class: center, middle, inverse, title-slide # Bayes’ Rule for Events ### Dr. Dogucu --- layout: true <div class="my-header"></div> <div class="my-footer"> From Bayes Rules! book Copyright &copy; Drs. Alicia Johnson, Miles Ott & Mine Dogucu. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div> --- ## Example An algorithm is written to detect cat images. When given a cat image the algorithm identifies the image as a cat image 80% of the time. However, when given an image without a cat, the algorithm falsely identifies it as a cat image 50% of the time. The algorithm is to be tested with a set of images of which are 7% cat images. What is the probability that an image is a cat image if the algorithm identifies it as a cat image? --- # Write out the given information Let event `\(A\)` represent an event of algorithm identifying an image as a cat image Let event `\(B\)` represent an event of an image being a cat image. --- ## Bayes' Rule for Events `$$P(B |A) = \frac{P(A \cap B)}{P(A)} = \frac{P(A|B)P(B)}{P(A)}$$` -- where, by the Law of Total Probability, `$$P(A) = P(A|B)P(B) + P(A|B^c)P(B^c)$$` --- ## Spam email Priya, a data science student, notices that her college's email server is using a faulty spam filter. Taking matters into her own hands, Priya decides to build her own spam filter. As a first step, she manually examines all emails she received during the previous month and determines that 40% of these were spam. --- ## Prior Let event B represent an event of an email being spam. `\(P(B) = 0.40\)` If Priya was to act on this prior what should she do about incoming emails? --- ## A possible solution Since most email is non-spam, sort all emails into the inbox. This filter would certainly solve the problem of losing non-spam email in the spam folder, but at the cost of making a mess in Priya's inbox. --- ## Features of spam emails Priya observes the following about her latest incoming email. Which of the following is more likely to be a feature of a spam email? - The email was sent at 1:03 am. - The email was sent by an email address which started with the letter _p_. - The email came from outside the college. - The email was written in all capital letters ("all caps") and its subject line. --- ## Data Focusing on the last feature, Priya decides to look at some data. In her one-month email collection, 20% of spam but only 5% of non-spam emails used all caps. -- Using notation: `\(P(A|B) = 0.20\)` `\(P(A|B^c) = 0.05\)` --- <div align = "center"> <div id="htmlwidget-74d1fdee336a0140d631" style="width:504px;height:504px;" class="grViz html-widget"></div> <script type="application/json" data-for="htmlwidget-74d1fdee336a0140d631">{"x":{"diagram":"\n digraph {\n # graph aesthetics\n graph [ranksep = 0.5]\n # node definitions with substituted label text\n 1 [label = \"Prior: \n Only 40% of emails are spam.\"]\n 2 [label = \"Data: \n All caps is more common among spam.\"]\n 3 [label = \"Posterior: \n Is the email spam or not?!\"]\n # edge definitions with the node IDs\n 1 -> 3\n 2 -> 3\n }\n","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script> </div> --- Which of the following best describes your posterior understanding of whether the email is spam? a. The chance that this email is spam drops from 40% to 20%. After all, the subject line might indicate that the email was sent by an excited professor that's offering Priya an automatic "A" in their course! b. The chance that this email is spam jumps from 40% to roughly 70%. Though using all caps is more common among spam emails, let's not forget that only 40% of Priya's emails are spam. c. The chance that this email is spam jumps from 40% to roughly 95%. Given that so few non-spam emails use all caps, this email is almost certainly spam. --- class: center middle # Calculating the posterior `\(P(B|A) = \frac{P(A|B) P(B)}{P(A)}\)` -- `\(P(B|A) = \frac{P(A|B) P(B)}{P(A|B) P(B)+P(A|B^c) P(B^c)}\)` `\(P(B|A) = \frac{0.20 \cdot 0.40}{(0.20 \cdot 0.40) + (0.05 \cdot 0.60)}\)` --- # From prior to posterior __Prior__ to the analysis any email was likely to end up in the inbox, `\(P(B) = 0.40\)` -- After having observed data ( __likelihood__ ) all caps, Priya's __posterior__ move should be putting this email to spam folder. `\(P(B|A) = 0.\overline{72}\)` --- # In words `$$P(B |A) = \frac{P(A \cap B)}{P(A)} = \frac{P(A|B)P(B)}{P(A)}$$` `$$P(B |A) = \frac{P(B)P(A|B)}{P(A)}$$` `$$\text{posterior} = \frac{\text{prior}\cdot\text{likelihood}}{\text{marginal probability}}$$`