Monte Carlo Methods: Estimation, Sampling, and Computational Reliability

All flashcards

What is a Monte Carlo method?

A computational method that repeatedly samples inputs, evaluates a model or rule, and aggregates the results to estimate a numerical quantity.

What are the four usual steps in a Monte Carlo procedure?

Define the input domain, generate random inputs from a suitable distribution, compute outputs for those inputs, and aggregate the outputs.

How does the quadrant example estimate π?

Uniformly sample points in a unit square, find the fraction inside the inscribed quadrant, and multiply that fraction by 4.

Why does uniform sampling matter in the π example?

The fraction of sampled points in the quadrant estimates its area fraction only when points represent the square fairly, which uniform sampling supports.

Does adding samples guarantee that each run improves the estimate?

No. More appropriate independent samples generally improve accuracy over the long run, but an individual run can still move farther from the true value.

Quiz questions

1. How does Markov chain Monte Carlo represent a complex target probability distribution?Answer: It constructs a Markov chain whose stationary distribution is the target.
2. What is the key idea of importance sampling?Answer: Sample influential regions more often and weight results to account for the changed distribution.
3. How do quasi-Monte Carlo methods seek to improve coverage?Answer: They use low-discrepancy sequences designed to cover a space more evenly.
4. What is a characteristic of pseudorandom numbers?Answer: A deterministic algorithm generates sequences designed to behave statistically like random values.
5. In computational modeling, what does validation assess?Answer: Whether the model is appropriate for the real problem it is meant to represent.