Reading 6: Monte Carlo and Importance Sampling

Reading 6: Monte Carlo and Importance Sampling#

For the class on Wednesday, January 29th

Reading assignments#

Read the following sections of [Owe13]:

  • Sec. 2.1 “Accuracy of simple Monte Carlo” (i.e., Pages 15–17 of Chap. 1-2 PDF)

  • Chap. 9 “Importance sampling” preamble and Sec. 9.1 “Basic importance sampling” up to Page 6 (i.e., Page 3–6 of Chap. 9 PDF)

    Hint

    This is a rather math-heavy reading; however, you don’t need to be able to follow the proofs. Notation used by [Owe13] that may not be immediately clear:

    • \(\mathbb{E}\) denotes the expectation operator

    • \(\boldsymbol{X} \sim q\) denotes a random variable \(\boldsymbol{X}\) that follows the distribution \(q\)

    • \(\mathcal{Q} = \{ x | q(x) > 0 \} \) means that \(\mathcal{Q}\) is the set of all \(x\)’s that satisfy \(q(x) > 0\)

    • \(\mathcal{Q}^c\) denotes the complement of the set \(\mathcal{Q}\)

    • \(\mathcal{D} \cap \mathcal{Q}^c\) denotes the intersection of the set \(\mathcal{D}\) and the complement of the set \(\mathcal{Q}\)

Questions#

Submit your answer on Canvas. Due at noon, Wednesday, January 29th.

  1. List anything from your reading that confuses you. Explain why they confuse you. If nothing confuses you, briefly summarize what you have learned from this reading assignment.

  2. Recite from the your reading (you can quote the textbook directly, or cite where it is defined):

    • What is the importance sampling estimate (estimator) of \(\mu = \mathbb{E}[f(\boldsymbol{X})]\)?

    • What is the variance of the importance sampling estimator?

    • What choice of the importance distribution, \(q(x)\), can minimize the variance of the importance sampling estimator?