Теория вероятностей

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Категории: Рациональность  

• Связанное:

  • Bayesian reasoning
    

    A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.

    • Bayes' rule
      

      Bayes’ rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.

      • Bayes' rule: Guide
        

        The Arbital guide to Bayes’ rule

        • Bayes' Rule and its different forms
          
          This is an arc that includes different ways to look at Bayes’ Rule.
        • Bayes' Rule and its implications
          
          This is an arc that includes implications of the Bayes’s Rule.
        • Comprehensive guide to Bayes' Rule
          
          This is an arc that includes all Bayes content.
        • Introduction to Bayes' Rule odds form
          
          This is an arc that includes just enough content to teach about Bayes’s Rule odds form.
        • Wants to get straight to Bayes
          
      • Bayes' rule: Beginner's guide
        
        Beginner’s guide to learning about Bayes’ rule.
      • High-speed intro to Bayes's rule
        
        A high-speed introduction to Bayes’s Rule on one page, for the impatient and mathematically adept.
      • Bayes' rule: Definition
        
      • Bayes' rule: Probability form
        

        The original formulation of Bayes’ rule.

        • Odds form to probability form
          
        • Proof of Bayes' rule: Probability form
          
      • Frequency diagram
        

        Visualizing Bayes’ rule by manipulating frequencies in large populations

        • Frequency diagrams: A first look at Bayes
          
          The most straightforward visualization of Bayes’ rule.
      • Path: Multiple angles on Bayes's Rule
        
        A learning-path placeholder page for learning multiple angles on Bayes’s Rule.
      • Shift towards the hypothesis of least surprise
        
        When you see new evidence, ask: which hypothesis is _least surprised?_
      • Bayes' rule: Odds form
        

        The simplest and most easily understandable form of Bayes’ rule uses relative odds.

        • Introduction to Bayes' rule: Odds form
          
          Bayes’ rule is simple, if you think in terms of relative odds.
      • Proof of Bayes' rule
        

        Proofs of Bayes’ rule, with graphics

        • Proof of Bayes' rule: Probability form
          
      • Bayes' rule examples
        

        Interesting problems solvable by Bayes’ rule

        • Introductory Bayesian problems
          

          Bayesian problems to try to solve yourself, before beginning to learn about Bayes’ rule.

          • Blue oysters
            
            A probability problem about blue oysters.
          • Diseasitis
            
            20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What’s the chance you have Diseasitis?
          • Sock-dresser search
            
            There’s a 4⁄5 chance your socks are in one of your dresser’s 8 drawers. You check 6 drawers at random. What’s the probability they’ll be in the next drawer you check?
          • Sparking widgets
            
        • Realistic (Math 1)
          
          Real-life examples of Bayesian reasoning
      • Waterfall diagram
        

        Visualizing Bayes’ rule as the mixing of probability streams.

        • Waterfall diagrams and relative odds
          
          A way to visualize Bayes’ rule that yields an easier way to solve some problems
      • Belief revision as probability elimination
        
        Update your beliefs by throwing away large chunks of probability mass.
      • Probability notation for Bayes' rule
        

        The probability notation used in Bayesian reasoning

        • Probability notation for Bayes' rule: Intro (Math 1)
          
          How to read, and identify, the probabilities in Bayesian problems.
      • Bayes' rule: Vector form
        
        For when you want to apply Bayes’ rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)
      • Bayes' rule: Log-odds form
        
        A simple transformation of Bayes’ rule reveals tools for measuring degree of belief, and strength of evidence.
      • Bayes' rule: Functional form
        
        Bayes’ rule for to continuous variables.
      • Bayes' rule: Proportional form
        
        The fastest way to say something both convincing and true about belief-updating.
    • Bayesian update
      

      Bayesian updating: the ideal way to change probabilistic beliefs based on evidence.

      • Extraordinary claims require extraordinary evidence
        

        The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don’t provide quantitatively enough evidence.

        • Extraordinary claims
          
          What makes something an ‘extraordinary claim’ that requires extraordinary evidence?
      • Bayesian view of scientific virtues
        
        Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes’ rule, of course.
      • Ordinary claims require ordinary evidence
        
        Extraordinary claims require extraordinary evidence, but ordinary claims _don’t_.
      • Path: Insights from Bayesian updating
        
        A learning-path placeholder page for insights derived from the Bayesian rule for updating beliefs.
      • Strength of Bayesian evidence
        
        From a Bayesian standpoint, the strength of evidence can be identified with its likelihood ratio.
    • Humans doing Bayes
      

      The human use of Bayesian reasoning in everyday life

      • Explicit Bayes as a counter for 'worrying'
        
        Explicitly walking through Bayes’s Rule can summarize your knowledge and thereby stop you from bouncing around pieces of it.
    • Ignorance prior
      

      Key equations for quantitative Bayesian problems, describing exactly the right shape for what we believed before observation.

      • Inductive prior
        

        Some states of pre-observation belief can learn quickly; others never learn anything. An “inductive prior” is of the former type.

        • Индукция Соломонова
          

          Как просто и сверхразумно предсказывать последовательности данных, имея неограниченную вычислительную мощность.

          • Solomonoff induction: Intro Dialogue (Math 2)
            
            An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math
        • Laplace's Rule of Succession
          
          Suppose you flip a coin with an unknown bias 30 times, and see 4 heads and 26 tails. The Rule of Succession says the next flip has a 5⁄32 chance of showing heads.
        • Universal prior
          
          A “universal prior” is a probability distribution containing _all_ the hypotheses, for some reasonable meaning of “all”. E.g., “every possible computer program that computes probabilities”.
    • Likelihood
      

      • Relative likelihood
        
        How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.
      • Likelihood notation
        
      • Likelihood function
        
      • Likelihood ratio
        
    • Subjective probability
      

      Probability is in the mind, not in the environment. If you don’t know whether a coin came up heads or tails, that’s a fact about you, not a fact about the coin.

      • Probability distribution: Motivated definition
        
        People keep writing things like P(sick)=0.3. What does this mean, on a technical level?
    • Empirical probabilities are not exactly 0 or 1
      
      “Cromwell’s Rule” says that probabilities of exactly 0 or 1 should never be applied to empirical propositions—there’s always some probability, however tiny, of being mistaken.
    • Finishing your Bayesian path on Arbital
      
      The page that comes at the end of reading the Arbital Guide to Bayes’ rule
    • Interest in mathematical foundations in Bayesianism
      
      “Want” this requisite if you prefer to see extra information about the mathematical foundations in Bayesianism.
    • Posterior probability
      
      What we believe, after seeing the evidence and doing a Bayesian update.
    • Prior
      
      A state of prior knowledge, before seeing information on a new problem. Potentially complicated.
    • Prior probability
      
      What we believed before seeing the evidence.
    • Strictly confused
      
      A hypothesis is strictly confused by the raw data, if the hypothesis did much worse in predicting it than the hypothesis itself expected.
  • Odds
    

    Odds express a relative probability.

    • Odds: Introduction
      
      What’s the difference between probabilities and odds? Why is a 20% probability of success equivalent to 1 : 4 odds favoring success?
    • Odds: Refresher
      
      A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).
    • Odds: Technical explanation
      
      Formal definitions, alternate representations, and uses of odds and odds ratios (like a 1 : 2 chance of drawing a red ball vs. green ball from a barrel).
  • Probability
    

    The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.

    • Conditional probability
      

      The notation for writing “The probability that someone has green eyes, if we know that they have red hair.”

      • Conditional probability: Refresher
        
        Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?
    • Интерпретации «вероятности»
      

      Что _означает_, что честная монета имеет 50% вероятность выпасть решкой?

      • Correspondence visualizations for different interpretations of "probability"
        
      • Probability interpretations: Examples
        
      • Subjective probability
        

        Probability is in the mind, not in the environment. If you don’t know whether a coin came up heads or tails, that’s a fact about you, not a fact about the coin.

        • Probability distribution: Motivated definition
          
          People keep writing things like P(sick)=0.3. What does this mean, on a technical level?
    • Report likelihoods, not p-values
      

      • Likelihood functions, p-values, and the replication crisis
        
        What’s the whole Bayesian-vs.-frequentist debate about?
      • Report likelihoods not p-values: FAQ
        
    • Joint probability
      
      The notation for writing the chance that both X and Y are true.
  • Sample space
    

    The set of possible things that could happen in a part of the world that you are uncertain about.

    • Sample spaces are too large
      
      Sample spaces are often large, so it is hard to do probabilistic computations using a raw distribution over the sample space.
    • Uncountable sample spaces are way too large
      
      We can’t define probability distributions over uncountable sample spaces by just assigning numbers to each point in the sample space.
  • Square visualization of probabilities on two events
    

    • Square visualization of probabilities on two events: (example) Diseasitis
      
      But it _seems_ like the patient with the black tongue depressor has diseasitis…
  • Two independent events
    

    What do pair of dice, pair of coins, and pair of people on opposite sides of the planet all have in common?

    • Two independent events: Square visualization
      
  • Mutually exclusive and exhaustive
    
    The condition needed for probabilities to sum to 1
  • Normalization (probability)
    
    That thingy we do to make sure our probabilities sum to 1, when they should sum to 1.
  • Probability distribution (countable sample space)
    
    A function assigning a probability to each point in the sample space.
  • Probability distribution: Motivated definition
    
    People keep writing things like P(sick)=0.3. What does this mean, on a technical level?
  • Well-calibrated probabilities
    
    Even if you’re fairly ignorant, you can still strive to ensure that when you say “70% probability”, it’s true 70% of the time.

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Оригинал: Probability theory (читать на GreaterWrong)