Sp18_P2.pptx

Sp18_P2.pptx

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  Md Abul Hayat GraduateAssistant Spring 2018 04/06/2018 Sampling Techniques and Their Uses
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  Contents 1. Motivation 2. Techniques •UoU • Rejection Sampling • Metropolis-Hestings • Gibbs sampling 3. Applications
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  Usefulness • AF447, aflight from Rio de Janeiro, Brazil to Paris, France crashed on 1 June 2009. • All 228 people died. • Brazil Navy and French BEA searched the wreckage for one year, but failed. • In July 2010, the U.S.-based search consultancy Metron, Inc. had been engaged to draw up a probability map of where to focus the search, based on prior probabilities from flight data and local condition reports, combined with the results from the previous searches. • It was described as "classic" Bayesian search methods, an approach that had previously been successful in the search for the submarine USS Scorpion and SS Central America. • July 2011, full report was published and belongings of the passengers were returned.
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  Universality of Uniform Plugin a RV to it’s CDF. We get a uniform RV. Alternative Statement Plug in a uniform RV to a ICDF. We get a RV with that CDF
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  Universality of Uniform: Intuition
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  Universality of Uniform: Requirements • To sample from a RV, we need to know the ICDF of the RV • We need samples from uniform distribution. • This is why all programming languages have a uniform sample generator. Problems • We do not always have a closed form of CDF and ICDF. • Gets complicated for multivariate case.
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  Rejection Sampling • Wedesire samples from • We are able to draw samples from , and we know a value c such that cQ∗(x) > P∗(x) for all x. • A point (x; u) is generated at random under the curve • If this point also lies below , then it is accepted. • Otherwise we reject the sample. Target density P(x) Proposal density Q(x)
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  Rejection Sampling :Algorithm 1. Get a proposal density cQ* that covers target density P* 2. Draw a sample x𝑖 from Q* 3. Draw a sample ui from uniform distribution of [0, 1] 4. Reject sample X𝑖, if ui > P* (x𝑖)/ cQ*(x𝑖) 5. Accept sample X𝑖, if ui ≤ P* (x𝑖)/ cQ*(x𝑖) 6. Go to step #2 x𝑖 P* (x𝑖) cQ*(x𝑖)
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  Rejection Sampling :Summary • We need a proposal density that covers the target distribution properly • Again we need sampling from a uniform distribution. • The acceptance rate of is 1/c • For n dimensions, acceptance rate is 1/cn • Even though we use unnormalized target density P*(X) , we get samples from P(X) • Generated samples are independent
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  Monte Carlo Method •The samples xi from the distribution of X should be independent. • Code name of Law of Large Numbers.
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  Markov Chain A sequenceof random variables X0, X1, X2, … taking values in the state space {1, 2, . . . , M} is called a Markov chain if for all n ≥ 0, (Markov Property) • Here, state space = {Left, Right} • Random variable Xn denotes position at time t. • P(Xn+1 = j| Xn = i) is called the transition probability from state i to state j • Markov Chain / Process is a Random Process
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  Markov Chain • qij= P(Xn+1 = j| Xn = i) is called the transition probability from state i to state j. • The M x M matrix Q = (qij) is called the transition matrix of the chain. • Each row of Q sum up-to 1 ;
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  • We cancreate a Markov process with state space same as the support of a random variable we want to sample from. Stationary distribution is same as PDF/PMF. • We have to properly design transition matrix Q • In this example if we want to sample from a Bernoulli RV with p=0.416 and 1-p=0.58, this can be done using the Markov chain but only after being stationary. • What happens if q12 = q21 = 1 ?
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  Markov Chain MonteCarlo • Metropolis-Hestings • Gibbs Sampling
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  Suppose, for astationary distribution s, we have to find out a transition matrix P. We create this Markov Chain with the help of another transition matrix Q = = = =
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  Metropolis-Hestings Algorithm :Example
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  Metropolis-Hestings Algorithm :Summary • MH produces dependent (linearly correlated) samples. • It requires a burn-in/warm-up period to reach stationarity, after that we can get samples. • No rule for getting the best proposal density and burn-in time. An illustrative example of Metropolis sampler can be found here. Some cases of Metropolis-Hestings Sampler. • Metropolis Sampler: If, (like Gaussian) then, . So, if then, the sampler samples more from high density area as expected.
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  Gibbs Sampling • Gibbs samplingcan be viewed as a Metropolis method in which a sequence of proposal distributions Q are defined in terms of the conditional distributions of the joint distribution P(x). • It is assumed that, whilst P(x) is too complex to draw samples from directly, its conditional distributions are tractable to work with.
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  Gibbs Sampling :Intuition
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  Gibbs Sampling :Example
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  Gibbs Sampling :Example
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  Questions ?
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  Thank you foryour patience!