6 Day 6 (February 6)

6.1 Announcements

• Please read (and re-read) Ch. 4 and 6 in BBM2L book.

• Final project is posted

• Questions about activity 2?

• Selected questions/clarifications from journals

  • Statistical significance, p-values, and other stuff
  • Learning to focus on the important things
  • Selecting the best model/distribution

6.2 Building our first statistical model

• The backstory
  • Bat
  • Rabies
• Building a statistical model using a Bayesian approach
  • Specify (write out) the likelihood/data model
  • Specify the parameter model (or prior) including hyper-parameters
  • Select an approach to obtain the posterior distribution
    • Analytically (i.e., pencil and paper)
    • Deterministic numerical algorithm
    • Simulation-based or stochastic algorithm (e.g., Metropolis-Hastings, MCMC, importance sampling, ABC, etc)

6.3 Monte Carlo Integration

• Deterministic vs stochastic methods to approximate integrals

  • Work well for high-dimensional multiple integrals
  • Easy to program

• Monte Carlo integration

  • $$\begin{eqnarray} \text{E}(g(y)) &=& \int g(y)[y|\theta]dy\\ &\approx& \frac{1}{Q}\sum_{q=1}^{Q}g(y_q) \end{eqnarray}$$
  • Examples:
  1. $$\text{E}(y) = \int_{-\infty}^\infty y\frac{1}{\sqrt{2\pi\sigma^2}}\textit{e}^{-\frac{1}{2\sigma^2}(y - \mu)^2}dy$$

  y <- rnorm(n = 10^6, mean = 2, sd = 3)
  mean(y)

  ## [1] 2.000739

  2. $$\text{E}((y-\mu)^2) = \int_{-\infty}^\infty (y-\mu)^2\frac{1}{\sqrt{2\pi\sigma^2}}\textit{e}^{-\frac{1}{2\sigma^2}(y - \mu)^2}dy$$

  y <- rnorm(n = 10^6, mean = 2, sd = 3)
  mean((y - 2)^2)

  ## [1] 8.987738

  3. $$\text{E}(\frac{1}{y} ) = \int_{-\infty}^\infty \frac{1}{y}\frac{1}{\sqrt{2\pi\sigma^2}}\textit{e}^{-\frac{1}{2\sigma^2}(y - \mu)^2}dy$$

  y <- rnorm(n = 10^6, mean = 2, sd = 4)
  mean(1/y)

  ## [1] -2.855014

• Questions about activity 2?

• Live example using bat model

  • Download R script