Collaborative Work with Git
Learning goal: First experiences with collaborative data science work with git
Step 1: git clone project-Y-USERNAMES
Team formation is mostly complete. Repositories project-A-USERNAMES, …, project-H-USERNAMES are created. You are to deliver your project reports in your repository.
Step 2: First Team member commits and pushes
The first team member1 does the following:
- Open the file
report.qmd
- Write your name in the first line at
author in the YAML
git add the file report.qmdgit commit (enter “Added name” as commit message)git push
Step 3: Other team members pull
- Do
git pull in the RStudio interface.
What does git pull do?
git pull does two things: fetching new stuff and merging it with existing stuff
- First it
git fetchs the commit from the remote repository (GitHub) to the local machine
- Then it
git merges the commit with the latest commit on your local machine.
- When we are lucky this works with no problems. (Should be the case with new files.)
Step 4: Merge independent changes
Git can merge changes in the same file when there are no conflicts. Let’s try.
- The second team member:
- Add your name in the author section of the YAML, save the file, add the file in the Git pane and make a commit.
- Save, add the file in the Git pane, commit with message “Next author name”, push.
- The third (or first) team member:
- Change the title in the YAML to something meaningful (and also add your name if third team member), save the file, add the file in the Git pane and make a commit.
- Try to push. You should receive an error. Read it carefully, often it tells you what to do. Here: Do
git pull first. You cannot push because remotely there is a newer commit (the one your colleague just made).
- Pull. This should result in message about a successfull auto-merge. Check that both are there: Your line and the line of your colleague. If you receive several hints instead, first read the next slide!
??? git configuration for divergent branches
If you pull for the first time in a local git repository, git may complain like this:
![]()
Read that carefully. It advises to configure with git config pull.rebase false as the default version.
How to do the configuration?
- Copy the line
git config pull.rebase false and close the window.
- Go to the Terminal pane (not the console, the one besides that). This is a terminal not for R but to speak with the computer in general. Paste the command and press enter. Now you are done and your next
git pull should work.
Step 5: Push and pull the other way round
- The third/first member:
- The successful merge creates a new commit, which you can directly push.
- Push.
- The second team member (and all others):
- Pull the changes of your colleague.
Practice a bit more pulling and pushing commits and check the merging.
Step 6: Create a merge conflict
- First and second team members:
- Write a different sentence after “Executive Summary.” in YAML
abstract:.
- Each
git add and git commit on local machines.
- First member:
git push
- Second member:
git pull. That should result in a conflict. If you receive several hints instead, first read the slide two slides before!- The conflict should show directly in the file with markings like this
>>>>>>>>
one option of text,
======== a separator,
the other option, and
<<<<<<<.
Step 7: Solve the conflict
- The second member
- You have to solve this conflict now!
- Solving is by editing the text
- Decide for an option or make a new text
- Thereby, remove the
>>>>>,=====,<<<<<<
- When you are done:
git add, git commit, and git push.
Now you know how to solve merge conflicts. Practice a bit in your team.
Working in VSCode: The workflow is the same because it relies on git not on the editor of choice.
Advice: Collaborative work with git
- Whenever you start a work session: First pull to see if there is anything new. That way you reduce the need for merges.
- Inform your colleagues when you pushed new commits.
- Coordinate the work, e.g. discuss who works on what part and maybe when. However, git allows to also work without full coordination and in parallel.
- When you finish your work session, end with pushing a nice commit. That means. The file should render. You made comments when there are loose ends and todo’s.
- You can also use the issues section of the GitHub repository for things to do.
- When you work on different parts of the file, be aware that also a successful merge can create problems. Example: Your colleague changed the data import, while you worked on graphics. Maybe after the merge the imported data is not what you need for your chunk. Then coordinate.
- Commit and push often. This avoids that potential merge conflicts become large.
The set of all events and the probability function
The set of all events
- The set of all events is the set of all subsets of a sample space \(S\).
- When the sample space has \(n\) atomic events, the set of all events has \(2^n\) elements.
- The set of all events is very large also for fairly simple examples!
Examples
- For 3 coin tosses: How many events exist? \(2^3=8\) atomic events \(\to\) \(2^8=256\) event
- How is it for four coin tosses? \(2^{(2^4)} = 65536\)
- Select two out of five people (without replacement)1?
Ten atomic events: 12, 13, 14, 15, 23, 24, 25, 34, 35, 45. Events: \(2^{10} = 1024\)
These are typical problems of combinatorics, the theory of counting.
Probability function
Definition: A set of all events a function \(\text{Pr}: \text{Set of all subsets of $S$} \to \mathbb{R}\) is a probability function when
- The probability of any event is between 0 and 1: \(0\leq \text{Pr}(A) \leq 1\). (So, actually a probability function is a function into the interval \([0,1]\).)
- The probability of the event coinciding with the whole sample space (the sure event) is 1: \(\text{Pr}(S) = 1\).
- For events \(A_1, A_2, \dots, A_n\) which are pairwise disjoint we can sum up their probabilities:
\[\text{Pr}(A_1 \cup A_2\cup\dots\cup A_n) = \text{Pr}(A_1) + \text{Pr}(A_2) + \dots + \text{Pr}(A_n) \]
This captures the essence of how we think about probabilities mathematically. Most important: We can only easily add probabilities when they do not share atomic events.
Example Probability Function
Example coin tosses: We can define a probability function \(\text{Pr}\) by assigning the same probability to each atomic event.
- \(\text{Pr}(\{H\}) = \text{Pr}(\{T\}) = 1/2\)
- \(\text{Pr}(\{HH\}) = \text{Pr}(\{HT\}) = \text{Pr}(\{TH\}) = \text{Pr}(\{TT\}) = 1/4\)
So, the probability one or zero HEADs is \(\text{Pr}(\{HT, TH, TT\}) = \text{Pr}(\{HT\}) + \text{Pr}(\{TH\}) + \text{Pr}(\{TT\}) = \frac{3}{4}\).
Example selection of two out of five people: We can define a probability function \(\text{Pr}\) by assigning the same probability to each atomic event.
- \(\text{Pr}(\{12\}) = \text{Pr}(\{13\}) = \dots = \text{Pr}(\{45\}) = 1/10\).
So, the probability that 1 is among the selected \(\text{Pr}(\{12, 13, 14, 15\}) = \frac{4}{10}\).
Some basic probability rules
We can compute the probabilities of all events by summing the probabilities of the atomic events in it. So, the probabilities of the atomic events are building blocks for the whole probability function.
\(\text{Pr}(\emptyset) = 0\)
For any events \(A,B \subset S\) it holds
- \(\text{Pr}(A \cup B) = \text{Pr}(A) + \text{Pr}(B) - \text{Pr}(A \cap B)\)
- \(\text{Pr}(A \cap B) = \text{Pr}(A) + \text{Pr}(B) - \text{Pr}(A \cup B)\)
- \(\text{Pr}(A^c) = 1 - \text{Pr}(A)\)
Recap from the motivation of logistic regression: When the probability of an event is \(A\) is \(\text{Pr}(A)=p\), then its odds (in favor of the event) are \(\frac{p}{1-p}\). The logistic regression model “raw” predictions are log-odds \(\log\frac{p}{1-p}\).
Conditional probability
Definition: The conditional probability of an event \(A\) given an event \(B\) (write “\(A | B\)”) is defined as
\[\text{Pr}(A|B) = \frac{\text{Pr}(A \cap B)}{\text{Pr}(B)}\]
We want to know the probability of \(A\) given that we know that \(B\) has happened (or is happening for sure).
Two coin flips: \(A\) = “first coin is HEAD”, \(B\) = “one or zero HEADS in total”. What is \(\text{Pr}(A|B)\)? \(A\) = {HH, HT}, \(B\) = {TT, HT, TH} \(\to\) \(A \cap B = \{HT\}\)
\(\to\) \(\text{Pr}(A\cap B) = \frac{3}{4}\), \(\text{Pr}(A\cap B) = \frac{1}{4}\)
\(\to\) \(\text{Pr}(A|B) = \frac{1/4}{3/4} = \frac{1}{3}\)
More examples of conditional probability
COVID-19 Example: What is the probability that a random person in the tested sample has COVID-19 (event \(P\) “positive”) given that she has a positive test result (event \(PP\) “predicted positive”)?
\[\text{Pr}(P|PP) = \frac{\text{Pr}(P \cap PP)}{\text{Pr}(PP)}\]
Definition p-value: Probability of observed or more extreme outcome given that the null hypothesis (\(H_0\)) is true.
\[\text{p-value} = \text{Pr}(\text{observed or more extreme outcome for test-statistic} | H_0)\]
Probability in the Confusion Matrix
4 probabilities in confusion matrix
Sensitivity and Specificity
\(\to \atop \ \) Sensitivity is the true positive rate: TP / (TP + FN)
\(\ \atop \to\) Specificity is the true negative rate: TN / (TN + FP)
Positive/negative predictive value
\(\scriptsize\downarrow \ \) Positive predictive value: TP / (TP + FP)
\(\scriptsize\ \downarrow\) Negative predictive value: TN / (TN + FN)
Here TP, TN, FP, FN are the numbers of true positives, true negatives, false positives, and false negatives.
As the set of atomic events \(\{TP\}, \{FP\}, \{FN\}, \{TN\}\) we can define the probabilities of the events like \(\text{Pr}(\{TP\}) = \frac{TP}{N}\) with \(N = TP + FP + FN + TN\).
… as conditional probabilities
Sensitivity and specificity are conditional probabilities:
Sensitivity is the probability of a positive test result given that the person has the condition: \(\text{Pr}(PP|P) = \frac{TP}{TP + FN}\)
Specificity is the probability of a negative test result given that the person does not have the condition: \(\text{Pr}(PN|N) = \frac{TN}{TN + FP}\)
Positive predictive value is the probability of the condition given that the test result is positive: \(\text{Pr}(P|PP) = \frac{TP}{TP + FP}\)
Negative predictive value is the probability of the condition given that the test result is negative: \(\text{Pr}(N|PN) = \frac{TN}{TN + FN}\)
Note: The positive predictive value \(\text{Pr}(P|PP)\) is sensitivity \(\text{Pr}(PP|P)\) with “flipped” conditionality.
Bayes’ Theorem
Bayes’ Theorem is a fundamental theorem in probability theory that relates the conditional probabilities \(\text{Pr}(A|B)\) and \(\text{Pr}(B|A)\) to the marginal probabilities \(\text{Pr}(A)\) and \(\text{Pr}(B)\):
\[\text{Pr}(A|B) = \frac{\text{Pr}(B|A) \cdot \text{Pr}(A)}{\text{Pr}(B)}\]
Example: What is the probability that a random person in the tested sample has COVID-19 (\(P\) = positive) given that she has a positive test result (\(PP\) = predicted positive)?
\[\text{Pr}(P|PP) = \frac{\text{Pr}(PP|P) \cdot \text{Pr}(P)}{\text{Pr}(PP)}\] So, we can compute the positive predictive value \(\text{Pr}(P|PP)\) from the sensitivity \(\text{Pr}(PP|P)\) and the rate (or probability) of positive conditions \(\text{Pr}(P)\) and the rate (or probability) of positive tests.
Prevalence
- The rate (or probability) of the positive conditions in the population is also called Prevalence:
\[\text{Pr}(P) = \frac{P}{N} = \frac{TP + FN}{N}\]
Sensitivity and Specificity are properties of the test (or classifier) and are independent of the prevalence of the condition in the population of interest.
The Positive/Negative Predictive Values are not!
How the positive predictive value (PPV) depends on prevalence
We assume a test with sensitivity 0.9 and specificity 0.99. \(N = 1000\) people were tested.
Sensitivity = TP / P
Specificity = TN / N
PPV = TP / PP
Prevalence = P / N
Prevalence = 0.1
\(P\) = 100 \(N\) = 900
PPV = 90 / (90 + 9) = 0.909
From the positive tests 90.9% have COVID-19.
Prevalence = 0.01
\(P\) = 10 \(N\) = 990
PPV = 9 / (9 + 9.9) = 0.476
From the positive tests only 47.6% have COVID-19!
