# libraries
library(bayesrules)
@@ -145,40 +160,24 @@ code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warni
What is the proportion of news articles that were labeled fake vs real.
-
+
fake_news |> glimpse()fake_news |> head()
-
Rows: 150
-Columns: 30
-$ title <chr> "Clinton's Exploited Haiti Earthquake ‘to Stea…
-$ text <chr> "0 SHARES Facebook Twitter\n\nBernard Sansaric…
-$ url <chr> "http://freedomdaily.com/former-haitian-senate…
-$ authors <chr> NA, NA, "Sierra Marlee", "Jack Shafer,Nolan D"…
-$ type <fct> fake, real, fake, real, fake, real, fake, fake…
-$ title_words <int> 17, 18, 16, 11, 9, 12, 11, 18, 10, 13, 10, 11,…
-$ text_words <int> 219, 509, 494, 268, 479, 220, 184, 500, 677, 4…
-$ title_char <int> 110, 95, 96, 60, 54, 66, 86, 104, 66, 81, 59, …
-$ text_char <int> 1444, 3016, 2881, 1674, 2813, 1351, 1128, 3112…
-$ title_caps <int> 0, 0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0…
-$ text_caps <int> 1, 1, 3, 3, 0, 0, 0, 12, 12, 1, 2, 5, 1, 1, 6,…
-$ title_caps_percent <dbl> 0.000000, 0.000000, 6.250000, 0.000000, 0.0000…
-$ text_caps_percent <dbl> 0.4566210, 0.1964637, 0.6072874, 1.1194030, 0.…
-$ title_excl <int> 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0…
-$ text_excl <int> 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0…
-$ title_excl_percent <dbl> 0.0000000, 0.0000000, 2.0833333, 0.0000000, 0.…
-$ text_excl_percent <dbl> 0.00000000, 0.00000000, 0.06942034, 0.00000000…
-$ title_has_excl <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE…
-$ anger <dbl> 4.24, 2.28, 1.18, 4.66, 0.82, 1.29, 2.56, 3.47…
-$ anticipation <dbl> 2.12, 1.71, 2.16, 1.79, 1.23, 0.43, 2.05, 1.74…
-$ disgust <dbl> 2.54, 1.90, 0.98, 1.79, 0.41, 1.72, 2.05, 1.35…
-$ fear <dbl> 3.81, 1.90, 1.57, 4.30, 0.82, 0.43, 5.13, 4.25…
-$ joy <dbl> 1.27, 1.71, 1.96, 0.36, 1.23, 0.86, 1.54, 1.35…
-$ sadness <dbl> 4.66, 1.33, 0.78, 1.79, 0.82, 0.86, 2.05, 1.93…
-$ surprise <dbl> 2.12, 1.14, 1.18, 1.79, 0.82, 0.86, 1.03, 1.35…
-$ trust <dbl> 2.97, 4.17, 3.73, 2.51, 2.46, 2.16, 5.13, 3.86…
-$ negative <dbl> 8.47, 4.74, 3.33, 6.09, 2.66, 3.02, 4.10, 4.63…
-$ positive <dbl> 3.81, 4.93, 5.49, 2.15, 4.30, 2.16, 4.10, 4.25…
-$ text_syllables <int> 395, 845, 806, 461, 761, 376, 326, 891, 1133, …
-$ text_syllables_per_word <dbl> 1.803653, 1.660118, 1.631579, 1.720149, 1.5887…# A tibble: 6 × 30
+ title text url authors type title…¹ text_…² title…³ text_…⁴ title…⁵
+ <chr> <chr> <chr> <chr> <fct> <int> <int> <int> <int> <int>
+1 Clinton's E… "0 S… http… <NA> fake 17 219 110 1444 0
+2 Donald Trum… "\n\… http… <NA> real 18 509 95 3016 0
+3 Michelle Ob… "Mic… http… Sierra… fake 16 494 96 2881 1
+4 Trump hits … "“Cr… http… Jack S… real 11 268 60 1674 0
+5 Australia V… "Whe… http… Blair … fake 9 479 54 2813 0
+6 It’s “Trump… "Lik… http… View A… real 12 220 66 1351 1
+# … with 20 more variables: text_caps <int>, title_caps_percent <dbl>,
+# text_caps_percent <dbl>, title_excl <int>, text_excl <int>,
+# title_excl_percent <dbl>, text_excl_percent <dbl>, title_has_excl <lgl>,
+# anger <dbl>, anticipation <dbl>, disgust <dbl>, fear <dbl>, joy <dbl>,
+# sadness <dbl>, surprise <dbl>, trust <dbl>, negative <dbl>, positive <dbl>,
+# text_syllables <int>, text_syllables_per_word <dbl>, and abbreviated
+# variable names ¹title_words, ²text_words, ³title_char, ⁴text_char, …fake_news |>
group_by(type) |>
@@ -245,12 +244,12 @@ Probability and Likelihood
gt::cols_width(everything() ~ px(100))
-
+
diff --git a/R/ch2.qmd b/R/ch2.qmd
index 4f2dd87..c4b1d05 100644
--- a/R/ch2.qmd
+++ b/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 $\pi$
:::
\ No newline at end of file
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+
+
+
+
+
+
@@ -855,12 +854,12 @@ Baye’s Rule
gt::cols_width(everything() ~ px(100))
-
+
+
+
@@ -1276,11 +1275,11 @@ Baye’s Rule
fake
-3941
-0.3941 4031
+0.4031
real
-6059
-0.6059 5969
+0.5969
@@ -1317,8 +1316,8 @@ Baye’s Rule
# Groups: usage [2]
usage fake real
<chr> <int> <int>
-1 no 2936 5932
-2 yes 1005 127
+1 no 2955 5845
+2 yes 1076 124
@@ -1345,8 +1344,46 @@ Baye’s Rule
+# A tibble: 2 × 3
type total prop
<chr> <int> <dbl>
-1 fake 1005 0.888
-2 real 127 0.112Binomial 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 | +