Rejection sampling - Acceptance probability. Review: How to sample from a multivariate normal in R. Review: Rejection sampling. Weighted resampling

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1 Rejection sampling - Acceptance probability Review: How to sample from a multivariate normal in R Goal: Simulate from N d (µ,σ)? Note: For c to be small, g() must be similar to f(). The art of rejection sampling is to find a g() that is similar to f() and which we know how to sample from. Let N d (0,I) y = µ+a i) y N(µ,AA ) Issues: c is generally large in high-dimensional spaces, and since α = 1/c, many samples will get rejected. Thus, if we choose A so that AA = Σ we are done. Note: The only distribution we actually sample from is a univariate normal distribution! Review: Rejection sampling Weighted resampling We want f() We know how to g() Assume f() g() c for all where f() > 0. Algorithm: repeat generate g() compute α = 1 c f() g() generate u U[0, 1] until u α A problem when using rejection sampling is to find a legal value for c. An approimation to rejection sampling is the following: Let, as before: f(): target distribution g(): proposal distribution The single trials are independent, so the number of trials up to the first success is geometrically distributed with parameter 1/c. The epected number of trials up to the first success is therefore c.

2 Algorithm Adaptive rejection sampling This method works only for log concave densities, i.e. end. Generate 1,..., n g() iid Compute weights w i = f( i ) g( i ) n f( i ) i=1 g( i ) For i = 1,...,m generate a sample from the discrete distribution on { 1,..., n } with probabilities w 1,...,w n. This approimate algorithm is sometimes called sampling importance resampling (SIR) algorithm. f() lnf() 0, for all ln(f()) Many densities are log-concave, e.g. the normal, the gamma (a > 1), densities arising in GLMs with canonical link. Basic idea: Form an upper envelope (the upper bound on f()) adaptively and use this in place of c g() in rejection sampling. Adaptive rejection sampling (2) Start with an initial grid of points 1, 2,..., m and construct the envelope using the tangents at ln(f( i )), i = 1,...,m. Draw a sample from the envelop function and if accepted the process is terminated. Otherwise, use it to refine the grid. ln(f()) Adaptive rejection sampling (3) The piecewise eponential distribution is defined as g() = k m λ m ep( λ m ( t m 1 )), with (t m 1,t m ],i = 1,...,m. Here, t 1,...,t m denotes the partition (intersection points of the tangent), λ m is the slope of the tangent at m and k m accounts for the corresponding offset. (t 0 = 0,k 1 = 1). ln(f())

3 Monte Carlo integration Assumption It is easy to generate independent samples (1),..., (M) from a distribution f() of interest. A Monte Carlo estimate of the mean E() = f()d is then given by Ê() = 1 M M (m). m=1 The strong law of large numbers ensures, that this estimate is consistent. This approach is called Monte Carlo integration Monte Carlo integration (II) Monte Carlo integration Suppose (1),..., (M) is an iid sample drawn from f(). Then the strong law of large numbers says: Eamples Ê(g()) = 1 M M m=1 g( (m) ) a.s g()f()d = E(g()) Using g() = 2 we obtain an estimate for E( 2 ). An estimate for the variance follows as Var() = Ê( 2 ) Ê() 2 Importance sampling One of the principal reasons for wishing to sample from complicated probability distributions f(z) is to be able to evaluate epectations with respect to some function p(z): E(p) = p(z)f(z)dz 1 L L p(z (l) ). The technique of importance sampling provides a framework for approimating epectations directly but does not itself provide a mechanism for drawing samples from a distribution. l=1 Importance sampling (2) Importance sampling is based on the use of a proposal distribution g() from which it is easy to draw samples. We can then epress the epectation in the form of a finite sum over samples (l) drawn from g(): E(p) = p(z)f(z)dz = p(z) f(z) g(z) g(z)dz 1 L L l=1 f(z (l) ) g(z (l) ) p(z(l) ). The quantities w l = f(z (l) )/g(z (l) ) are known as importance weights. The importance weights correct the bias from a wrong distribution.

4 Importance sampling (3) Bayesian concept As with rejection sampling, the success of importance sampling depends crucially on how well the proposal distribution g() matches the target distribution f()....the essence of the Bayesian approach is to provide a mathematical rule eplaining how you change your eisting beliefs in the light of new evidence. In other words, it allows scientists to combine new data with their eisting knowledge or epertise.... The Economist, September 30th 2000 Bayes Theorem I named after the English theologian and mathematician Thomas Bayes [ ] The theorem relies on the asymmetry of the definition of conditional probabilities: P(A B) = P(A B) P(B) P(B A) = P(A B) P(A) P(A B) = P(B) P(A B) (1) P(A B) = P(A) P(B A) (2) for any two events A and B under regularity conditions, i.e. P(B) 0 in (1) and P(A) 0 in (2). Bayes Theorem II Thus, from P(A B) P(B) = P(B A) P(A) follows Bayes Theorem P(A B) = P(B A) P(A) P(B) Law of tot. prob. = P(B A) P(A) P(B A)P(A)+P(B Ā) P(Ā) More general, let A 1,...,A n be eclusive and ehaustive events, then Interpretation P(A i B) = P(A i ) prior probabilities P(A i B) posterior probabilities P(B A i ) P(A i ) n i=1 P(B A i) P(A i ) After observing B the prob. of A i changes from P(A i ) to P(A i B).

5 Towards inference A more general formulation of Bayes theorem is given by f(x = Y = y) = where X and Y are random variables. f(y = y X = )f(x = ) f(y = y) (Note: Switch of notation from P(.) to f(.) to emphasise that we do not only relate to probabilities of events but to general probability functions of the random variables X and Y.) Even more compact version f( y) = f(y )f(). f(y) Posterior distribution The posterior distribution is the most important quantity in Bayesian inference. It contains all information about the unknown parameter θ after having observed the data X =. Let X = denote the observed realisation of a random variable or random vector X with density function f( θ). Specification of a prior distribution with density function f(θ) allows to compute the density function of the posterior distribution using Bayes theorem: f(θ ) = f( θ)f(θ) f() = f( θ)f(θ). f( θ)f(θ)dθ For discrete parameter space the integral has to be replaced with a sum. Posterior distribution (II) Since the denominator in f(θ ) = f( θ)f(θ) f() does not depend on θ, the density of the posterior distribution is proportional to f(θ ) f( θ) p(θ) }{{}}{{}}{{} Posterior Likelihood Prior where 1/ f( θ)f(θ)dθ is the corresponding normalising constant to ensure f(θ )dθ = 1. Reminder: A likelihood approach uses only the likelihood and calculated Maimum Likelihood estimate (MLE), defined as the particular value of θ that maimises the likelihood. Bayesian point estimates Statistical inference about θ is based solely on the posterior distribution f(θ ). Suitable point estimates are location parameters, such as: Posterior mean E(θ ): E(θ ) = Posterior mode Mod(θ ): θf(θ )dθ. Mod(θ ) = arg maf(θ ) θ Posterior median Med(θ ) is defined as the value a which satisfies a f(θ )dθ = 0.5 and a f(θ )dθ = 0.5

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