PHYS366: Statistical Methods in Astrophysics

Lesson 2: Understanding From Data

Goals for this session:

• Be able to describe the Bayesian view of models, parameters and uncertainty

• Know how to set up and perform simple Bayesian inferences, including the assignment of probability distributions

• Be able to draw simple PGMs and understand their connection to probability expressions

Related reading:

• MacKay Chapter 2, sections 2.1, 2.1, 2.3, and Chapter 3, sections 3.1, 3.2
• Ivezic Chapter 3, sections 3.1, 3.3 and Chapter 5, sections 5.1, 5.2, 5.3

Generative Models and Posterior Inferences

A good way to start understanding a dataset is to try to recreate it.

• Let's look at another forward model, with the X-ray Image dataset

Now, let's look at the inverse problem, and learn the model parameters from the data.

• A short diversion about models in science

• Inferring cluster model parameters from the X-ray Image

Probability

• In Bayesian analysis we use probability to quantify degree of belief.

• In this framework, both data and parameters have probability distributions, that quantify our state of knowledge of them.

• Let's remind ourselves of the rules of probability theory.

Example: The Cepheid Period-Luminosity Relation

• Let's get some more practice in setting up a Bayesian inference.

• This boils down to two tasks: 1) assigning probability distributions (for both the data and the parameters) and 2) doing integrals.

• Let's work through a simple model for some of our Cepheid data.

Assigning Priors

In the previous examples, we assigned various prior PDFs for our model parameters.

• Let's look at some general principles, and some health warnings, about priors

Recap

• What did we learn?