Please enable JavaScript.

Coggle requires JavaScript to display documents.

Statistical Methods (2 Probability Distributions (Binomial…

- - - - Central Limit Theorem
- - - - set S: "sample space":
        \(A\subset S: \exists P(A)\in R\):
        P>=0 positive
        if A,B disjunct: P(A)+P(B) additive
        P(S)=1normalized
- - - - if analytic difficult: MC method
        simulate experiments
        compute ML estimates
        use e.g. sampling variance for estimating
      - if MC toilsome: graphical method
        taylor logL
        => \(logL(\theta)=logL_{max}-\frac{(\theta-\hat\theta)^2}{2\hat{\sigma^2}_{\hat\theta}}\)
        =>\(logL(\hat\theta\pm\hat\sigma_{\hat\theta})=logL_{max}-\frac{1}{2}\)
        
        use RCF lower bound :red_cross:
      - analytic method
        \(V[\hat\theta]=E[\hat\theta^2]-(E[\hat\theta])^2\)
        insert in sampling distribution
    - - with binned data
        \(logL(\nu_{tot},\theta)=-\nu_{tot}+\sum_{i=1}^Nn_ilog\nu_i(\nu_{tot},\theta)\)
    - - Gaussian
      - Exponential
    - - Bayesian
        knowledge contained in \(p(\theta|x)\)
        =>report mode
      - problem with uniform prior:
        normalization (usually not a problem)
        transformation of variables! Bayesian not invariant in general
  - - - estimator
        
        unbiased
        \(b=E[\hat\theta]-\theta\)
        if b=0 for all n
        
        MSE (mean square error)
        \(E[(\hat\theta-\theta)^2]=V[\hat\theta]+b^2\)
        
        why?
        combination of different experiments
        
        consistent: \(\hat\theta \xrightarrow {n\rightarrow\infty} \theta\) (minimum requirement)
        
        asymptotically unbiased
        b=0 for \(n\rightarrow\infty\)
        
        efficient
        =minimum variance
        
        MVB (minimum variance bound)
        RCF inequality:
        \(V[\hat\theta]\geq (1+\frac{\partial b}{\partial \theta})^2/E[-\frac{\partial^2logL}{\partial \theta^2}]\)
        
        sampling distribution is PD of estimator
        \(E[\theta(\vec x)]=\int\hat\theta g(\hat\theta;\theta)d\hat\theta=\int...\int\hat\theta (x)f(x_1;\theta)...f(x_n;\theta)dx_1...dx_n\) :!?:=likelihood?
    - - sample mean normal
        variance: \(\frac{1}{n-1}\)
- - - - \(E[y(\vec x)]=y(\vec \mu)\)
      - \(\sigma_y^2=\sum \frac{\partial y}{\partial x_i}\frac{\partial y}{\partial x_j} V_{ij}\)
- - - - chi2 statistic: \(\chi^2=\sum_{i=1}^N\frac{(n_i-\nu_i)^2}{\nu_i}\)
        for poisson data N with mean \(\nu\)
        
        follows chi2 distribution with N-1 degrees of freedom (distribution free) \(n_d\) (in general N-m m:parameters estimated from data)
        P-Value: \(P=\int_{\chi^2}^{\infty}f(z;n_d)dz\)
        interpretation of P-Value: probability of observing n_obs or more events under H_0
        vb=0.5 n_obs=5 give v_s=4.5±2.2 misleading, error on
      - depends on binning!:
        large binning->info loss
        but at least 5 entries per bin otherwise not distribution free
        =>for small data samples: Kolmogorov-Smirnov (no binning)
    - - each bin Poisson
- - - - sensitivity: separation of signal and background
- - - - Fisher separation
        define separation: \(J(a)=\frac{(\tau_0-\tau_1)^2}{\Sigma_0^2+\Sigma_1^2}\)
        optimal for Gaussian with common covariance
    - - NN
        
        structure
        
        single layer perceptron
        +hidden layer =>multilayer perceptron
        +>1hidden layers=>DeepNN
        
        neighboring layers only->feed-forward network
        
        \(t(x)=s(a_0+\sum_{i=1}^na_ix_i)\)
        s: activation function (eg. logistic sigmoid)
        
        supervised learning: minimize error function
      - BDT
        
        DT:
        define Impurity gain: 2p(1-p) Gini
        physically motivated, all events retained
        but (-) sensitive to statistical fluctuation