more work

R/ch2.qmd

@@ -11,12 +11,18 @@ format:
         css: styles.css
         callout-icon: false
         callout-apperance: simple
+        toc: true
 ---
 
+*Note: these notes are a work in progress*
+
 In this chapter we step through an example 
 of "fake" vs "real" news to build a framework to determine the probability 
 of real vs fake of a new news article titled "The President has a secret!"
 
+We then go on to build a probability known as the Binomial model using the 
+Bayesian framework
+
 ```{r}
 #| message: false
 #| warning: false
@@ -34,7 +40,7 @@ fake_news <- tibble::as_tibble(fake_news)
 What is the proportion of news articles that were labeled fake vs real. 
 
 ```{r}
-fake_news |> glimpse()
+fake_news |> head()
 
 fake_news |>
     group_by(type) |> 
@@ -316,6 +322,33 @@ articles_sim |>
     )
 ```
 
+## Binomial Model and the chess example
+
+The example used here is the case of a chess match between a human 
+and a computer "Deep Blue". The set up is such that we know the two
+faced each other in 1996, in which the human won. There is a rematch
+scheduled for the next 1997. We would like to model the number of games
+out of 6 that the human can win. 
+
+Let $\pi$ be the probability that the human wins any one match against
+the computer. To simplify things greatly we assume that $\pi$ takes on 
+values of .2, .5, .8. We also assume the following prior (we are told
+in the book that we will learn how to build these later on):
+
+| $\pi$  | .2 | .5 | .8 | total |
+|--------|----|----|----|-------|
+|$f(\pi)$|.10 |.25 |.65 | 1 |
+
+:::{.callout-caution}
+## Note
+
+its important to note here that the sum of the values of $\pi$ **do
+not** add up to 1. $\pi$ represents the chances of winning any single
+game, we would expect $\pi$ to take on any value in $\mathbb{R}$. On 
+the other hand $f$ is a function that maps $\pi$ into a space of
+probabilities, this is next. 
+:::
+
 
 :::{.callout-note}
 ## Discrete Probability Model
@@ -336,4 +369,27 @@ and has the following properties
 what does this mean? well its very straightforward a pmf is a function that takes
 in a some value y and outputs the probability that the random variable 
 $Y$ equals $y$. 
+:::
+
+### The Binomial Model
+
+:::{.callout-note}
+## Conditional probability model of data $Y$
+
+Let $Y$ be a discrete random variable that depends on some parameter
+$\pi$. We define the conditional probability model of $Y$ as the 
+conditional pmf, 
+
+$$f(y|\pi) = P(Y = y | \pi)$$
+
+and has the following properties, 
+
+1. $0 \leq f(y|\pi) \leq 1\;\; \forall y$
+2. $\sum_{\forall y}f(y|\pi) = 1$
+:::
+
+:::{.callout-caution}
+## in emanuel's words
+this is essentially the same probability model had defined above, except
+now we are condition probabilities by some parameter $\pi$
 :::