diff --git a/lab/lab_week_05.Rmd b/lab/lab_week_05.Rmd
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+---
+title: "EEEB UN3005/GR5005  \nLab - Week 05 - 24 and 26 February 2020"
+author: "USE YOUR NAME HERE"
+output: pdf_document
+fontsize: 12pt
+---
+
+```{r setup, include = FALSE}
+
+knitr::opts_chunk$set(echo = TRUE)
+
+library(rethinking)
+```
+
+
+# Statistical Distributions and Summary Statistics
+
+
+## Exercise 1: Grid Approximation, Our Old Friend
+
+Imagine that the globe tossing example from the *Statistical Rethinking* text and class resulted in 8 water observations out of 15 globe tosses.
+
+With this set of data, use grid approximation (with 101 grid points) to construct the posterior for *p* (the probability of water). Assume a flat prior.
+
+Plot the posterior distribution.
+
+```{r}
+
+```
+
+
+## Exercise 2: Sampling From a Grid-Approximate Posterior
+
+Now generate 10,000 samples from the posterior distribution of *p*. Call these samples `post.samples`. Visualize `post.samples` using the `dens()` function. 
+
+For your own understanding, re-run your sampling and plotting code multiple times to observe the effects of sampling variation.
+
+```{r}
+
+```
+
+
+## Exercise 3: Summarizing Samples
+
+Return the mean, median, and mode (using `chainmode()`) of `post.samples`. Then calculate the 80%, 90%, and 99% highest posterior density intervals of `post.samples`.
+
+```{r}
+
+```
+
+
+## Exercise 4: Model Predictions
+
+Using `post.samples`, generate 10,000 simulated model predictions (you can call these `preds`) for a binomial trial of size 15. Visualize the model predictions using the `simplehist()` function. 
+
+Based on these posterior predictions, what is the probability of observing 8 waters in 15 globe tosses?
+
+```{r}
+
+```
+
+
+## Exercise 5: More Model Predictions
+
+Using the *same* posterior samples (i.e., `post.samples`), generate 10,000 posterior predictions for a binomial trial of size 9.
+
+Using these new predictions, calculate the posterior probability of observing 8 waters in 9 tosses.
+
+```{r}
+
+```
diff --git a/lecture/lecture_week_04.R b/lecture/lecture_week_04.R
index afb411c..e37282b 100644
--- a/lecture/lecture_week_04.R
+++ b/lecture/lecture_week_04.R
@@ -1,24 +1,27 @@
 
 
-# Demonstrate grid approximation of the posterior for a globe tossing problem
-# with 6 water observations out of 9 globe tosses
+# 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
+# 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
+# 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)
@@ -26,42 +29,46 @@ 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
+# 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
+# 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
+# 4) Compute the unstandardized posterior at each 
+# parameter value on the grid
 
-# The unstandardized posterior is simply the product of the likelihood and 
-# prior
+# 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
+# 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
+# 5) Standardize the posterior
 
 posterior <- unstd.posterior/sum(unstd.posterior)
 
-# This standardized posterior is a now proper probability distribution
+# This standardized posterior is a now proper probability 
+# distribution
 sum(posterior)
 
 # Visualize the posterior
diff --git a/lecture/lecture_week_05.R b/lecture/lecture_week_05.R
new file mode 100644
index 0000000..5d909b6
--- /dev/null
+++ b/lecture/lecture_week_05.R
@@ -0,0 +1,292 @@
+
+
+library(rethinking)
+
+#==========================================================
+
+
+# Probability mass function for the binomial distribution
+
+# Parameters: 
+#
+# 1) number of trials = 10 
+# (Note: This is a parameter that we normally will NOT 
+# need to estimate in our statistical models. Rather, it's 
+# something we observe. But it is a parameter in the sense 
+# that it's a numeric characteristic used to define
+# the binomial distribution.)
+#
+# 2) probability of success = 0.3
+
+# If we have 10 binomial trials, then only 11 different
+# outcomes are possible. We could have anywhere from 0
+# to 10 successes
+
+# Let's get the probability masses associated with each 
+# of these 11 outcomes, given our arbitrarily chosen 
+# parameter values
+prob.masses <- dbinom(0:10, size = 10, prob = 0.3)
+
+prob.masses
+
+# We can plot these probability masses
+plot(
+  x = 0:10, y = prob.masses,
+  xlab = "Number of successful binomial trials",
+  ylab = "Probability",
+  pch = 19
+)
+
+# Note that the probability mass of all outcomes 
+# sums to 1
+sum(prob.masses)
+# In other words, if we conduct 10 binomial trials, one
+# of these 11 outcomes MUST be observed. But some outcomes
+# are more likely than others
+
+
+# Random generation function for the binomial distribution
+
+# 10 simulated draws (observations) from the binomial 
+# distribution
+# Note that each of these observations is composed of 
+# 10 binomial trials. The "size" and "prob" parameters
+# have not changed from the examples above
+draws.10 <- rbinom(10, size = 10, prob = 0.3)
+
+simplehist(
+  draws.10, xlim = c(0, 10),
+  xlab = "Number of successes observed"
+)
+
+# 100 simulated draws (observations)
+draws.100 <- rbinom(100, size = 10, prob = 0.3)
+
+simplehist(
+  draws.100, xlim = c(0, 10),
+  xlab = "Number of successes observed"
+)
+
+# 10,000 simulated draws (observations)
+draws.10000 <- rbinom(10000, size = 10, prob = 0.3)
+
+simplehist(
+  draws.10000, xlim = c(0, 10),
+  xlab = "Number of successes observed"
+)
+
+#==========================================================
+
+
+# Probability density function for the normal distribution
+
+# Parameters: 
+#
+# 1) mean = 2.5
+#
+# 2) standard deviation = 1.5
+
+# Plot the probability density function for the
+# normal distribution
+curve(
+  dnorm(x, mean = 2.5, sd = 1.5), 
+  from = -10, to = 10,
+  ylab = "Probability density"
+)
+
+# 10 simulated draws (observations) from the 
+# normal distribution
+rnorm(10, mean = 2.5, sd = 1.5)
+
+
+# Probability density function for the Cauchy distribution
+
+# Parameters: 
+#
+# 1) location = 2.5
+#
+# 2) scale = 1.5
+
+# Plot the probability density function for the
+# Cauchy distribution
+curve(
+  dcauchy(x, location = 2.5, scale = 1.5),
+  from = -10, to = 10,
+  ylab = "Probability density"
+)
+
+# 10 simulated draws (observations) from the 
+# Cauchy distribution
+rcauchy(10, location = 2.5, scale = 1.5)
+
+#==========================================================
+
+
+# Sampling from a posterior distribution
+
+# First, let's generate a posterior distribution using
+# grid approximation (with 101 points). The data observed 
+# are 6 successes out of 9 binomial trials
+p_grid <- seq(from = 0, to = 1, length.out = 101)
+prior <- rep(1, 101)
+likelihood <- dbinom(6, size = 9, prob = p_grid)
+unstd.posterior <- likelihood * prior
+posterior <- unstd.posterior / sum(unstd.posterior)
+
+# Plot the posterior
+plot(
+  p_grid, posterior,
+  pch = 19, type = "b",
+  xlab = "Proportion of water on globe",
+  ylab = "Posterior probability"
+)
+
+# Generate 10,000 samples of p (the proportion of water
+# parameter) from the grid-approximate posterior
+samples <- sample(
+  p_grid, size = 10000, 
+  prob = posterior, replace = TRUE
+)
+# Reading the arguments within "samples()" would go
+# something like this: I want 10,000 samples (with
+# replacement) from the values in "p_grid", taking values
+# according to the probability assigned to them by the
+# "posterior"
+
+# Plot the posterior samples
+plot(samples, pch = 19, col = alpha("black", 0.2))
+dens(samples, xlim = c(0, 1),
+     xlab = "Proportion of water on globe")
+
+
+# Grid-approximate posterior for 5 successes in 5
+# binomial trials (leads to an extremely skewed 
+# posterior distribution for p)
+p_grid2 <- seq(from = 0, to = 1, length.out = 101)
+prior2 <- rep(1, 101)
+likelihood2 <- dbinom(5, size = 5, prob = p_grid2)
+unstd.posterior2 <- likelihood2 * prior2
+posterior2 <- unstd.posterior2 / sum(unstd.posterior2)
+
+# Plot the posterior
+plot(
+  p_grid2, posterior2,
+  pch = 19, type = "b",
+  xlab = "Proportion of water on globe",
+  ylab = "Posterior probability"
+)
+
+# Generate 10,000 samples of p (the proportion of water
+# parameter) from the grid-approximate posterior
+samples2 <- sample(
+  p_grid2, size = 10000, 
+  prob = posterior2, replace = TRUE
+)
+
+# Plot the posterior samples
+plot(samples2, pch = 19, col = alpha("black", 0.2))
+dens(samples2, xlim = c(0, 1),
+     xlab = "Proportion of water on globe")
+
+#==========================================================
+
+
+# Calculate percentile and highest posterior density
+# intervals from our posterior samples
+
+# 50% PI
+PI(samples2, prob = 0.5)
+
+dens(samples2, xlim = c(0, 1), xaxs = "i",
+     xlab = "Proportion of water on globe",
+     main = "50% PI shaded")
+shade(
+  density(samples2, adjust = 0.5), 
+  PI(samples2, prob = 0.5),
+  col = "darkgrey"
+)
+
+# 50% HPDI
+HPDI(samples2, prob = 0.5)
+
+dens(samples2, xlim = c(0, 1), xaxs = "i",
+     xlab = "Proportion of water on globe",
+     main = "50% HPDI shaded")
+shade(
+  density(samples2, adjust = 0.5), 
+  HPDI(samples2, prob = 0.5),
+  col = "darkgrey"
+)
+
+
+# 70% PI
+PI(samples2, prob = 0.7)
+
+dens(samples2, xlim = c(0, 1), xaxs = "i",
+     xlab = "Proportion of water on globe",
+     main = "70% PI shaded")
+shade(
+  density(samples2, adjust = 0.5), 
+  PI(samples2, prob = 0.7),
+  col = "darkgrey"
+)
+
+# 70% HPDI
+HPDI(samples2, prob = 0.7)
+
+dens(samples2, xlim = c(0, 1), xaxs = "i",
+     xlab = "Proportion of water on globe",
+     main = "70% HPDI shaded")
+shade(
+  density(samples2, adjust = 0.5), 
+  HPDI(samples2, prob = 0.7),
+  col = "darkgrey"
+)
+
+
+# With a less skewed posterior, the PI and HPDI are
+# often going to be very, very similar
+
+# 70% PI
+PI(samples, prob = 0.7)
+
+dens(samples, xlim = c(0, 1),
+     xlab = "Proportion of water on globe",
+     main = "70% PI shaded")
+shade(
+  density(samples, adjust = 0.5), 
+  PI(samples, prob = 0.7),
+  col = "darkgrey"
+)
+
+# 70% HPDI
+HPDI(samples, prob = 0.7)
+
+dens(samples, xlim = c(0, 1),
+     xlab = "Proportion of water on globe",
+     main = "70% HPDI shaded")
+shade(
+  density(samples, adjust = 0.5), 
+  HPDI(samples, prob = 0.7),
+  col = "darkgrey"
+)
+
+#==========================================================
+
+
+# Generate a posterior predictive distribution for 
+# binomial trials of size 10, given the posterior in
+# samples2
+preds <- rbinom(10000, size = 10, prob = samples2)
+
+plot(preds, pch = 19, col = alpha("black", 0.2))
+simplehist(preds, xlim = c(0, 10))
+
+
+# Note that even if we fix the probability of success value
+# there would still be variation in our outcomes because
+# the data-generating process is subject to outcome 
+# uncertainty
+test.outcomes <- rbinom(10000, size = 10, prob = 0.8)
+
+simplehist(test.outcomes, xlim = c(0, 10))
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