♎
Limited AI
  • Machine Learning
    • Linear Model Cheating Sheet
    • Nonlinear Model Cheating Sheet
    • General Linear Model 1
    • General Linear Model 2
    • General Linear Model 3
    • Tree Based Methods
    • Tree Based Methods Supplement
    • XG,Cat,Light__Boosting
    • KNN&PCA
    • Model Performance
    • Model Evaluation
    • Code Practice
      • KNN
      • Decision Tree Python Code
    • Data and Feature Engineering
      • Handle Bias Data
      • Cold Start Problem
  • Deep Learning
    • Summary v2
    • Basic Neural Network
      • From Linear to Deep
      • Perceptron and Activation Function
      • NN network Details
      • Backpropagation Details
      • Gradient Vanishing vs Gradient Exploding
    • Basic CNN
      • Why CNN
      • Filter/ Convolution Kernel and Its Operation
      • Padding& Stride
      • Layers
      • Extra:From Fully Connected Layers to Convolutions
      • Extra: Multiple Input and Multiple Output Channels
    • Advance CNN
      • Convolutional Neural Networks(LeNet)
      • Deep Convolution Neural Networks(AlexNet)
      • Networks Using Blocks (VGG)
      • Network in Network(NiN)
      • Multi-Branch Networks(GoogLeNet&I mageNet)
      • Residual Networks(ResNet) and ResNeXt
      • Densely Connected Networks(DenseNet)
      • Batch Normalization
    • Basic RNN
      • Seq Model
      • Raw Text to Seq
      • Language Models
      • Recurrent Neural Networks(RNN)
      • Backpropagation Through Time
    • Advance RNN
      • Gated Recurrent Units(GRU)
      • Long Short-Term Memory(LSTM)
      • Bidirectional Recurrent Neural Networks(BRNN)
      • Encoder-Decoder Architecture
      • Seuqence to Sequence Learning(Seq2Seq)
    • Attention Mechanisms and Transformers
      • Queries, Keys, and Values
      • Attention is all you need
        • Attention and Kernel
        • Attention Scoring Functions
        • The Bahdanau Attention Mechanism
        • Multi-Head Attention
        • Self-Attention
        • Attention的实现
      • The Transformer Architecture
        • Extra Reading
        • 最短的最大路径长度
      • Large-Scaling Pretraning with Transformers
        • BERT vs OpenAI GPT vs ELMo
        • Decoder Model框架
        • Bert vs XLNet
        • T5& GPT& Bert比较
        • 编码器-解码器架构 vs GPT 模型
        • Encoder vs Decoder Reference
      • Transformers for Vision
      • Transformer for Multiomodal
    • NLP Pretraining
      • Word Embedding(word2vec)
        • Extra Reading
      • Approximate Training
      • Word Embedding with Global Vectors(GloVe)
        • Extra Reading
        • Supplement
      • Encoder(BERT)
        • BERT
        • Extra Reading
      • Decoder(GPT&XLNet&Lamma)
        • GPT
        • XLNet
          • XLNet架构
          • XLNet特点与其他比较
      • Encoder-Decoder(BART& T5)
        • BART
        • T5
  • GenAI
    • Introduction
      • GenAI Paper Must Read
      • GenAI六个阶段
    • Language Models Pre-training
      • Encoder-Decoder Architecture
      • Encoder Deep Dive
      • Decoder Deep Dive
      • Encoder VS Decoder
      • Attention Mechanism
      • Transformers
    • Example: Llama 3 8B架构
    • Fine-Tuning Generation Models
    • RAG and Adavance RAG
    • AI Agent
  • Statistics and Optimization
    • A/B testing
    • Sampling/ABtesting/GradientMethod
    • Gradient Decent Deep Dive
  • Machine Learning System Design
    • Extra Reading
    • Introduction
  • Responsible AI
    • AI Risk and Uncertainty
      • What is AI risk
      • General Intro for Uncertainty Quantification
      • Calibration
      • Conformal Prediction
        • Review the linear regression
        • Exchangeability
        • Split Conformal Prediction
        • Conformalized Quantile Regression
        • Beyond marginal coverage
        • Split Conformal Classification
        • Full Conformal Coverage
        • Cross-Validation +
        • Conformal Histgram Regression
    • xAI
      • SHAP value
  • Extra Research
    • Paper Reading
    • Reference
Powered by GitBook
On this page
  • Exchangeable random variables
  • Prediction without covariates
  • One-sided prediction interval without covariates
  1. Responsible AI
  2. AI Risk and Uncertainty
  3. Conformal Prediction

Exchangeability

PreviousReview the linear regressionNextSplit Conformal Prediction

Last updated 1 year ago

Exchangeable random variables

Prediction without covariates

Lemma 1

Suppose Z1,⋯ZnZ_1, \cdots Z_nZ1​,⋯Zn​ are exchangeable random variables. For any α∈{0,1}\alpha \in \{0, 1\}α∈{0,1}, P[Zn≤Q^n(α)]≥αP[Z_n \leq \hat{Q}_n(\alpha)] \geq \alphaP[Zn​≤Q^​n​(α)]≥α

Moreover, if Z1,⋯ZnZ_1, \cdots Z_{n}Z1​,⋯Zn​ are a.s distinct, P[Zn≤Q^n+1(α)]≤α+1nP[Z_{n} \leq \hat{Q}_{n+1}(\alpha)] \leq \alpha + \frac{1}{n}P[Zn​≤Q^​n+1​(α)]≤α+n1​

  • key proof: using the uniform distribution or these order statistics?

Lemma 2

Suppose Z1,⋯Zn+1Z_1, \cdots Z_{n+1}Z1​,⋯Zn+1​ are exchangeable random variables. For any α∈{0,1}\alpha \in \{0, 1\}α∈{0,1}, define αn\alpha_nαn​ as αn=(1+1n)α\alpha_n = (1 + \frac{1}{n})\alphaαn​=(1+n1​)α. Then, P[Zn+1≤Q^n+1(αn)]≥αP[Z_{n+1} \leq \hat{Q}_{n+1}(\alpha_n)] \geq \alphaP[Zn+1​≤Q^​n+1​(αn​)]≥α

Moreover, if Z1,⋯Zn+1Z_1, \cdots Z_{n+1}Z1​,⋯Zn+1​ are a.s distinct, P[Zn+1≤Q^n+1(αn)]≤α+1nP[Z_{n+1} \leq \hat{Q}_{n+1}(\alpha_n)] \leq \alpha + \frac{1}{n}P[Zn+1​≤Q^​n+1​(αn​)]≤α+n1​.

  • Key of the proof:

    • use lemma 1 and change n points to (n+1) points of these order statistics

    • and you cannot jump for two points since you only added one point into the empirical distribution

One-sided prediction interval without covariates