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STDSR, Bandit Algorithms, Poisson Distribution Discrete distr. Number…

- - - - Given our sample, the likelihood function L(parameter) represents the joint density of all observations, assuming they are independent:
  - - - if you run 100 experiments a 95%-confidence interval will contain the value of parameter 95 times
  - - - ERRORS
        
        Type I
        
        Occurs when we reject null hypothesis and state the difference between data and study
        
        Avoid by using p-value. Lower p-value -less chance to make a Type I error
        
        P-value: given a chance model, results are as extreme as described
        
        Type II
        
        Occurs when we state that there is no difference, but in fact there is and H0 is false
        
        Avoid by using power. Higher power allows to detect Type II Errors
        
        Power: probability that it will be able to find the effect it is
        looking for, assuming that the effect is there
    - - Population Standard Deviation is known?
        
        Yes
        
        Sample size >= 30?
        
        yes
        
        Z-test
        
        compares whether the sample mean (X) is significantly different from a known population mean (μ) when the population standard deviation (σ) is known
        
        no
        
        T-test
        
        determines whether a sample mean differs significantly from a known population mean.
        
        no
        
        T-test
      - Chi-Square GoF Test
        
        One sample test
        Used when you have counts of values for a categorical variable.
        
        H0 - assumes that there is no significant difference between the observed and the expected value.
        
        Ha - assumes that there is a
        significant difference between the observed and the expected value.
      - ANOVA
        
        tests for a statistically
        significant difference among the groups(3 or more)
        
        H0 - average of the groups is equal
        
        Ha - Difference between group average
  - - - function that maps a variable's observed values to their cumulative probabilities.
        
        1 indicator function
        1(Xi <= X) = 1 if Xi <= X
        0 otherwise
  - - - assesses the quality of a sample and establishes the existence and consistency of estimators
        
        Where Fn(x) is ECDF
  - - - H0 population medians are equal
      - H1 population medians are not equal
    - - used to decide if a sample comes from a population with a specific distribution
        
        Two sample Version
    - - sign test
        
        X is continious r.v and m is the median of X
        
        Y is number of negative values for
        X1-m, X2-m, ... Xn - m
        
        Y has Bin(n, 1/2) and is a statistic
      - Signed ranked test
        
        Same idea but |X1-m|, |X2-m| ...
        
        After calculating, sort the values and value of the magnitude is R - rank
        
        If Difference is negative multiply R by -1
        
        Statistic would be the sum of all R's
        
        H0 In the population, the median difference of the two dependent samples is equal
        
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        H1 In the population, the median difference is positive (one-sided)
        
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    - - ADVANTAGES:
        
        No assumtions about underlying distribution
        2.Less sensitive to outliers and skewed distributions
        
        More robust and less affected by violations of assumptions
      - DISADVANTAGES:
        
        Lower Power
        
        Less informative results
        
        Limited applications
        
        Less interpretability
- - - - We are left with B statistics of interest which we can use to calculate anything we need
    - - Parametric Bootstrap: We know that F belongs to a parametric family of distributions and we just estimate its parameters from the sample. We generate samples from F using the estimated parameters.
      - Non-parametric Bootstrap: We do not know the form of F and we estimate it by the empirical distribution Fn obtained from the data
    - - it is a cross-validation technique and, therefore, a form of resampling. It is especially useful for bias and variance estimation.
        
        Given a sample of size n, a jackknife estimator can be built by aggregating the parameter estimates from each subsample of size (n-1) obtained by omitting one observation
        
        Linear approximation of Bootstrap
  - - - Consist from:
        
        M Bins with width b 2.Anchor point
        
        A histogram density estimator is a step function with a constant value within each of the bins, where the constant is given by the proportion of data points Xi which fall in the bin divided by the bin volume.
    - - Let X1..Xn be univariate i.i.d sample drawn from some distrib. with unknown density f
        
        k - kernel >= 0
        h>0 - bandwidth, smoothing parameter
        
        For a function to be a kernel:
        
        f integrates to 1
        2.E(f) = 0
        
        Var(f) is finite
        
        h - bandwidth as small as data allows
        Beware of bias-variance tradeoff
        
        More dimensions for KDE
        
        Kh - is a scaled kernel build using a kernel function
        
        H - pos. def symmetric matrix of smoothing h
        
        Silverman rule-of-thumb
        
        if Gaussian Basis function is used, h can be estimated as follows:
        
        !!!
        If density not close to normal it can yield widely inaccurate estimates
        
        sigma is the estimate of SD samples
  - - - function that maps probabilities to values of a random variable
        
        for a given probability p, the inverse CDF returns the value x such that the cumulative distribution function (CDF) of x is equal to p
        
        transforms uniformly distributed random variables into random variables with a specific distribution
    - - allows sampling from a target distribution f(x) by utilizing a simpler proposal distribution g(x) from which samples are easier to draw.
        
        method involves generating samples from g(x) and then probabilistically accepting or rejecting each sample based on how well it approximates f(x).
    - - improve the efficiency and accuracy of statistical estimates by using a different distribution. Focuses on sampling from areas that contribute more significantly to the value of the integral or estimation objective
        
        Importance Sampling can dramatically reduce variance in the estimates compared to simple random sampling
  - - - Transition Matrix
        
        Stationary Distribution - p
        Matrix - A
  - - - Random Walk
        uses a random walk proposal distribution. It is designed to generate a sequence of sample values from a probability distribution by creating a Markov chain that has the desired distribution as its equilibrium distribution.
        
        1.Initialize: Start with an arbitrary point x0 initial state of MC
        
        Iterate: For each step t, perform the following:
        A)Create a candidate for the next sample
        using a symmetric proposal distribution
        B)Calculate Acceptance Probability: Compute the acceptance ratio
        C) Accept or Reject: Generate a uniform random number u from U(0;1) and accept or reject based on u and a
        
        Metropolis Hasting
        generates a sequence of samples from a target distribution by constructing a Markov chain that reaches equilibrium at the desired distribution :
        
        1) Generate a Proposal: From the current state xt generate state using proposal dist.
        2) Compute the Acceptance Ratio: Calculate the acceptance probability a
        3) Accept or Reject the Proposal
- - - - theta - prob of heads
        D - prob of getting certain amount of heads and tails(h and t resp.)
        1 - theta - prob ot tails
        
        Lets take beta-distribution where
        alpha =2 and Beta = 2
        
        Find Posterior distrbution
        
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- - - - Data and sampling distributions
        
        Binomial Distribution
        
        2 outcomes per trial
        
        Probabilities are the same
        
        Number of trials is fixed
        
        Each trial is independent
        
        Normal Distribution
        Describes data that clusters around a mean or average. The shape of the curve is symmetrical and bell-shaped, defined by its mean and variance
      - Central Limit Theorem
        
        Let (X1, X2 ... Xn) be i.i.d r.v.
        with PDF, mean and variance
        
        CLT states that distribution of Z(r.v.) converges to standart normal deviation as n -> inf
        
        Simple example: Sum of 2 dices rolled some amount of times would be distributed like following
      - Law of Large Numbers
        
        The more sample size is, its mean is approaching true mean(mean of the population
        
        or
        
        As the number of trials or observations increases the actual or observed probability approaches theoretical or expected probability
  - - - Random Variables
        
        Discrete - value
        
        Continious - value on interval
        
        Mass of random animal on planet
      - Probability Density Functions
        
        Calculate the Def integral to get P(X) where X is cont.
        
        CDF(Cumulative DF)
        
        Def. integral where we are interested in area of curve to the left from x
      - Expected Value - mean of X(r.v.)
        
        Value of E(X) based on some probability distribution
        
        Properties
      - Varience Var(X)
      - Sample Varience
        
        Standart Deviation
      - Statistic - function of >= 1 r.v.(no unknown parameter)
        
        Population - all elements of group
        
        Sample: represents main properties of population
        
        Estimator - function that allows to calculate estimate of the parameter of a sample
        
        Constant estimator converges to parameter as sample increases
        
        Bias of an estimator:
        Bias = E(Bias) - parameter
        if = 0 unbiased
        otherwise under/over estimated
        
        1) solves backwards problem
        2)Get data and observe how it was generated
        3)quantifies the uncertainty
- - - - Select arm that maximizes this function, where xj -average reward of an arm jhas been played, and t is the total number of plays
    - - Uses Bayesian approach; each arm has a prior distribution of rewards which updates as feedback (rewards) is received. At each step, sample a random value from the current distribution for each arm and choose the arm with the highest sample.
- - - - Chi-distribution
        A distribution of the sum of the squares of k independent standard normal random variables, primarily used in hypothesis testing.