5 Day 5 (February 4)

5.1 Announcements

• Please read (and re-read) Ch. 3 and 4 in BBM2L book.

• Selected questions/clarifications from journals

  • How to choose/select a distribution
    • Definition of a model
    • Combining data and models/assumptions gives use prediction/forecasts/inference.
  • Sample size questions
    • n = Inf
    • n = 0
    • Power analysis
    • Anxiety/statistical therapy
    • Adaptive designs

• Good reading from The American Statistician link

5.2 Building our first statistical model

• The backstory
  • Bat
  • Rabies
• Building a statistical model using a likelihood-based (classical) approach
  • Specify (write out) the likelihood
  • Select an approach to estimate unknown parameters (e.g., maximum likelihood)
  • Quantify uncertainty in unknown parameters (e.g., using normal approximation, see here)
• 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)
    • Simulation-based (e.g., Metropolis-Hastings, MCMC, importance sampling, ABC, etc)

5.3 Numerical Integration

• Why do we need integrals to do Bayesian statistics?

  • Example using Bayes theorem to estimate prevalence rate of rabies
  • Why it is important to keep track of what we are calculating (i.e., clarity in what is being estimated)

• Numerical approximation vs. analytical solutions

• Definition of a definite integral $$\int_{a}^b f(z)dz = \lim_{Q\to\infty} \sum_{q=1}^{Q}\Delta q f(z_q)$$ where $\Delta q =\frac{b-a}{Q}$ and $z_q = a + \frac{q}{2}\Delta q$.

• Riemann approximation (midpoint rule) $$\int_{a}^b f(z)dz \approx \sum_{q=1}^{Q}\Delta q f(z_q)$$ where $\Delta q =\frac{b-a}{Q}$ and $z_q = a + \frac{2q - 1}{2}\Delta q$.

• Using similar approach in R (Adaptive quadrature)

  fn <- function(y){dnorm(y,0,1)}
  integrate(f=fn,lower=-4,upper=4,subdivisions=10)

  ## 0.9999367 with absolute error < 4.8e-12

5.4 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] 1.999428

  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] 9.014978

  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] 0.4597593

• Questions about activity 2?

• Live example using bat and coin data/model