Statistical Cosmology, Likelihoods, and Simulation-Based Inference
From Data to Posteriors
Likelihood-Based Inference
Cosmological inference usually proceeds by defining a likelihood for parameters , combining it with priors to obtain the posterior . Because modern surveys are high-dimensional and systematics-limited, a major research focus is robust uncertainty quantification, model comparison, and computational scalability.
Common Inference Tools
| Tool | Strength | Limitation |
|---|---|---|
| MCMC | Accurate posterior sampling | Can be expensive in high dimensions |
| Nested sampling | Evidence estimation | May struggle with complex posteriors |
| Approximate Bayesian computation | Likelihood-free inference | Often requires summary statistics |
| Normalizing flows | Flexible density estimation | Needs careful validation |
Simulation-Based Inference
When the likelihood is intractable, forward simulations plus machine learning can approximate posteriors directly from data, provided calibration and coverage are validated rigorously.
What is the posterior proportional to?
Bayes' theorem gives posterior proportional to prior multiplied by likelihood.
Correct answer: Prior times likelihood
Likelihood Pitfalls in Cosmology
Non-Gaussian posteriors, parameter degeneracies, selection effects, missing data, and unknown systematics can invalidate naive Gaussian approximations. Hierarchical Bayesian modeling, nuisance-parameter marginalization, emulators, and end-to-end simulations are increasingly central to credible cosmological inference.
Why can summary statistics limit information recovery?
Choosing insufficient statistics can make different cosmologies look artificially similar.
Correct answer: Because compression may discard non-Gaussian or higher-order information present in the full data set.
Frequentist vs Bayesian Cosmology
Frequentist approach
- Focuses on estimators, confidence intervals, and test statistics.
- Useful for coverage and pipeline validation.
- Can be awkward for complex nuisance structures.
Bayesian approach
- Provides coherent posterior inference.
- Handles priors and hierarchical structure naturally.
- Requires careful prior sensitivity analysis.
What is one advantage of normalizing flows in cosmological inference?
Normalizing flows are flexible density estimators useful for approximating complicated posteriors.
Correct answer: They can approximate complex probability densities efficiently
A Modern Bayesian Workflow
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1
Specify the forward model and nuisance parameters.
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2
Choose priors and the data likelihood or simulator.
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3
Sample the posterior and assess convergence.
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4
Validate with posterior predictive checks and coverage tests.