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%HW2-Sivia N=10000; alphas=rand(N,1); alphas=(alphas-1/2)*pi; xs=tan(alphas); xs=sort(xs); figure; plot(xs) meanest=mean(xs) medest=median(xs) llh=[]; for x0=-1:0.01:1 llh=[llh sum(-log((1+(x-x0).^2)))]; end figure; plot(llh)

HW2-Lighthouse problem

meanest =-0.1099 medest =0.0047

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Statistical Data models, Non-parametrics,

Dynamics

Non-informative, proper and improper priors

•  For real quantity bounded to interval, standard prior is uniform distribution

•  For real quantity, unbounded, standard is uniform - but with what density?

•  For real quantity on half-open interval, standard prior is f(s)=1/s - but integral diverges!

•  Divergent priors are called improper - they can only be used with convergent likelihoods

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Dirichlet Distribution- prior for discrete distribution

Mean of Dirichlet - Laplaces estimator

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Occurence table probability

Occurence table probability Uniform prior:

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Non-parametric inference

•  How to perform inference about a distribution without assuming a distribution family?

•  A distribution over reals can be approximated by a piecewise uniform distribution a mixture of real distributions

•  But how many parts? This is non-parametric inference

Non-parametric inference Change-points, Rao-Blackwell

•  Given times for events (eg coal-mining disasters) Infer a piecewise constant intensity function (change-point problem)

•  State is set of change-points with intensities inbetween •  But how many pieces? This is non-parametric inference •  MCMC: Given current state, propose change in segment

bounadry or intensity •  But it is possible to integrate out intensities proposed

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Probability ratio in MCMC

For a proposed merge of intervals j and j+1, with sizes proportional to (α,1-α), were the counts and obtained by tossing a ‘coin’ with success probability or not? Compute model probability ratio as in HW1. Also, the total number of breakpoints has prior distribution Poisson with parameter (average) . Probability ratio in favor of split :

!

n j

!

n j+1

!

"

!

"

Averging MCMC run, positions and number of breakpoints

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Averging MCMC run, positions with uniform test data

Mixture of Normals

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Mixture of Normals elimination of nuisance parameters

Mixture of Normals elimination of nuisance parameters

(integrate using normalization constant of Gaussian and Gamma distributions)

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Matlab Mixture of Normals, MCMC (AutoClass method)

function [lh,lab,trlpost,trm,trstd,trlab,trct,nbounc]= mmnonu1(x,N,k,labi,NN); %[lh,lab,trlpost,trm,trstd,trlab,trct,nbounc]= % MMNONU1(x,N,k,labi,NN); %inputs % 1D MCMC mixture modelling, % x - 1D data column vector % N - MCMC iterations. % k - number of components %lab,labi - component labelling of data vector) % NN - thinning (optional)

Matlab Mixture of Normals, MCMC

function [lab,trlh,trm,trstd,trlab,trct,nbounc]= mmnonu1(x,N,k,labi,NN); %[lh,lab,trlpost,trm,trstd,trlab,trct,nbounc]= % MMNONU1(x,N,k,labi,NN); %outputs %trlh - thinned trace of log probability (optional) %trm - thinned trace of means vector (optional) %trstd - thinned vector of standard deviations (optional) %trlab - thinned trace of labels vector (size(x,1) by N/NN (optional) %trct - thinned trace of mixing proportions

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Matlab Mixture of Normals, MCMC

N=10000; NN=100; x=[randn(100,1)-1;randn(100,1)*3;randn(100,1)+1]; % 3 components synthetic data k=2; labi=ceil(rand(size(x))*2); [llhc,lab2,trl,trm,trstd,trlab,trct,nbounc]= … mmnonu1(x,N,k,labi,NN); [llhc2,lab2,trl2,trm2,trstd2,trlab2,trct2,nbounc]=… mmnonu1(x,N,k,lab2,NN); … (k=3, 4, 5)

Matlab Mixture of Normals, MCMC

The three components and the joint empirical distr

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Matlab Mixture of Normals, MCMC Putting them

together makes the identification seem harder.

Matlab Mixture of Normals, MCMC

K=2:

std

mean

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Matlab Mixture of Normals, MCMC

K=3:

std

mean

Burn in progressing

Matlab Mixture of Normals, MCMC

K=3:

std

mean

Burnt in

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Matlab Mixture of Normals, MCMC

K=4: Low prob

std

mean

No focus- No interpretation as 4 clusters

Matlab Mixture of Normals, MCMC

K=5: Low prob

std

mean

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Matlab Mixture of Normals, MCMC

X sample: 1-100 : (-1 1) 101:200: (0 3) 201:300: (1 1)

Trace of state labels

Unsorted sample label trace sorted

Mixtures of multivariate normals

•  This works the same way, but instead of a Gamma distribution for the variance we use the Wishart distribution, a matrix-valued distribution over covariance matrices.

•  Competes well with both clustering and Expectation Maximization, which are prone to overfitting (clustering cannot handle overlapping components)

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Dynamic Systems, time series

•  An abundance of linear prediction models exists

•  For non-linear and Chaotic systems, method was developed in 1990:s (Santa Fe)

•  Gershenfeld, Weigend: The Future of Time Series

•  Online/offline: prediction/retrodiction

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Berry and Linoff have eloquently stated their preferences with ���the often quoted sentence: "Neural networks are a good choice for most classification problems when the results of the model are more important than understanding how the model works".��� “Neural networks typically give the right answer”

Dynamic Systems and Taken’s Theorem

•  Lag vectors (xi,x(i-1),…x(i-T), for all i, occupy a submanifold of E^T, if T is large enough

•  This manifold is ‘diffeomorphic’ to original state space and can be used to create a good dynamic model

•  Taken’s theorem assumes no noise and must be empirically verified.

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Dynamic Systems and Taken’s Theorem

Santa Fe 1992 Competition

Unstable Laser

Intensive Care Unit Data, Apnea

Exchange rate Data

Synthetic series with drift

White Dwarf Star Data

Bach’s unfinished Fugue

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Stereoscopic 3D view of state space manifold, series A (Laser) The points seem to lie on a surface, which means that a lag-vector of 3 gives good prediction of the time series. The surface is either produced for a training batch, or produced on-the-fly from neighboring data points (possibly downweighing very old points)

Figure in book misleading: Origin where surface touches ground

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Variational Bayes

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True trajectory in state space (Valpola-Karhunen 2002)

Reconstructed trajectory in inferred state space

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Hidden Markov Models

•  Given a sequence of discrete signals xi •  Is there a model likely to have produced xi

from a sequence of states si of a Finite Markov Chain?

•  P(.|s) - transition probability in state s •  S(.|s) - signal probability in state s •  Speech Recognition, Bioinformatics, …

Hidden Markov Models function [Pn,Sn,stn,trP,trS,trst,tll]=… hmmsim(A,N,n,s,prop,Po,So,sto,NN); %[Pn,Sn,stn,trP,trS,trst]=HMMSIM(A,N,n,s,prop,Po,So,sto,NN); % Compute trace of posterior for hmm parameters % A - the sequence of signals % N - the length of trace % n - number of states in Markov chain % s - number of signal values % prop - proposal stepsize % optional inputs: % Po - starting transition matrix (each of n columns a discrete pdf % in n-vector % So - starting signal matrix (each of n columns a discrete pdf

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Hidden Markov Models function [Pn,Sn,stn,trP,trS,trst,tll]=… hmmsim(A,N,n,s,prop,Po,So,sto,NN); % in s-vector % sto - starting state sequence (congruent to vector A) % NN - thining of trace, default 10 % outputs % Pn - last transition matrix in trace % Sn - last signal emission matrix % stn - last hidden state vector (congruent to A) % trP - trace of transition matrices % trS - trace of signal matrices % trace of hidden state vectors

Hidden Markov Models

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Hidden Markov Models

Hidden Markov Models

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Hidden Markov Models Over 100000 iterations, burnin is visible 2 states, 2 signals P-transition matrix S-signaling

Chapman Kolmogorov version of Bayes’ rule

f (!t | Dt ) " f (dt | !t)# f (!t |!t$1) f (!t$1 | Dt$1 )d!t$1

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Chapman Kolmogorov version of Bayes’ rule

f (!t | Dt ) " f (dt | !t)# f (!t |!t$1) f (!t$1 | Dt$1 )d!t$1

Observation and video based particle filter tracking

Defence: tracking with heterogeneous observations

Crowd analysis: tracking from video

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Cycle in Particle filter

Importance (weighted) sample Resampled ordinary sample Diffused sample Weighted by likelihood X- state Z - Observation

Time step cycle

Particle filter- general tracking