ASTR 474 Problem Set 06: Monte Carlo and Parameter Inference

Objectives and instructions

Practice computational reasoning, quantitative analysis, validation, and scientific communication for lectures 09-12.

Use SI or explicitly stated astronomical units consistently; show conversions and significant figures.
Cite all data, software, and adapted code. Distinguish supplied measured data from synthetic records.
Include runnable code and record environment, command, and seed where used.
Collaboration on concepts is permitted; each submitted derivation, code, and interpretation must be the student's own and contributors must be named.

Revisions and resubmissions are encouraged to improve mastery and grade under the course policy.

Problem 6.1

For tau=2, calculate the analytic transmission and compare each seeded N estimate with its standard error. Explain why one seed does not establish a universal confidence interval.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Problem 6.2

Derive the likelihood ratio for the tilted Bernoulli proposal and show that its weighted estimator is unbiased. Compare its computed standard error with ordinary sampling at N=10000.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Problem 6.3

Fit the synthetic radial-velocity data with the stated model. Report K, phase, gamma, standard errors, weighted chi-square, degrees of freedom, residual RMS, and one residual pattern that would invalidate the model.

Deliver: derivation or code, units/conventions, a reproducible result, uncertainty or validation evidence, and a short physical interpretation.

Reproducibility check

Use this generated reference result as one consistency check, not as a substitute for showing reasoning: At N=10000, ordinary-sampling standard error is 0.0013723; the likelihood-weighted proposal gives 0.00036502. One run is evidence for this seed, not a universal variance guarantee.

No textbook exercise is adopted for this course; problems are original and cite the primary software/data sources in the reference log.

Submission

Submit a readable PDF report, source code, environment/version record, and any generated tables/figures. The assignment is complete only when another student can rerun the command and identify the inputs and validation test.