Bayesian Techniques for Parameter Estimation. He has Van Gogh s ear for music, Billy Wilder

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1 Bayesian Techniques for Parameter Estimation He has Van Gogh s ear for music, Billy Wilder

2 Statistical Inference Goal: The goal in statistical inference is to make conclusions about a phenomenon based on observed data. Frequentist: Observations made in the past are analyzed with a specified model. Result is regarded as confidence about state of real world. Probabilities defined as frequencies with which an event occurs if experiment is repeated several times. Parameter Estimation: o Relies on estimators derived from different data sets and a specific sampling distribution. o Parameters may be unknown but are fixed Bayesian: Interpretation of probability is subjective and can be updated with new data. Parameter Estimation: Parameters described as density

3 Bayesian Inference Framework: Prior Distribution: Quantifies prior knowledge of parameter values. Likelihood: Probability of observing a data if we have a certain set of parameter values. Posterior Distribution: Conditional probability distribution of unknown parameters given observed data. Joint PDF: Quantifies all combination of data and observations Bayes Relation: Specifies posterior in terms of likelihood, prior, and normalization constant Problem: Evaluation of normalization constant typically requires high dimensional integration.

4 Bayesian Inference Uninformative Prior: No a priori information parameters Informative Prior: Use conjugate priors; prior and posterior from same distribution Evaluation Strategies: Analytic integration --- Rare Classical quadrature; e.g., p = 2 Monte Carlo quadrature Techniques Markov Chains

5 Example: Bayesian Inference

6 Bayesian Inference Example: Note: 1 Head, 0 Tails 5 Heads, 9 Tails 49 Heads, 51 Tails

7 Bayesian Inference Example: Now consider 5 Heads, 5 Tails 50 Heads, 50 Tails Note: Poor informative prior incorrectly influences results for a long time.

8 Parameter Estimation Problem Likelihood: Assumption: Note:

9 Parameter Estimation: Example Example: Consider the spring model

10 Ordinary Least Squares: Here Parameter Estimation: Example Sample Distribution

11 Parameter Estimation: Example Bayesian Inference: The likelihood is Posterior Distribution: Strategy: Create Markov chain using random sampling so that created chain has the posterior distribution as its limiting (stationary) distribution.

12 Markov Chains Definition: Note: A Markov chain is characterized by three components: a state space, an initial distribution, and a transition kernel. State Space: Initial Distribution: (Mass) Transition Probability: (Markov Kernel)

13 Markov Chains Example: Chapman-Kolmogorov Equations:

14 Markov Chains: Limiting Distribution Example: Raleigh weather -- Tomorrow s weather conditioned on today s weather rain sun Question: Definition: This is the limiting distribution (invariant measure)

15 Example: Raleigh weather Markov Chains: Limiting Distribution Solve rain sun

16 Reducible Markov Chain: Irreducible Markov Chains p1 p2 Note: Limiting distribution not unique if chain is reducible. Irreducible:

17 Periodic Markov Chains Example: Periodicity: A Markov chain is periodic if parts of the state space are visited at regular intervals. The period k is defined as

18 Example: Periodic Markov Chains

19 Stationary Distribution Theorem: A finite, homogeneous Markov chain that is irreducible and aperiodic has a unique stationary distribution and the.chain will converge in the sense of distributions from any initial distribution. Recurrence (Persistence): Example: State 3 is transient Ergodicity: A state is termed ergodic if it is aperiodic and recurrent. If all states of an irreducible Markov chain are ergodic, the chain is said to be ergodic.

20 Matrix Theory Definition: Lemma: Example:

21 Matrix Theory Theorem (Perron-Frobenius): Corollary 1: Proposition:

22 Stationary Distribution Corollary: Proof: Convergence: Express

23 Stationary Distribution

24 Detailed Balance Conditions Reversible Chains: A Markov chain determined by the transition matrix is reversible if there is a distribution that satisfies the detailed balance conditions Proof: We need to show that Example:

25 Markov Chain Monte Carlo Methods Strategy: Markov chain simulation used when it is impossible, or computationally prohibitive, to sample directly from Note: In Markov chain theory, we are given a Markov chain, P, and we construct its equilibrium distribution. In MCMC theory, we are given a distribution and we want to construct a Markov chain that is reversible with respect to it.

26 General Strategy: Markov Chain Monte Carlo Methods Intuition: Recall that

27 Markov Chain Monte Carlo Methods Intuition: Note: Narrower proposal distribution yields higher probability of acceptance.

28 Metropolis Algorithm Metropolis Algorithm: [Metropolis and Ulam, 1949]

29 Metropolis-Hastings Algorithm: Metropolis-Hastings Algorithm Examples: Note: Considered one of top 10 algorithms of 20th century

30 Proposal Distribution Proposal Distribution: Significantly affects mixing Too wide: Too many points rejected and chain stays still for long periods; Too narrow: Acceptance ratio is high but algorithm is slow to explore parameter space Ideally, it should have similar shape to posterior (target) distribution. Problem: Anisotropic posterior, isotropic proposal; Efficiency nonuniform for different parameters Result: Recovers efficiency of univariate case

31 Proposal Distribution and Acceptance Probability Proposal Distribution: Two basic approaches Choose a fixed proposal function o Independent Metropolis Random walk (local Metropolis) o Two (of several) choices: Acceptance Probability:

32 Random Walk Metropolis Algorithm for Parameter Estimation

33 Random Walk Metropolis Algorithm for Parameter Estimation

34 Markov Chain Monte Carlo: Example Example: Consider the spring model

35 Example: Single parameter c Markov Chain Monte Carlo: Example

36 Markov Chain Monte Carlo: Example Example: Consider the spring model Case i:

37 Markov Chain Monte Carlo: Example Case i: Note:

38 Markov Chain Monte Carlo: Example Case i:

39 Markov Chain Monte Carlo: Example Example: SMA-driven bending actuator -- talk with John Crews Model: Estimated Parameters:

40 Markov Chain Monte Carlo: Example Example: SMA-driven bending actuator -- talk with John Crews

41 Markov Chain Monte Carlo: Example Example: SMA-driven bending actuator -- talk with John Crews

42 Markov Chain Monte Carlo: Example Example: SMA-driven bending actuator -- talk with John Crews

43 Transition Kernel and Detailed Balance Condition Transition Kernel: Recall that Detailed Balance Condition:

44 Transition Kernel and Detailed Balance Condition Detailed Balance Condition: Here Note: Transition Kernel: Definition

45 Sampling Error Variance Strategy: Treat error variance as parameter to be estimated. Recall: Assumption that errors are normally distributed yields Goal: Determine posterior distribution Strategy: Choose prior so that posterior is from same family --- termed conjugate prior. For normal distribution with unknown variance, conjugate prior is inverse Gamma distribution which is equivalent to inverse

46 Definition: Sampling Error Variance Strategy: Note: Note:

47 Example: Consider the spring model Sampling Error Variance

48 Related Topics Note: This is an active research area and there are a number of related topics Burn in and convergence Adaptive algorithms Population Monte Carlo methods Sequential Monte Carlo methods and particle filters Gaussian mixture models Development of metamodels, surrogates and emulators to improve implementation speeds References: A. Solonen, Monte Carlo Methods in Parameter Estimation of Nonlinear Models, Masters Thesis, H. Haario, E. Saksman, J. Tamminen, An adaptive Metropolis algorithm, Bernoulli, 7(2), pp , C. Andrieu and J. Thomas, A tutorial on adaptive MCMC, Statistics and Computing, 18, pp , M. Vihola, Robust adaptive Metropolis algorithm with coerced acceptance rate, arxiv: v2.

Tutorial on Markov Chain Monte Carlo

Tutorial on Markov Chain Monte Carlo Kenneth M. Hanson Los Alamos National Laboratory Presented at the 29 th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Technology,