SOCI 620: Quantitative methods 2

Agenda

Multi-parameter posterior

  1. Administrative
  2. Modeling income
    with two parameters
  3. MAP estimation
  4. Hands on:
    Estimation in R

Slides are licensed under CC BY-NC-SA 4.0

Modeling income

Cartoon duck in a vault of gold coins, throwing them over his head. In the background, two younger cartoon ducks are dumping bags of coins into the vault.

Modeling income

Annual family income in Canada,
2021

(n=3195)

Data source:
Canadian Income Survey: Public Use Microdata

Modeling income

Annual family income in Canada,
2021

(n=3183)

Data source:
Canadian Income Survey: Public Use Microdata

Modeling income

Annual family income in Canada,
2021

(n=3183)

Data source:
Canadian Income Survey: Public Use Microdata

Plan: Model log income as a normally distributed random variable

A two-parameter model

Model of normally distributed log income:

Two Parameters:
μ determines ‘location’
σ determines ‘width’

A two-parameter model

Model of normally distributed log income:

A two-parameter model

Model of normally distributed log income:

A two-parameter model

Model of normally distributed log income:

A two-parameter model

Multi-parameter posteriors

Pr(μ) prior

Pr(μ | Y) posterior

Pr(σ) prior

Pr(σ | Y) posterior

Multi-parameter posteriors

Pr(μ,σ | Y) posterior

Maximum
a posteriori estimation
(MAP/QUAP)

Photo of a hiker about to reach a summit

MAP estimation

(Marginal) posterior
distribution of μ


Pr(μ | Y)

Maximum a posteriori
(quadratic approximation)

  1. Find the ‘peak’ of the posterior and call it mμ
  2. Estimate the ‘spread’ of the posterior and call it sμ
  3. Approximate the posterior with Norm(mμ, sμ)

Normals all the way down

So many normal distributions!
This can get confusing!

Pr(Y | μ)

  • First we modeled income using a normal distribution with unobserved parameters μ and σ:

Pr(μ)

  • Then we gave σ a normal prior:

Pr(μ | Y)

  • And then we used a normal distribution to approximate the posterior distributions of μ and σ:

Image credit

Figures by Peter McMahan (source code)

Cartoon duck in a vault of gold coins, throwing them over his head. In the background, two younger cartoon ducks are dumping bags of coins into the vault.

Image copyright Disney

Photo of a hiker about to reach a summit

Photo by Paulo Etxeberria

- Nearly normal in shape - Very narrow bounds - Lumpy because built from posterior