Google

The fundamental principle for making a prediction in many machine learning models is to select the output that has the highest probability given an input. This is formally expressed as: $$\hat{y} = \underset{y \in Y}{\arg\max}\, \text{Pr}(y|x)$$ In this formula, $x$ is the input, $Y$ is the set of all possible outputs, $y$ is a candidate output from that set, and $\hat{y}$ is the final predicted output. The prediction $\hat{y}$ is chosen because it maximizes the conditional probability $\text{Pr}(y|x)$.

General Formula for Prediction via Maximum Probability

The prediction of a Large Language Model, denoted as $$\hat{y}$$, when provided with a full context $$c$$ and an input $$z$$, is determined by selecting the output $$y$$ that maximizes the conditional probability. This process is formally expressed by the formula: $$\hat{y} = \underset{y}{\arg\max}\, \text{Pr}(y|c, z)$$ This prediction often serves as the target or 'gold standard' when learning a compressed representation of the context.

LLM Prediction with Full Context

Prompt ensembling involves using a set of $$K$$ distinct prompts, $$\mathbf{x}_1, ..., \mathbf{x}_K$$, to perform a task. The process consists of two main steps. First, for each prompt $$\mathbf{x}_i$$, a Large Language Model (LLM) is used to find the best prediction, $$\hat{\mathbf{y}}_i$$, by maximizing the conditional probability: $$\hat{\mathbf{y}}_i = \arg\max_{\mathbf{y}_i} \Pr(\mathbf{y}_i|\mathbf{x}_i)$$. Second, these $$K$$ individual predictions are combined to form a new prediction. This formalizes the approach of generating multiple outputs and then combining them to improve results.

Mathematical Formulation of Prompt Ensembling

A simple method to score a reasoning path is to count the number of steps classified as 'correct'. This can be represented by the formula: 

$$r(\mathbf{x}, \mathbf{y}) = \sum_{k=1}^{n_s} \delta(\text{correct}, C(\mathbf{x}, \bar{\mathbf{y}}_{\le k}))$$

where:
- $$r(\mathbf{x}, \mathbf{y})$$ is the total reward for the reasoning path $$\mathbf{y}$$ given the input $$\mathbf{x}$$.
- $$n_s$$ is the total number of steps in the reasoning path.
- $$C(\mathbf{x}, \bar{\mathbf{y}}_{\le k})$$ is the classification output for step $$k$$, determined by selecting the label with the maximum probability.
- $$\delta(a, b)$$ is the Kronecker delta function, which is $${}1$$ if $$a=b$$ and $${}0$$ otherwise. In this context, it equals $${}1$$ if the classification for step $$k$$ is 'correct'.

Formula for Scoring Reasoning Paths by Counting Correct Steps

A spam detection model is designed to classify emails as either 'Spam' or 'Not Spam'. For a particular email input, the model calculates the following probabilities: `Pr(y='Spam' | x) = 0.48` and `Pr(y='Not Spam' | x) = 0.52`. The model predicts 'Not Spam', but this turns out to be incorrect as the email was actually spam. Based on the principle of selecting the output with the maximum probability, analyze and explain why the model made this specific prediction, even though it was factually wrong.

Model Prediction vs. Ground Truth

A language model is tasked with completing a sentence. For a given input context, it calculates the following probabilities for the next word: `Pr('mat'|context) = 0.4`, `Pr('chair'|context) = 0.3`, `Pr('roof'|context) = 0.2`, and `Pr('dog'|context) = 0.1`. Based on the principle of selecting the output with the maximum probability, identify the model's prediction. Then, explain why a model strictly following this principle would *not* choose 'chair', even though it is a plausible real-world completion.

Analyzing a Model's Prediction Choice

The prediction of a Large Language Model, denoted as $$\hat{y}_{\sigma}$$, when using a soft prompt $$\sigma$$ (a compressed context) and an input $$z$$, is determined by selecting the output $$y$$ that maximizes the conditional probability. This is formally expressed as: $$\hat{y}_{\sigma} = \underset{y}{\arg\max}\, \text{Pr}(y|\sigma, z)$$ This prediction is compared against the prediction from the full context to optimize the soft prompt.

Formula for LLM Prediction with Compressed Context

A classification model is given an input, `x`, and must choose an output, `y`, from the set of possible classes {A, B, C, D}. The model's decision rule is to select the class that has the highest conditional probability, `Pr(y|x)`. Given the following probabilities calculated by the model for the input `x`, what will its final prediction be?

- `Pr(y=A | x)` = 0.15
- `Pr(y=B | x)` = 0.55
- `Pr(y=C | x)` = 0.25
- `Pr(y=D | x)` = 0.05

Learn Before

Related