of iterations
for the Metropolis Hastings algorithm.
• You need to evaluate the density function (1) with different θ at each data point to implement the
Metropolis-Hastings algorithm. Observe however that you need to evaluate the beta components
only once, which may reduce computation time.
• Make a 2 by 2 figure just like the one in the lecture slides with a relevant title for each diagram
and labels for the vertical and horizontal axes, when they are informative.
• For the 95% credible band on panel (d), use the 2.5 and 97.5 percentiles at each point in the unit
interval. You may use 100 equidistant grid points for this exercise. Make sure you use different
line stiles for each different objects in the diagram so that they are visually distinguished.
- Similarly, replicate the figure on page 8 of the lecture slides on “Bayesian Methods – Computation 2.”
• If your code for Problem 2 works well, you only need to modify your code slightly; see page 7 of
the lecture note.