 
Bayesian Analysis of the Cosmic Microwave Background
There is a wealth of cosmological information
encoded in the spatial power spectrum of temperature anisotropies of the cosmic
microwave background. Originally detected by the COBE satellite, the brightness fluctuations in the microwave
sky have now been mapped by ground, balloon, and satellite instruments to a spatial resolution smaller than 1 degree.
These brightness fluctuations trace small density perturbations in the early universe (roughly 300,000 years after the Big Bang), which later grow through gravitational instability to the largescale structure seen in redshift
surveys. The details of the physics in the early universe leaves a telltale signature on the statistical structure of hot and cold spots, with more details
of the physics encoded at subangular degree spatial scales. With the push to map the microwave sky at
higher spatial resolution has come a flood of data, with maps containing
millions of pixels observed at several different frequencies (from 30 to 900
GHz), all with slightly different resolutions and noise properties. The
resulting analysis challenge is to estimate, and quantify our uncertainty in,
the spatial power spectrum of the cosmic microwave background given the
complexities of "missing data", foreground emission, and complicated
instrumental noise. In this talk, I
will review a Bayesian formulation of this problem and its numerical implementation
with Gibbs sampling from the posterior given the data. In addition I will discuss the limitations
of Gibbs sampling, and report on the development of an adaptive Monte Carlo
approach, allowing the entire past history of computation to be used in
generating the next generation of samples from the posterior.
