Bayesian Estimation with Sparse Grids

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1 Bayesian Estimation with Sparse Grids Kenneth L. Judd and Thomas M. Mertens Institute on Computational Economics August 7, 27 / 48

2 Outline Introduction 2 Sparse grids Construction Integration with sparse grids 3 Bayesian Estimation 4 Numerical results 5 Conclusion/Future Research 2/ 48

3 Introduction Bayesian estimation involves integration of the posterior. We present a numerical quadrature approach for Bayesian estimation. We use sparse grids to deal with high dimensionality. It is an alternative to the use of simulations. 4/ 48

4 Bayesian Estimation Get data Y. Obtain likelihood function L (Y θ). Use prior information p(θ). Posterior: p(θ Y) L (Y θ) p(θ) We then compute Bayesian estimators: M = h(θ)l (Y θ)p(θ)dθ 5/ 48

5 Introduction Main difficulty is dimensionality of the problem. Monte-Carlo methods converge at a rate independent of dimension but slowly. We want a faster method. There is hope: the function is very smooth. 6/ 48

6 Introduction Pseudo-random schemes give you convergence O(N 2). Equidistributional sequences give convergence of order for C -functions. There is convergence of O(N k ) for periodic C k -functions. For very smooth functions, there is essentially no curse of dimensionality. 7/ 48

7 Related literature MCMC: Peter Rossi, Bayesian Statistics and Marketing Sparse grids: Bungartz and Griebel: Sparse grids, Acta Numerica (24). Sparse grids have been used in economics and finance. Examples in economics are: Kuebler and Krueger, and Winschel. 8/ 48

8 Approximation Goal: Approximate function f (x) with basis functions. We can the write the approximation ˆf (x) as: ˆf (x) = (l,i) u (l,i) φ (l,i) (x) f (x) Basis functions could for instance be: / 48

9 Approximation / 48

10 Approximation / 48

11 Approximation / 48

12 Approximation / 48

13 Exponents Why can we truncate the sum at level l? Look at the approximation again. The absolute value of coefficients shrinks to zero. More precisely: u (l,i) c 2 2 l. 5/ 48

14 Sparse grids Construction The goal is to generalize the one-dimensional approximation. This requires to specify: the basis function the grid 6/ 48

15 Sparse grids Construction Choice of basis functions and associated grid. n= n= φ (l,i) = j φ (l j,i j ) 7/ 48

16 Sparse grids Construction n= n=2 n= Curse of Dimensionality for full grids grid size grows at n d. 8/ 48

17 Sparse grids Construction Sparse grids: n= n=2 n= / 48

18 Approximation Note: We can now still write the approximation as ˆf (x) = (l,i) u (l,i) φ (l,i) (x) f (x) where l = (l,...,l d ), i = (i,...,i d ) φ (l,i) = j φ (l j,i j ) It is a generalization of the -d concept. But: sparse grids come at a cost: you have to have bounded second mixed derivatives. 2/ 48

19 Features Full grids Sparse grids # grid points n d O(n log(n) d ) Accuracy L 2 -error O(N 2 ) O(N 2 log(n) d ) L -error O(N 2 ) O(N 2 log(n) d ) 2/ 48

20 Why is that such a big thing? Just imagine: 32 grid points per dimension, 3 dimensions. Sparse grids: 4, 58, 8 grid points. That fits into a computer s RAM points in full grids don t! 22/ 48

21 Integration and moments Having the approximation, we can now integrate Just sum up! (l,i) u (l,i) φ (l,i) (x) = (l,i) u (l,i) φ (l,i) (x) How about computing moments? Use the same grid: since f ˆf < δ x 2 (f (x) ˆf (x))dx < δ x 2 dx 23/ 48

22 Bayesian Estimation Let s look at the problem again. Compute expected value of functions of θ M = h(θ)l (Y θ)p(θ)dθ Problem: cannot sample from L (Y θ)p(θ) directly. Use MCMC methods: Gibbs or Metropolis-Hastings. 25/ 48

23 Technique: MCMC Create Markov-chain that has true distribution as stationary distribution. First: Gibbs sampling. Pick starting point. Pick successively draw from each marginal distribution to get to the next step. 26/ 48

24 Technique: MCMC Now: Metropolis-Hastings. Pick starting point. Draw sample ψ from proposal distribution Q(θ t θ t ) (i.e. Q(θ t θ t ) N(θ t,σ)). Accept draw only if u < otherwise θ t = θ t. p(ψ)l (Y ψ) p(θ t )L (Y θ t ) where u U([,]) 27/ 48

25 Technique: MCMC Both Gibbs and Metropolis-Hastings samplers have a burn-in phase. These are the samples you need to get to the stationary distribution. These draws are not used to compute the integral. 28/ 48

26 Technique: MCMC There are further complications with MCMC doesn t use all information about the density at a draw. takes long to converge. error estimation is comparably hard. no possibility to check for identification. 29/ 48

27 Our approach Take burn-in samples. Find peak of the posterior using samples as starting points. Compute Hessian at the peak. Take out Gaussian function. Repeat until only little mass is left. Treat rest with sparse grids. 3/ 48

28 In pictures Treat remainder with sparse grids / 48

29 Advantages We then have the approximation: p(θ Y ) = G(θ) + (l,i) u (l,i) φ (l,i) (θ) We can then integrate since we know h(θ)g(θ) and h(θ)u (l,i) φ (l,i) (θ) (l,i) 32/ 48

30 Advantages Precise error estimation for normalized posterior. Otherwise have to rely on relative accuracy. From one approximation, you can compute all moments. You get a normal approximation for free. Check for identification. 33/ 48

31 f (x) = [ d i= 2s + cos ( )] x(i) µ s π Numerical results / 48

32 Numerical results / 48

33 Integration problem # points Sparse grids e-4.4e-6.3e-9 MC Table: L -error quadrature versus MC, 2D 37/ 48

34 Numerical results 2 L error log scale Grid points log scale 3 38/ 48

35 Numerical results Sum of normals / 48

36 Integration problem # points Sparse grids e-6.2e-.4e-4 MCMC Table: Error for MCMC versus integration, 2D 4/ 48

37 Numerical results 2 4 L error log scale Mean Integral MCMC 4 4 Grid points log scale 4/ 48

38 Numerical results f (x) = a e 2 (x µ)σ(x µ) + b e (x µ)2 Σ(x µ) / 48

39 Numerical results / 48

40 Numerical results Integral Mean MCMC L 2 error log scale Grid points log scale 44/ 48

41 Result This solves three problems:. Method is fast. 2. Error estimation is accurate. 3. Get posterior to check for identification. 45/ 48

42 Conclusion Frank Schorfheide said at the macro annual: This is heavy computing: Don t try this at home! We ll put a Matlab code on the web. Please do try it at home! We can improve along many dimensions: polynomial exactness, better code, and adaptivity. 47/ 48

43 Applications Sparse grids have been used in several contexts. Numerical Integration Projection methods (e.g. Kuebler and Krueger) Solution to partial differential equations 48/ 48

High-dimensional Problems in Finance and Economics. Thomas M. Mertens

High-dimensional Problems in Finance and Economics Thomas M. Mertens NYU Stern Risk Economics Lab April 17, 2012 1 / 78 Motivation Many problems in finance and economics are high dimensional. Dynamic Optimization: