PRML CHAPTER1

• for input may be preprocessed
  
  • like fixed size of the image
  • also called feature extraction
  • speed up computation
  • easier to solve problem
• supervised learning
  
  • classification- discrete output
  • regression- continues output

• unsupervised learning

  • visualization
  • clustering
  • density estimation
    ...

• technique of reinforcement learning : find suirable actions to take in a given stituation in order to maximized the reward

• function which is linear in unknow parameters are called linear model

  • for the polynomial model: $y(x, \mathbf{\beta}) = \beta^0 + \beta_1 x + \beta_2 x^2 + .... + \beta_M x^M$ is nonliear in input x but linear in unknow parameter $\beta$
  • implement issue:
    
    • choose the value of coefficents/parameters/weights is determined by fitting data - minimizing error function - cost function
    • choose the order M - model selection/comparision
      
      • overfitting
      • FACT: as M increases, the magnitude of the coefficients typically gets larger
      • regulazation to fight overfitting
        
        • involves adding a penalty term to the error function (1.2) in order to discourage the coefficients from reaching large values
        • add $\lambda / 2 \| \mathbf{\beta} \|^2$ $\lambda$ is the importance of the regulazation term;
          
          • if use quadric regularizer - call ridge regression
        • weight decay? in neural network
        • $\lambda$ can suppressed over-fitting, reduce value of weight, but if $\lambda$ too large, weight goes to 0, will lead to poor fit

probability theory

• 提出probability 是想更加科学一些
• expectation
  
  • def: weighted averages of funtions
  • if we are given a finite number N of points drawn from the probability distribution or probability density
    
    • $E[f] ~- 1/N sum_{n=1,2...N}f(x_n)$
  • consider expectation of functions of several variables eg f(x,y)
    
    • expectation have subscript with repesct to different variable:
    • $E_x[f(x,y)]$ and $E_y[f(x,y)]$
• variance: $Var[x] = E[x^2] - E[x]^2$
  
  • consider functions of several variables eg f(x,y)
    
    • covariance: $cov[x,y] = E_{x,y}[{x - E[x]}{y^T - E[y^T]}]$

interpretation of probabilities

• popular: classical or frequentist way
  
  • the probability P of an uncertain event A, P(A) is defined by the frequency of that event based on previous observations
• another point of view: Bayesian view - probability provide a quanrification of uncertainty
  
  • for future event, we do not have historical database thus can not count the frequency.
  • but can measure the belief in a statement $a$ based on some 'knowledge', denote as P(a|K), different K can generate different P(a|K) and even same K can have different P(a|K) -- the belief is subjective
• Bayes rule
  
  • consider conditional probabilities
  • $P(A|B) = P(B|A) P(A)/P(B)$
    
    • interpretation: updating our belief about a hypothesis A in the light of new evidence B
      
      • in likelihood, it is, output brief of y/A given B/input values+paramters
    • P(A|B): posterior belief
    • P(A): prior belief
    • P(B|A): likelihood, ie the B(our model) will occur if A(the output value of the sample data) is true.
    • P(B) is computed by: $sum_{i=1,2,...}P(B|A_i)P(A_i)$ by marginalisation.
  • in machine learning, Bayes theorem is used to convert a priot ptobability $P(A) = P(\mathbf{\beta})$ into a porterior probability $P(A|B) = P(\mathbf{\beta}|y)$ by incorpoating the evidence provided by the observed data
    
    • for $\mathbf{\beta}$ in the polynormial curve fitting model, we can take an approach with Bayes theorem:
    • $P(\mathbf{\beta} | \mathbf{y}) =\frac{ P(\mathbf{y}|\mathbf{\beta}) p(\mathbf{\beta})}{P(\mathbf{y})}$
      
      • given data {y_1,y_2,...}, we want to know the $\beta$, cant get directly. $P(\mathbf{\beta} | \mathbf{y})$:= posterior probability
      • $P(\beta)$:= prior probability; our assumption of $\beta$
      • ${P(\mathbf{y})}$:= normalization constant since the given data is fixed
      • $P(\mathbf{y}|\mathbf{\beta})$:= likelihood function;
        
        • can be view as function of parameter $\beta$
        • not a probability distrubution, so intergral w.r.t $\beta$ not nessary = 1
      • state Bayes theorem as : posterior 8 likelihood × prior, consider all of these as function of parameters $\beta$
      • intergrate both side base on $\beta$: $p(y)= \intergral p(y|\beta)p(\beta)d\beta$
      • issue: particularly the need to marginalize (sum or integrate) over the whole of parameter space

different view of likelihood function

• likelihood function: $P(\mathbf{y}|\mathbf{\beta})$
• from frequentist way of interpretation:
  
  • parameter $\beta$ is a fixed parameter, the value is determined by 'estimator'
  • A widely used frequentist estimator is maximum likelihood, in which wis set to the value that maximizes the likelihood function
  • ie. choosing $\beta$ s.t. probability of the observed data is maximized
  • in practice, use negative log of likelihood function = log-likelihood:= error function(monotonically decreasing)
  • One approach to determining frequentist error bars is the bootstrap,
    
    • s1: 就是在已有的dataset(size N)里面random弄出L个dataset(size N) by drawing data from 已有的dataset(抽取方式是，可以重复抽，可以有的没有被抽中)
    • s2: looking at the variability of predictions between the different bootstrap data sets. then evaluate the accuracy of the estimates of the parameter
  • drawback: may lead to extreme conclusion if the dataset is bad, eg, a fair-looking coin is tossed three times and lands heads each time. in this case, we will generate parameter $\beta$ to make P(lands head) = 1
• from Bayesian viewpoint:
  
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