Likelihood of a Single Observation For Logistic Regression Model

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How are you arriving the expression for Likelihood Function for a Single Observation?

The expression for the likelihood function of a single observation is derived based on the following reasoning:
1. Understanding the Logistic Function:
In logistic regression, we model the probability that a binary outcome $y^{(i)}$ equals 1 (i.e., the event happens) using the logistic function:
$$\phi(z) = \frac{1}{1 + e^{-z}}$$
Where $z = \mathbf{w}^T \mathbf{x}^{(i)}$, with $\mathbf{w}$ being the weight vector and $\mathbf{x}^{(i)}$ the feature vector for the $i$-th observation.
2. Probability of Class 1 and Class 0:
The probability that $y^{(i)} = 1$ (the event happens) given the input features $\mathbf{x}^{(i)}$ is:
$$P(y^{(i)} = 1 \mid \mathbf{x}^{(i)}; \mathbf{w}) = \phi(\mathbf{w}^T \mathbf{x}^{(i)})$$
The probability that $y^{(i)} = 0$ (the event does not happen) is:
$$P(y^{(i)} = 0 \mid \mathbf{x}^{(i)}; \mathbf{w}) = 1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})$$
3. Likelihood Function for a Single Observation:
The likelihood for a single observation $i$ is the probability of observing $y^{(i)}$ given the input $\mathbf{x}^{(i)}$ and model parameters $\mathbf{w}$. Since $y^{(i)}$ can be either 0 or 1, we can combine the two cases into a single expression using the following formulation:
$$P(y^{(i)} \mid \mathbf{x}^{(i)}; \mathbf{w}) = \phi(\mathbf{w}^T \mathbf{x}^{(i)})^{y^{(i)}} \cdot \left(1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})\right)^{1 - y^{(i)}}$$
4. Explanation of the Formula:
When $y^{(i)} = 1$:
The expression simplifies to:
$$P(y^{(i)} = 1 \mid \mathbf{x}^{(i)}; \mathbf{w}) = \phi(\mathbf{w}^T \mathbf{x}^{(i)})$$
This is because $y^{(i)} = 1$ makes the first term $\phi(\mathbf{w}^T \mathbf{x}^{(i)})^{y^{(i)}}$ become $\phi(\mathbf{w}^T \mathbf{x}^{(i)})$ and the second term $\left(1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})\right)^{1 - y^{(i)}}$ becomes 1.
When $y^{(i)} = 0$:
The expression simplifies to:
$$P(y^{(i)} = 0 \mid \mathbf{x}^{(i)}; \mathbf{w}) = 1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})$$
This is because $y^{(i)} = 0$ makes the first term $\phi(\mathbf{w}^T \mathbf{x}^{(i)})^{y^{(i)}}$ become 1 (since anything raised to the power of 0 is 1) and the second term $\left(1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})\right)^{1 - y^{(i)}}$ becomes $1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})$.
5. General Case:
By combining both cases (i.e., whether $y^{(i)}$ is 0 or 1) into a single expression, we arrive at the following likelihood function for a single observation:
$$P(y^{(i)} \mid \mathbf{x}^{(i)}; \mathbf{w}) = \phi(\mathbf{w}^T \mathbf{x}^{(i)})^{y^{(i)}} \cdot \left(1 - \phi(\mathbf{w}^T \mathbf{x}^{(i)})\right)^{1 - y^{(i)}}$$
This expression allows us to compute the probability of observing the outcome $y^{(i)}$ given the input features and the model's parameters, regardless of whether $y^{(i)}$ is 0 or 1.
Summary:
The key idea is that this formulation captures both possible outcomes for $y^{(i)}$ in a single expression, by raising the probability to the power of $y^{(i)}$. This way, when $y^{(i)}$ is 1, the first term dominates, and when $y^{(i)}$ is 0, the second term dominates, effectively covering both scenarios.

Reference
Python Machine Learning
Sebastian Raschka 
Packt Publishing (2015)

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Questions from Course 'Pair Programming With a Large Language Model'

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Course by DeepLearning.AI: Pair Programming with a Large Language Model

1. What are the four parts of a prompt?

Ans:
1. Priming of the Bot / System Role
2. Instruction / Query / Problem
3. Decoration
4. Input (Optional: Depends on whether it is Zero Shot Prompting or Few Shot Prompting)

2. What is zero shot prompting?

3. What is few shot prompting?

4. What things should a developer take care of when using an LLM as a Pair Programmer?

When using a Large Language Model (LLM) as a pair programmer, a developer should be mindful of several key considerations to ensure productive collaboration and to mitigate potential risks. Here are some important things to keep in mind:
1. Understanding the Model’s Strengths and Limitations
Strengths: LLMs are excellent at generating code snippets, suggesting improvements, providing explanations, and automating repetitive tasks. They can also assist in learning new languages or frameworks by providing quick examples and explanations.
Limitations: LLMs lack real-world understanding, may generate incorrect or inefficient code, and can make errors in complex logic or architecture. They might not understand the specific context of a project or the nuances of a development environment.
2. Code Review and Verification
Manual Review: Always review and test code suggestions from an LLM. The model can produce code that looks correct but has subtle bugs or security vulnerabilities.
Testing: Implement proper unit tests and integration tests to verify the correctness of any generated code before merging it into production.
3. Security and Privacy
Sensitive Data: Avoid sharing sensitive or proprietary information with the LLM. Be cautious about inputting code that contains secrets, credentials, or personally identifiable information (PII).
Data Handling: Understand how the LLM processes and stores data, especially if using a third-party API. Ensure compliance with data protection regulations and company policies.
4. Bias and Ethical Considerations
Bias: LLMs can sometimes reflect biases present in their training data, leading to suggestions that are discriminatory or unethical. Be vigilant about such biases and correct them if they appear in code or comments.
Ethical Use: Use the LLM ethically, ensuring it contributes positively to the project and does not introduce unethical practices, such as generating code for malicious purposes.
5. Context Awareness
Project Context: The LLM may not be fully aware of the project’s broader context, such as architectural patterns, long-term goals, or existing codebase nuances. Supplement its suggestions with your contextual knowledge.
Documentation: Ensure that generated code fits well within the existing codebase, follows project conventions, and is well-documented to maintain consistency.
6. Model Dependency
Avoid Over-Reliance: While LLMs are powerful tools, avoid becoming overly reliant on them. Maintain and develop your own problem-solving and coding skills.
Critical Thinking: Use the LLM as a tool to enhance your productivity, not as a replacement for critical thinking. Always question and evaluate the model’s suggestions.
7. Collaboration and Communication
Human Collaboration: Balance the use of LLMs with collaboration among human team members. Human insight and experience are invaluable, especially in complex or novel situations.
Feedback: Provide feedback to the LLM when possible, to improve its suggestions over time (if using a system that allows learning or customization based on feedback).
8. Legal and Compliance Considerations
Licensing Issues: Ensure that any code generated by the LLM does not violate software licenses or intellectual property rights.
Compliance: Make sure that the code and practices suggested by the LLM comply with industry regulations and standards applicable to your project.
9. Performance and Efficiency
Code Quality: Evaluate the performance and efficiency of the code generated by the LLM. It may generate suboptimal solutions that require refinement.
Optimization: Be prepared to optimize or refactor the generated code to meet performance requirements or to improve readability and maintainability.
10. Continual Learning
Stay Updated: Continuously learn and stay updated on the capabilities and limitations of LLMs as they evolve. This will help you use them more effectively and responsibly.
Skill Development: Use the LLM as a learning tool to deepen your understanding of new languages, frameworks, or algorithms by exploring its suggestions and explanations.
By considering these aspects, developers can effectively and responsibly integrate LLMs into their workflow, enhancing productivity while minimizing potential risks.

5. If I ask an LLM to add comments to the code it provides to explain its work:
Which prompt engineering technique am I using? Direction Specific Prompting or Chain of Thought

When you ask an LLM to add comments to the code it provides in order to explain its work, you are primarily using Direction Specific Prompting.
Here's why:
Direction Specific Prompting: You are guiding the model to produce an output that not only includes the code but also adds explanatory comments. This is a form of directing the model to follow a specific set of instructions or format—namely, to include comments that explain the code. The prompt is specific about the desired format and content, instructing the model to produce code that is annotated with explanations.
Chain of Thought (CoT) Prompting (for contrast):
Chain of Thought Prompting involves encouraging the model to explicitly articulate its reasoning process as it arrives at a solution or answer. In the context of coding, this might involve asking the model to describe each step of the problem-solving process before writing the code, rather than just annotating the code after it has been written.
Example Comparison:
Direction Specific Prompting:
Prompt: "Please write a Python function that sorts a list of numbers and add comments to explain each part of the code."
This is direction-specific because you're telling the model to generate the code and include comments in a specific format.
Chain of Thought Prompting:
Prompt: "Explain your thought process for sorting a list of numbers, then write the Python function based on that explanation."
This would be Chain of Thought, as you're asking the model to articulate its reasoning before generating the final output.
So, in your scenario, Direction Specific Prompting is the technique being used.

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