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\@ifundefined{c@chapter}{\newfloat{codelisting}{h}{lop}}{\newfloat{codelisting}{h}{lop}[chapter]} \floatname{codelisting}{Listing} \newcommand*\listoflistings{\listof{codelisting}{List of Listings}} \makeatother \makeatletter \@ifpackageloaded{caption}{}{\usepackage{caption}} \@ifpackageloaded{subcaption}{}{\usepackage{subcaption}} \makeatother \makeatletter \@ifpackageloaded{tcolorbox}{}{\usepackage[many]{tcolorbox}} \makeatother \makeatletter \@ifundefined{shadecolor}{\definecolor{shadecolor}{rgb}{.97, .97, .97}} \makeatother \makeatletter \makeatother \ifLuaTeX \usepackage{selnolig} % disable illegal ligatures \fi \IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}} \IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available \urlstyle{same} % disable monospaced font for URLs \hypersetup{ pdftitle={Chapter 2 Notes}, pdfauthor={Emanuel Rodriguez}, colorlinks=true, linkcolor={blue}, filecolor={Maroon}, citecolor={Blue}, urlcolor={Blue}, pdfcreator={LaTeX via pandoc}} \title{Chapter 2 Notes} \author{Emanuel Rodriguez} \date{} \begin{document} \maketitle \ifdefined\Shaded\renewenvironment{Shaded}{\begin{tcolorbox}[interior hidden, enhanced, breakable, frame hidden, sharp corners, boxrule=0pt, borderline west={3pt}{0pt}{shadecolor}]}{\end{tcolorbox}}\fi 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!'' \begin{Shaded} \begin{Highlighting}[] \CommentTok{\# libraries} \FunctionTok{library}\NormalTok{(bayesrules)} \FunctionTok{library}\NormalTok{(dplyr)} \FunctionTok{library}\NormalTok{(tidyr)} \FunctionTok{library}\NormalTok{(gt)} \FunctionTok{data}\NormalTok{(fake\_news)} \NormalTok{fake\_news }\OtherTok{\textless{}{-}}\NormalTok{ tibble}\SpecialCharTok{::}\FunctionTok{as\_tibble}\NormalTok{(fake\_news)} \end{Highlighting} \end{Shaded} What is the proportion of news articles that were labeled fake vs real. \begin{Shaded} \begin{Highlighting}[] \NormalTok{fake\_news }\SpecialCharTok{|\textgreater{}} \FunctionTok{glimpse}\NormalTok{()} \end{Highlighting} \end{Shaded} \begin{verbatim} Rows: 150 Columns: 30 $ title "Clinton's Exploited Haiti Earthquake ‘to Stea~ $ text "0 SHARES Facebook Twitter\n\nBernard Sansaric~ $ url "http://freedomdaily.com/former-haitian-senate~ $ authors NA, NA, "Sierra Marlee", "Jack Shafer,Nolan D"~ $ type fake, real, fake, real, fake, real, fake, fake~ $ title_words 17, 18, 16, 11, 9, 12, 11, 18, 10, 13, 10, 11,~ $ text_words 219, 509, 494, 268, 479, 220, 184, 500, 677, 4~ $ title_char 110, 95, 96, 60, 54, 66, 86, 104, 66, 81, 59, ~ $ text_char 1444, 3016, 2881, 1674, 2813, 1351, 1128, 3112~ $ title_caps 0, 0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 0~ $ text_caps 1, 1, 3, 3, 0, 0, 0, 12, 12, 1, 2, 5, 1, 1, 6,~ $ title_caps_percent 0.000000, 0.000000, 6.250000, 0.000000, 0.0000~ $ text_caps_percent 0.4566210, 0.1964637, 0.6072874, 1.1194030, 0.~ $ title_excl 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0~ $ text_excl 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0~ $ title_excl_percent 0.0000000, 0.0000000, 2.0833333, 0.0000000, 0.~ $ text_excl_percent 0.00000000, 0.00000000, 0.06942034, 0.00000000~ $ title_has_excl FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE~ $ anger 4.24, 2.28, 1.18, 4.66, 0.82, 1.29, 2.56, 3.47~ $ anticipation 2.12, 1.71, 2.16, 1.79, 1.23, 0.43, 2.05, 1.74~ $ disgust 2.54, 1.90, 0.98, 1.79, 0.41, 1.72, 2.05, 1.35~ $ fear 3.81, 1.90, 1.57, 4.30, 0.82, 0.43, 5.13, 4.25~ $ joy 1.27, 1.71, 1.96, 0.36, 1.23, 0.86, 1.54, 1.35~ $ sadness 4.66, 1.33, 0.78, 1.79, 0.82, 0.86, 2.05, 1.93~ $ surprise 2.12, 1.14, 1.18, 1.79, 0.82, 0.86, 1.03, 1.35~ $ trust 2.97, 4.17, 3.73, 2.51, 2.46, 2.16, 5.13, 3.86~ $ negative 8.47, 4.74, 3.33, 6.09, 2.66, 3.02, 4.10, 4.63~ $ positive 3.81, 4.93, 5.49, 2.15, 4.30, 2.16, 4.10, 4.25~ $ text_syllables 395, 845, 806, 461, 761, 376, 326, 891, 1133, ~ $ text_syllables_per_word 1.803653, 1.660118, 1.631579, 1.720149, 1.5887~ \end{verbatim} \begin{Shaded} \begin{Highlighting}[] \NormalTok{fake\_news }\SpecialCharTok{|\textgreater{}} \FunctionTok{group\_by}\NormalTok{(type) }\SpecialCharTok{|\textgreater{}} \FunctionTok{summarise}\NormalTok{(} \AttributeTok{total =} \FunctionTok{n}\NormalTok{(),} \AttributeTok{prop =}\NormalTok{ total }\SpecialCharTok{/} \FunctionTok{nrow}\NormalTok{(fake\_news)} \NormalTok{ ) } \end{Highlighting} \end{Shaded} \begin{verbatim} # A tibble: 2 x 3 type total prop 1 fake 60 0.4 2 real 90 0.6 \end{verbatim} If we let \(B\) be the event that a news article is ``fake'' news, and \(B^c\) be the event that a news article is ``real'', we can write the following: \[P(B) = .4\] \[P(B^c) = .6\] This is the first ``clue'' or set of data that we have to build into our framework. Namely, majority of articles are ``real'', therefore we could simply predict that the new article is ``real''. This updated sense or reality now becomes our priors. Getting additional data, and updating our priors, based on additional data. The new observation we make is the use of exclamation marks ``!''. We note that the use of ``!'' is more frequent in news articles labeled as ``fake''. We will want to incorporate this into our framework to decide whether the new incoming should be labelled as real or fake. \hypertarget{likelihood}{% \subsubsection{Likelihood}\label{likelihood}} \begin{tcolorbox}[enhanced jigsaw, coltitle=black, rightrule=.15mm, bottomtitle=1mm, breakable, colframe=quarto-callout-note-color-frame, toprule=.15mm, colback=white, opacityback=0, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Probability and Likelihood}, bottomrule=.15mm, opacitybacktitle=0.6, toptitle=1mm, titlerule=0mm, arc=.35mm, leftrule=.75mm, left=2mm, colbacktitle=quarto-callout-note-color!10!white] When the event \(B\) is known, then we can evaluate the uncertainy of events \(A\) and \(A^c\) given \(B\) \[P(A|B) \text{ vs } P(A^c|B)\] If on the other hand, we know event \(A\) then we can evaluate the relative compatability of data \(A\) with \(B\) and \(B^c\) using likelihood functions \[L(B|A) \text{ vs } L(B^c|A)\] \[=P(A|B) \text{ vs } P(A|B^c)\] \end{tcolorbox} So in our case, we don't know whether this new incoming article is real or not, but we do know that the title has an exclamation mark. This means we can evaluate how likely this article is real or not given that it contains an ``!'' in the title using likelihood functions. We can formualte this as: \[L(B|A) \text{ vs } L(B^c|A)\] And perform the computation in R as follows: \begin{Shaded} \begin{Highlighting}[] \CommentTok{\# if fake, what are the proprotions of ! vs no{-}!} \NormalTok{prop\_of\_excl\_within\_type }\OtherTok{\textless{}{-}}\NormalTok{ fake\_news }\SpecialCharTok{|\textgreater{}} \FunctionTok{group\_by}\NormalTok{(type, title\_has\_excl) }\SpecialCharTok{|\textgreater{}} \FunctionTok{summarise}\NormalTok{(} \AttributeTok{total =} \FunctionTok{n}\NormalTok{()} \NormalTok{ ) }\SpecialCharTok{|\textgreater{}} \FunctionTok{ungroup}\NormalTok{() }\SpecialCharTok{|\textgreater{}} \FunctionTok{group\_by}\NormalTok{(type) }\SpecialCharTok{|\textgreater{}} \FunctionTok{summarise}\NormalTok{(} \AttributeTok{has\_excl =}\NormalTok{ title\_has\_excl,} \AttributeTok{prop\_within\_type =}\NormalTok{ total }\SpecialCharTok{/} \FunctionTok{sum}\NormalTok{(total)} \NormalTok{ ) } \end{Highlighting} \end{Shaded} \begin{Shaded} \begin{Highlighting}[] \NormalTok{prop\_of\_excl\_within\_type }\SpecialCharTok{|\textgreater{}} \FunctionTok{pivot\_wider}\NormalTok{(}\AttributeTok{names\_from =} \StringTok{"type"}\NormalTok{, }\AttributeTok{values\_from =}\NormalTok{ prop\_within\_type) }\SpecialCharTok{|\textgreater{}} \FunctionTok{gt}\NormalTok{() }\SpecialCharTok{|\textgreater{}} \NormalTok{ gt}\SpecialCharTok{::}\FunctionTok{cols\_label}\NormalTok{(} \AttributeTok{has\_excl =} \StringTok{"Contains Exclamtion"}\NormalTok{,} \AttributeTok{fake =} \StringTok{"Fake"}\NormalTok{, } \AttributeTok{real =} \StringTok{"Real"}\NormalTok{) }\SpecialCharTok{|\textgreater{}} \NormalTok{ gt}\SpecialCharTok{::}\FunctionTok{fmt\_number}\NormalTok{(}\AttributeTok{columns=}\FunctionTok{c}\NormalTok{(}\StringTok{"fake"}\NormalTok{, }\StringTok{"real"}\NormalTok{), }\AttributeTok{decimals =} \DecValTok{3}\NormalTok{) }\SpecialCharTok{|\textgreater{}} \NormalTok{ gt}\SpecialCharTok{::}\FunctionTok{cols\_width}\NormalTok{(}\FunctionTok{everything}\NormalTok{() }\SpecialCharTok{\textasciitilde{}} \FunctionTok{px}\NormalTok{(}\DecValTok{100}\NormalTok{))} \end{Highlighting} \end{Shaded} \begin{longtable}{crr} \toprule Contains Exclamtion & Fake & Real \\ \midrule FALSE & $0.733$ & $0.978$ \\ TRUE & $0.267$ & $0.022$ \\ \bottomrule \end{longtable} The table above also shows the likelihoods for the case when an article does not contain exclamation point in the title. \end{document}