Adding week 04 lecture and lab

lab/lab_week_04.Rmd

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+---
+title: "EEEB UN3005/GR5005  \nLab - Week 04 - 17 and 19 February 2020"
+author: "USE YOUR NAME HERE"
+output: pdf_document
+fontsize: 12pt
+---
+
+```{r setup, include = FALSE}
+
+knitr::opts_chunk$set(echo = TRUE)
+```
+
+
+# Bayesian Basics
+
+
+## Exercise 1: Applying Bayes' Theorem using Grid Approximation
+
+Imagine if the series of observations in the globe tossing example from the *Statistical Rethinking* text and class were: W L W W, where "W" corresponds to water and "L" corresponds to land.
+
+With this set of observations, use grid approximation (with 11 grid points) to construct the posterior for the parameter *p* (the proportion of water on the globe). Assume a flat prior for *p*.
+
+Plot the posterior distribution.
+
+```{r}
+
+```
+
+
+## Exercise 2: Thinking Deeper with Bayes' Theorem
+
+Suppose in the globe tossing scenario there are actually two globes, one for Earth and one for Mars. The Earth globe is 30% land. The Mars globe is 100% land. Further suppose that one of these globes—-you don’t know which—-was tossed in the air and produced a “land” observation. Assume that each globe was equally likely to be tossed. Show that the posterior probability that the globe was Earth, conditional on seeing “land” (Pr(Earth|land)), is 0.23.
+
+Note, this problem might seem like it has a lot of information to consider, but it is actually a direct application of Bayes' Theorem. If you're having problems getting started, write out Bayes' Theorem. Also, R is not strictly necessary for this problem. You could do the math by hand, so R is really just a glorified calculator here.
+
+```{r}
+
+```

lecture/lecture_week_04.R

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+
+
+# Demonstrate grid approximation of the posterior for a globe tossing problem
+# with 6 water observations out of 9 globe tosses
+
+#==============================================================================
+
+
+# 1) Define the grid to be used to compute the posterior
+
+# In this case, let's say we choose 20 grid points
+# Since the parameter of interest (proportion of water on the globe) is
+# bounded by 0 and 1, our grid should have those limits as well
+p_grid <- seq(from = 0, to = 1, length.out = 20)
+
+p_grid
+
+#==============================================================================
+
+
+# 2) Compute/define the value of the prior at each parameter value on the grid
+
+# In this case, simply choose a flat prior
+prior <- rep(1, 20)
+
+# You could visualize this prior
+plot(p_grid, prior, pch = 19, type = "b")
+
+#==============================================================================
+
+
+# 3) Compute the likelihood at each parameter value on the grid
+
+# This requires the use of a likelihood function applied to the observed data,
+# evaluated at each potential parameter value on the grid
+likelihood <- dbinom(6, size = 9, prob = p_grid)
+
+# You could also visualize the likelihood
+plot(p_grid, likelihood, pch = 19, type = "b")
+
+#==============================================================================
+
+
+# 4) Compute the unstandardized posterior at each parameter value on the grid
+
+# The unstandardized posterior is simply the product of the likelihood and 
+# prior
+unstd.posterior <- likelihood * prior
+
+# Again, could visualize
+plot(p_grid, unstd.posterior, pch = 19, type = "b")
+
+# Note that this unstandardized posterior is not a proper probability 
+# distribution since it does not add to 1
+sum(unstd.posterior)
+
+#==============================================================================
+
+
+# 4) Standardize the posterior
+
+posterior <- unstd.posterior/sum(unstd.posterior)
+
+# This standardized posterior is a now proper probability distribution
+sum(posterior)
+
+# Visualize the posterior
+plot(p_grid, posterior, pch = 19, type = "b",
+     xlab = "proportion of water on globe",
+     ylab = "posterior probability")