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Bayesian Processing of cDNA Microarray Images

Bayesian Processing of cDNA Microarray Images. Neil Lawrence Machine Learning Group Department of Computer Science The University of Sheffield. Overview. cDNA Microarray Review Bayesian Model for Image Processing ( Bioinformatics , In press) cDNA Summary Tracking Toy application

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Bayesian Processing of cDNA Microarray Images

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  1. Bayesian Processing of cDNA Microarray Images Neil Lawrence Machine Learning Group Department of Computer Science The University of Sheffield

  2. Overview • cDNA Microarray Review • Bayesian Model for Image Processing (Bioinformatics, In press) • cDNA Summary • Tracking • Toy application • Real application • Tracking Summary

  3. cDNA Microarrays • Genes -> RNA -> protein (Central Dogma). • Proteins made determines role of cell. • Gene expression • want to measure which genes are expressing themselves. • important in cell differentiation & interaction networks. • Collaborators: • Biologists: Pen Rashbass, Matthew Holley, Alireza Fazeli. Sheffield Medicine & Biomedical Sciences. • Comp Sci: Marta Milo, Mahesan Niranjan.

  4. Microarray Slides • Slides consist of genetic material (DNA) placed by a robot. • Slides are then • hybridised with fluorescent mutant and wild-type DNA samples. • scanned by a laser system. • Mutant is one colour wild-type another.

  5. Microarray Slides • Slides consist of genetic material placed by a robot. Different solutions DNA in solution

  6. One Grid

  7. Microarray Slides • Expression ratio • the ratio of red/green. • Software packages allow • manual placement of grids/spots. • extraction of mean/median pixel intensities. • We wish to • automate placement of spots. • extract with uncertainty measures.

  8. The Model likelihood prior x z is measurements (e.g. pixel intensities) z x is latent variable (e.g. a state space of interest)

  9. The Model • x – x, y position and radii of spot. • prior – taken from known grid layout. • likelihood – expected pixel values given spot shape and location. • Rx - region specified by x • p(zi | F) and p(zi | B) – foreground and background models.

  10. Model Inference • We wish to extract gene expression ratios. • Ratio value depends on spot location. • Can also find variance etc. • How do we get p(x|z)?

  11. Importance Sampling • Obtain samples from proposal, r(x), {xs}Ss=1. • Associate weights with samples to obtain estimates of moments under posterior.

  12. Importance Sampling • If proposal is the prior, r(x) = p(x). • The weights become the likelihood • Demo of approach:

  13. Approach Demo • msrcOne.m

  14. Fixes • `Anneal’ the likelihood values • Either raise it to power of a fraction (don’t like this!) • Or use simulated annealing (annealing schedule) • Take more samples • May need very many more. • Our approach: • Hierarchical Bayesian model likelihood prior

  15. The New Model likelihood prior P  x x is latent variable p()andp(P)are hyper-priors - priors on the prior z

  16. The Plan • The prior is now • We will use hyper-priors to form `hyper-posteriors’. • These `hyper-posteriors’ will govern the proposal. • The proposal will then adapt to z.

  17. New approach – Adaptive Proposal expectations under `hyper-posteriors’ likelihood p(z|x) p(x|<>,<P-1>)

  18. Model Specification • We take • Given • Marginalised prior is intractable. • `multi-variate Student-t with Gaussian prior on its mean.’

  19. Variational Inference • We have chosen distributions from conjugate exponential family. • This means we can use variational inference to approximate the posterior. • Variational inference: assume posterior factorises

  20. Variational Inference • What are the best forms for these factors? • Answer: • Relies on <P>, <xxT> and <x>. • Relies on <>, <T>, <xxT> and <x>. • q(x) however is more difficult. Notaition <> denotes expectation.

  21. Variational Inference • Relies on <P>, <>. • Finding constant of proportionality now intractable. • Turn to sampling to compute <x>, <xxT>.

  22. VIS • Variational importance sampler: • Estimate moments <x> and <xxT> by importance sampling. • Use N(x|, <>, <P>-1) as proposal. • Comparison with vanilla method: • Proposal distribution mean and covariance reflect data. • Iterative method where:

  23. Computational Requirements • Computation dominated by likelihood evaluation. • Vanilla importance sampling: • Requirement rises with S. • Variational importance sampling: • Requirement rises with S x number of iterations. • Approach is worthwhile if accuracy improves with iterations quicker than it improves with number of samples.

  24. Back to Demo • msrcTwo.m

  25. Microarray Demo 2

  26. Numerical Results Mean squared error for duplicate to duplicate prediction.

  27. variance vs intensity

  28. Simple Downstream Analysis • Aphakia – a developmental defect affecting the lens of the eye. • Samples of mutant and wild-type mice from E10.5, E11.5 and E12.5. • Here results are plotted on a 2-D graph. • x-axis is E10.5 – E11.5 • y-axis is E11.5 – E12.5

  29. Summarisation of Replicates

  30. Summarisation of Replicates

  31. cDNA Summary • Automatic processing of cDNA images. • Improved consistency of image processing • Used uncertainty in combination of replicates. • Future work: • Propagate uncertainty to more complex downstream analyses. • Automate initial rough grid layout.

  32. Dynamic Model • The variational importance sampler is a general methodology. • Here we apply it in tracking. • Joint work with: • Jaco Vermaak, Cambridge Engineering • Patrick Perez, Microsoft Research • Presented at CVPR 03

  33. Dynamic Model Pt Pt+1 t t+1 xt xt+1 zt zt+1

  34. Dynamic Model • `Prior’ at step t+1 is taken as

  35. Toy Problem

  36. Results

  37. Tracking Problem

  38. Tracking Results

  39. Contour Tracking

  40. Overall Summary • Bayesian Inference on images: • No need to artificially adjust likelihood. • Considerably better performance. • Developments: • Multi-modality through mixtures.

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