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Genomic Signal Processing: Issues in Engineering Molecular Medicine

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Title: Genomic Signal Processing: Issues in Engineering Molecular Medicine


1
Genomic Signal Processing Issues in Engineering
Molecular Medicine
  • Edward R. Dougherty
  • Department of Electrical and Computer
    Engineering, Texas AM University
  • Division of Computational Biology, Translational
    Genomics Research Institute
  • Department of Pathology, University of Texas,
    M.D. Anderson Cancer Center

2
Systems Medicine
  • Systems Biology Understanding the manner in
    which the parts of an organism interact in
    complex networks.
  • Systems Medicine Translation of systems biology
    into medicine.
  • Translational Genomics The part of systems
    medicine that deals with genome-based systems
    engineering.

3
Goals of Translational Genomics
  • Screen for key genes and gene families that
    explain specific cellular phenotypes (disease).
  • Use genomic signals to classify disease on a
    molecular level.
  • Build model networks to study dynamical genome
    behavior and derive intervention strategies to
    alter undesirable behavior.

4
Translational Genomics Tools
  • Signal Processing
  • Pattern Recognition
  • Information Theory
  • Control Theory
  • Network Theory
  • Communication Theory

5
Genomic Signal Processing
  • GSP The analysis, processing, and use of genomic
    signals for gaining biological knowledge and
    translation of that knowledge into systems-based
    applications.
  • Signals generated by the genome must be processed
    to characterize their regulatory effects and
    their relationship to changes at both the
    genotypic and phenotypic levels.

6
Central Dogma of Molecular Biology
DNA
Transcription
RNA
Translation
Protein
7
Transcription Factors
8
Gene Regulation
E1A
Rb
Gene regulatory controls
DNA damage
Myc
E2F
p53
MDM2
Hypoxia
transcription
Gene expression the process by which gene
products (proteins) are made
translation
protein
9
Gene Expression
  • Central Dogma of Molecular Biology Information
    flows from DNA to RNA to protein.
  • Transcription DNA ? RNA
  • Translation RNA ? protein
  • It is not possible to fully separate the three
    levels.
  • But the high level of interaction insures a
    significant amount of system information present
    at each level.
  • Measure gene expression by mRNA abundance.

10
Microarrays
  • Expression microarrays result from a complex
    biochemical-optical system incorporating robotic
    spotting and computer image formation and
    analysis.
  • They facilitate large-scale surveys of gene
    expression in which transcript levels can be
    determined for thousands of genes simultaneously.
  • cDNA Arrays Expressed Sequence Tags (ESTs).
  • Oligo Arrays Synthetic oligonucleotides.
  • Involve image processing and signal extraction.

11
Microarray Process
12
Classification of Diseases
  • Find a feature set of expression profiles to
    classify disease.
  • Diagnose cancer
  • Type
  • Stage
  • Prognosis

13
BRCA Classification
14
Small-Sample Issues
  • Imprecise classifier design designed classifier
    can be a poor estimate of the optimal classifier.
  • Poor error estimation owing to no test data.
  • Poor feature selection.
  • Dougherty, E. R., "Small Sample Issues for
    Microarray-Based Classification," Comparative and
    Functional Genomics, Vol. 2, 28-34, 2001.
  • Dougherty, E. R., Datta, A., and C. Sima,
    Dougherty, E. R., Datta, A., and C. Sima,
    Research Issues in Genomic Signal Processing,
    IEEE Signal Processing Magazine, 22 (6), 46-68,
    2005.

15
Classifier Design
  • From a sample form an estimate ?n of ?opt.
  • Design cost ?n ?n ? ?opt
  • Key issue good design often requires large
    samples and it is often impossible to get large
    enough samples to sufficiently reduce E?n.

16
Overfitting
  • If we apply a complex classification rule to a
    small sample, the rule is likely to conform to
    the data too closely.
  • We constrain classifier complexity to avoid
    overfitting, thereby restricting ourselves to
    easy problems.

17
Constraint
  • To lower design cost, optimization is constrained
    to a subclass C.
  • Constraint cost ?C ?C ? ?d.
  • The savings in design error must exceed the cost
    of constraint.
  • Key problem find appropriate constraints.
  • A constraint may be defined in accordance with a
    model, or maybe experience has shown a certain
    constraint works well in a given setting.

18
Classifier Design Error
19
Small-Sample Error Estimation
  • Train and test classifier on same data.
  • Basic Approaches
  • Resubstitution Count errors on training data
    (usually low biased).
  • Re-sampling Design on sub-samples and test on
    left-out data.
  • Regularization Enhance the data or estimate the
    distribution.

20
Cross-Validation
  • Error rate estimated by iteratively leaving
    out data points, testing on the deleted points,
    and averaging. 
  • Cross-validation unbiased in the following sense
  • ExpectationCV estimate ? error ? 0
  • This says little about the number we care about,
  • ExpectationCV estimate ? error
  • unless CV variance is small not for small
    samples.

21
Cross-validation Mythology
  • Myth Cross-validation is good for small samples.
  • CV is good for moderate to large samples because
    it allows all data to be used for design.
  • ? Myth CV always outperforms resubstitution.
  • Resub performs as well or better for estimating
    predictor error in low connectivity Boolean
    networks.
  • Resub can outperform CV for feature set ranking.
  • Resubstitution is much faster to compute.

22
Deviation Distributions
Experiment 1 (LDA, p2)
Experiment 3 (3NN, p2)
Experiment 5 (CART, p2)
Resubs
leave one out
cv10
cv5
cv10r
bbc
b632
  • Braga-Neto, U. M., and E. R. Dougherty, Is
    Cross-Validation Valid for Small-Sample
    Microarray Classification, Bioinformatics, 20
    (3), 374-380, 2004.

23
Bolstered Error Estimation
  • Estimate classifier error by spreading the data
    via Bolstering Kernels
  • Error estimate results from integrating kernels
    over the domain to which points should not be
    included.
  • Braga-Neto, U. M., and E. R. Dougherty,
    Bolstered Error Estimation, Pattern
    Recognition, 37 (6), 1267-1281, 2004.

24
Bolstering Properties
  • Error can be computed via integration with closed
    form for LDA and Monte Carlo integration
    otherwise.
  • Choosing variance of bolstering kernel is key
    because it affects both bias and variance of the
    bolstered estimator.
  • A method for choosing the variance has been
    proposed.
  • Resubstitution results from zero bolstering
    variance.

25
Deviation Distributions CART, 5 Genes
26
Feature Selection Impacts Cross-Validation
  • Feature selection increases the already large
    deviation variance of cross-validation.
  • Coefficient of Relative Increase in Deviation
    Dispersion
  • ?opt true error using best features.
  • ?cv true error using selected features.
  • Xiao, Y., Hua, J. and E. R. Dougherty,
    Quantification of the Impact of Feature
    Selection on Cross-validation Error Estimation,
    EURASIP J. Bioinformatics and Systems Biology,
    2007.

27
How Many Features?
  • Peaking Phenomenon Overfitting.

28
Feature-Selection Problem
  • Select a subset of k features from a set of n
    features with minimum error among all subsets of
    size k.
  • Cover and van Campenhout Theorem All k-element
    subsets must be checked.
  • Heuristic suboptimal algorithms have been
    proposed to circumvent the full combinatorial
    search.
  • Issues
  • Mathematical analysis of algorithms
  • Impact of error estimation
  • Impact of sample size

29
Optimal Number of Features
  • Optimal number of features depends on sample
    size, classification rule and feature-label
    distribution.
  • Top LDA, linear model, slightly correlated
    features.
  • Bottom LDA, linear model, highly correlated
    features.
  • Hua, J., Xiong, Z., Lowey, J., Suh, E., and E. R.
    Dougherty, Optimal Number of Features as a
    Function of Sample Size for Various
    Classification Rules, Bioinformatics, 21(8),
    1509-1515, 2005.

30
Peaking Phenomenon is Nontrivial
  • Peaking can be later for smaller samples.
  • Top 3NN, nonlinear model, modestly correlated
    features.
  • Bottom Linear SVM, nonlinear model, modestly
    correlated features.
  • Hua, J., Xiong, Z., Lowey, J., Suh, E., and E. R.
    Dougherty, Optimal Number of Features as a
    Function of Sample Size for Various
    Classification Rules, Bioinformatics, 21(8),
    1509-1515, 2005.

31
Impact of Error Estimation on Feature Selection
  • Choice of error estimator can be more important
    than choice of algorithm.
  • LDA, Gaussian model, n 50, 5 features from 20.
  • Sima, C., Attoor, S., Braga-Neto, U., Lowey, J.,
    Suh, E., and E. R. Dougherty, Impact of Error
    Estimation on Feature-Selection Algorithms,
    Pattern Recognition, 38 (12), 2472-2482, 2005.

32
What Can We Expect from Feature Selection?
  • Top Regression of selected FS error on best FS
    error.
  • Bottom Regression of best FS error on selected
    FS error.
  • Sima, C., and E. R. Dougherty, What Should One
    Expect from Feature Selection in Small-Sample
    Settings, Bioinformatics, 22 (19), 2430-2436,
    2006.

33
Decorrelation of True and Estimated Errors
  • With feature selection, the problem is
    decorrelation of the error estimate from the true
    error, not increased estimator variance.
  • Selecting 5 features from 200 with sample size
    50.
  • With feature selection
  • Without feature selection
  • Hanczar, B., Hua, J., and E. R. Dougherty, Is
    There Correlation between the Estimated and True
    Classification Errors in Small-Sample Settings?
    IEEE Statistical Signal Processing Workshop,
    Madison, August, 2007.

34
Error Bounds
  • Distribution-free bounds exist on the RMS between
    the error and error estimate.
  • Typically, they are useless for small samples.
  • For n 100, RMS ? 0.435.

35
Salient Points for Small Samples
  • Beware of complex classifiers.
  • Keep feature sets small.
  • Avoid cross-validation where possible.
  • Recognize the heavy influence of the
    feature-label distribution and classification
    rule.
  • Report a list of classifiers and feature sets for
    analysis.
  • Issues Analysis of classifier and
    feature-selection performance
  • Better error estimation
  • Mathematical analysis of error estimators
  • Braga-Neto, U., and E. R. Dougherty, Exact
    Performance of Error Estimators for Discrete
    Classifiers, Pattern Recognition, 38 (11)
    1799-1814, 2005.

36
Is Knowledge Possible?
  • The scientific meaning of a classifier and its
    error estimate relate to the properties of the
    error estimator.
  • Choice 1 Estimate population density
    impossible.
  • Choice 2 Distribution-free error bounds
    useless.
  • Answer Model-Based Analysis
  • Pattern Recognition ? Data Mining
  • Knowledge is possible with proper epistemology.

37
Apparent Clusters in Microarray Data
Relationship?
patterns
38
What Are Good Clusters?
  • Example
  • 2 or 3 clusters?
  • What is the best separation?

39
The Clustering Problem
  • Apply a clustering algorithm to data and form
    clusters, as every clustering algorithm does.
  • Say, Gee Whiz! There are known related genes in
    a cluster.
  • Where is the possibility for verification by
    prediction? Indeed, what is to be verified?

40
Clustering and Scientific Knowledge
  • A scientific theory requires a model and a
    predictive methodology to test model validity.
  • Classification
  • Model classifier and error
  • Validity rests on the accuracy of error
    estimation
  • Model inferred (learned from data)
  • Clustering as historically used
  • Model (algorithm)
  • No framework for predictive model testing
  • No learning
  •  
  •  

41
Probabilistic Theory of Clustering
  • Clustering theory in the context of random sets
  • Probabilistic error measure based on points being
    clustered correctly
  • Bayes clusterer (optimal clustering algorithm)
  • Learning theory for clustering algorithms
  • Dougherty, E. R., and M. Brun, A Probabilistic
    Theory of Clustering, Pattern Recognition, 37
    (5), 917-925, 2004.

42
Example of Clustering Error
  • Left Realization of point process
  • Right Output of hierarchical clustering
  • Error 40

43
Validation Indices
  • Validation indices are meant to judge the
    validity of a clustering output.
  • They can be based on a number of heuristic
    considerations and methodologies.
  • Do they correspond to scientific validity?
  • Do validation indices correlate to clustering
    error?
  • Brun, M., Sima, C., Hua, J., Lowey, J., Carroll,
    B., Suh, E., and E. R. Dougherty, Model-Based
    Evaluation of Clustering Validation Measures,
    Pattern Recognition, 40 (3), 807-824, 2007.

44
Kendalls Correlation for Indices
  • Top Realization of point process
  • Bottom Kendalls correlation for different
    indices across different clustering algorithms

45
Regulatory Modeling
  • Find analytical tools for genomic data that can
    detect multivariate influences on decision-making
    produced by complex genetic networks.
  • Construct the minimal complexity network that can
    model sufficient information transfer to achieve
    goal.
  • Less computation
  • Less data required for inference
  • Given a model, discover ways to intervene in its
    dynamics to obtain desired behavior.

46
Gene Interaction
  • Genes interact via multi-protein complexes,
    feedback regulation, and pathway networks.
  • Complex molecular networks underlie biological
    function.
  • Most diseases do not result from a single gene
    product.
  • These interrelationships among genes constitute
    gene regulatory networks.

47
Muscle Network (Drosophila)
  • A gene network shows regulatory interaction.
  • msp-300 is a hub gene that regulates genes
    encoding motor proteins responsible for muscle
    contraction.
  • Zhao, W., Serpedin, E., and E. R. Dougherty,
    Inferring Gene Regulatory Networks from Time
    Series Data Using the Minimum Description Length
    Principle, Bioinformatics, 22 (17, 2129-2135,
    2006.

48
Desirable Model Properties
  • Incorporate rule-based dependencies between
    genes.
  • Rule-based dependencies may constitute important
    biological information.
  • Allow systematic study of global network
    dynamics.
  • In particular, individual gene effects on
    long-run network behavior.
  • Cope with uncertainty.
  • Small sample size, noisy measurements, robustness
  • System must be open to external latent variables

49
Infer Regulatory Genetic Function?
If gene X1 is active and gene X2 is suppressed,
gene Y would be activated
Can we infer regulatory genetic function from the
cDNA microarray data, for both known and unknown
functions?
50
Inference From Data
  • Key issues
  • Complex model
  • Limited data
  • Lack of appropriate time-course data for dynamics
  • Fundamental Principle Use simplest model that
    provides sufficient information to accomplish the
    task at hand and which is compatible with the
    data.
  • Formalize inference by postulating criteria that
    constitute a solution space for the inverse
    problem.
  • Constraint criteria are composed of restrictions
    on the form of the network biological,
    complexity.
  • Operational criteria are composed of relations
    that must be satisfied between the model and the
    data.

51
Regulatory Logic
  • Jacques Monod The logic of biological
    regulatory systems abideslike the workings of
    computers, by the propositional algebra of George
    Boole.
  • Shmulevich I., and E. R. Dougherty, Genomic
    Signal Processing, Princeton University Press,
    Princeton, 2007.

52
Boolean Predictive Relationships
  • Boolean Relationships in the NCI 60 ACDS
    (Anti-Cancer Drug Screen).
  • MRC1 VSNL1 ? HTR2C
  • SCYA7 CASR ? MU5SAC
  • Capture switch-like (ON/OFF) behavior.
  • Pal, R., Datta, A., Fornace, A. J., Bittner, M.
    L., and E. R. Dougherty, Boolean Relationships
    Among Genes Responsive to Ionizing Radiation in
    the NCI 60 ACDS, Bioinformatics, 21(8),
    1542-1549, 2005.

53
Basic Structure of Boolean Networks
1 means active/expressed 0 means
inactive/unexpressed
A
B
Boolean function A B X 0 0 1 0 1 1 1 0 0 1
1 1
X
In this example, two genes (A and B) regulate
gene X. In principle, any number of input genes
are possible. Positive/negative feedback is also
common (and necessary for homeostasis).
54
Network Dynamics
A
B
C
D
E
F
Time
0
1
1
0
0
1
At a given time point, all the genes form a
genome-wide gene activity pattern (GAP). Consider
the state space formed by all possible GAPs.
55
State Space of Boolean Networks
  • Similar GAPs lie close together.
  • There is an inherent directionality in the state
    space.
  • Some states are attractors (or limit-cycle
    attractors). The system may alternate between
    several attractors.
  • Other states are transient.

Picture generated using the program DDLab.
56
Probabilistic Boolean Networks
  • A PBN is composed of a collection of BNs.
  • At any time point, state transitions are
    controlled according to one of the BNs. With
    some probability, the PBN can switch to a
    different BN at a time point.
  • So long as there is no switch the PBN acts like a
    BN.
  • Allows for random gene perturbations.
  • Shmulevich, I., Dougherty, E. R., Kim, S., and W.
    Zhang, Probabilistic Boolean Networks A
    Rule-based Uncertainty Model for Gene Regulatory
    Networks, Bioinformatics, 18, 261-274, 2002.
  • Shmulevich, I., Dougherty, E. R., and W. Zhang,
    From Boolean to Probabilistic Boolean Networks
    as Models of Genetic Regulatory Networks,
    Proceedings of the IEEE, 90(11), 1778-1792, 2002.

57
PBN State Space
Attractors BN 2
BN 1
BN 2
Attractors BN 1
58
Properties of PBNs
  • Share the rule-based properties of Boolean
    networks.
  • Models uncertainty.
  • Dynamic behavior studied via Markov Chains.
  • Close relationship to Bayesian networks.
  • Attractors of a PBN are the attractors of the
    constituent BNs.
  • Can leave a BN attractor cycle when BN switches.
  • Brun, M., Dougherty, E. R., and I. Shmulevich,
    Steady-State Probabilities for Attractors in
    Probabilistic Boolean Networks, Signal
    Processing, in press, 2005.
  • Lahdesmaki, H., Hautaniemi, S., Shmulevich, I.,
    and Yli-Harja, O., Relationships Between
    Probabilistic Boolean Networks and Dynamic
    Bayesian Networks as Models of Gene Regulatory
    Networks, Signal Processing, in press, 2005.

59
Various Design Methods Proposed
  • Find genes with predictive capability for target
    gene (CoD).
  • Use mutual-information to find related genes.
  • Use MDL principle.
  • Optimize connectivity in a Bayesian framework
    relative to the gene profiles in the data.
  • Find networks satisfying biologically related
    constraints such as limited attractor structure,
    transient time, and connectivity.
  • Assuming steady-state data, require data states
    to be attractors.
  • Assuming biological determinism within a given
    cellular context, design a PBN under the
    assumption that constituent BNs produce
    consistent data subsets in the sample data.

60
Network Reduction
  • Network reduction is often desirable for
    computational reasons the state space is too
    large.
  • Delete genes reconstruct connectivity and rules.
  • Preserve probability structure.
  • Dougherty, E. R., and I. Shmulevich, Mappings
    Between Probabilistic Boolean Networks, EURASIP
    J. Signal Processing, 83 (4), 799-809, 2003.
  • Preserve steady-state distribution.
  • Ivanov, I., and E. R. Dougherty, Reduction
    Mappings Between Probabilistic Boolean Networks,
    EURASIP J. Applied Signal Processing, 4 (1),
    125-31, 2004.
  • Preserve dynamical structure.
  • Ivanov, I., Pal, R., and E. R. Dougherty,
    Reduction Mappings between Probabilistic Boolean
    Networks that Preserve Dynamical Structure, IEEE
    Trans. Signal Processing, to appear.

61
Intervention
  • A key goal of network modeling is to determine
    intervention targets (genes) such that the
    network can be persuaded to transition into
    desired states.
  • We desire genes that are the best potential
    lever points in the sense of having the
    greatest possible impact on desired network
    behavior.
  • Shmulevich, I., Dougherty, E. R., and W. Zhang,
    Gene Perturbation and Intervention in
    Probabilistic Boolean Networks, Bioinformatics,
    18, 1319-1331, 2002.

62
Dynamics
  • Dynamics of PBNs can be studied using Markov
    Chain theory.
  • We can ask the question In the long run, what
    is the probability that some given gene(s) will
    be ON/OFF?

63
Medical Benefits of Network Intervention
  • Prediction of new targets based on pathway
    context.
  • Stress and toxic response mechanisms.
  • Off-target effects of therapeutic compounds.
  • Characterization of disease states by dynamic
    behavior.
  • Gene- and protein-expression signatures for
    diagnostics.
  • Regulatory analysis for therapeutic intervention.

64
Possible Intervention Goals
  • Minimize the mean first passage time to a
    desirable state.
  • Maximize the probability of reaching a desirable
    state before a certain fixed time.
  • Minimize the time needed to reach a desirable
    state with a given fixed probability.
  • Shmulevich, I., Dougherty, E. R., and W. Zhang,
    Gene Perturbation and Intervention in
    Probabilistic Boolean Networks, Bioinformatics,
    Vol. 18, 1319-1331, 2002.
  • Shmulevich, I., Dougherty, E. R., and W. Zhang,
    Control of Stationary Behavior in Probabilistic
    Boolean Networks by Means of Structural
    Intervention, Biological Systems, Vol. 10,
    431-446, 2002.

65
Where and How to Intervene?
suppress or activate?
MBP-1
IAP-1
FRA-1
p21
ATF3
SSAT
REL-B
MDM2
PC-1
BCL3
p53
RCH1
66
External Control
  • Consider an external control variable and a cost
    function depending on state desirability and cost
    of action.
  • Minimize the cost function by a sequence of
    control actions over time control policy.
  • Application Design optimal treatment regime to
    drive the system away from undesirable states.
  • Datta, A., Choudhary, A., Bittner, M. L., and E.
    R. Dougherty, External Control in Markovian
    Genetic Regulatory Networks, Machine Learning,
    52, 169-181, 2003.
  • Pal. R., Datta, A., and E. R. Dougherty, Optimal
    Infinite Horizon Control for Probabilistic
    Boolean Networks, IEEE Transactions on Signal
    Processing, 54 (6), 2375-2387, 2006.
  •   

67
Optimal Control
  • Key Objective Optimally manipulate the external
    controls to move the GAP from an undesirable
    pattern to a desirable pattern.
  • Use available information, e.g., phenotypic
    responses, tumor size, etc.
  • Require a paradigm for modeling the evolution of
    the GAP under different controls.
  • PBN is one such paradigm.
  • Use the associated Markov chain.

68
Control in PBNs
  • Transition Probabilities depend on external
    control inputs e.g. chemotherapy, radiation, etc.
  • Assume m control inputs u1,u2......, um.
  • Each input can take on the values 0 ( not
    applied) or 1 (applied).
  • The values of the control inputs can be changed
    with time.

69
Control Setting
  • Control input vector at time k u1(k)
    ,......,um(k)
  • Change both GAP and control vector to integers,
    z(k) and v(k).
  • Then z(k) and v(k) can take on 2m values.
  • We have a system
  • w(k 1) w(k)A(v(k))
  • A is a stochastic matrix dependent on the control
    input.
  • We have a controlled homogeneous Markov chain.

70
Costs of Applying Control
  • Choose v(0), v(1), ..... to minimize a particular
    cost function.
  • Choice of cost function?
  • Consider finite treatment horizon k 0, 1,,M
    1.
  • Let Ck(z(k),v(k)) denote the cost of applying
    control v(k) at state z(k). (Input from
    biologists)
  • Cost of control over M 1 time steps

71
Terminal Costs
  • Net result of control action ends up in z(M).
  • Penalize z(M) in the cost to reduce chances of
    ending up in an undesirable state.
  • Define CM(z(M)) to be the terminal cost of ending
    up in state z(M).
  • Partition states into equivalence classes.
  • Assign higher penalties to states associated with
    rapid cell proliferation or reduced apoptosis and
    lower penalties for states associated with normal
    cell cycle. (Input from biologists)

72
Total Cost
73
Optimal Control Problem
74
WNT5A Network
  • Up-regulated WNT5A associated with increased
    metastasis.
  • Cost function penalizes WNT5A being up-regulated.
  • Optimal control policy with Pirin as control
    gene.

75
Shift of Steady-State Distribution
  • Optimal (infinite horizon) control with pirin has
    shifted the steady-state distribution to states
    with WNT5A down-regulated (a) with control (b)
    without control.

76
Robust Control
  • Owing to model uncertainty, we desire control
    policies that perform well under perturbations
    from model.
  • Performance bounds design policy on estimated
    transition matrix, P, and bound difference
    between the true and estimated controlled
    steady-state distribution
  • ?c ? ?c 1 ? kP ? P row-max
  • Pal, R., Datta, A., and E. R. Dougherty, Robust
    Intervention in Probabilistic Boolean Networks,
    IEEE Transactions on Signal Processing, to
    appear.

77
Robust Control Policies
  • Mini-Max Robust Policies Find optimal worst-case
    performance over uncertainty class of networks.
  • Pal, R., Datta, A., and E. R. Dougherty,
    Robustness of Intervention Strategies for
    Probabilistic Boolean Networks, IEEE GENSIPS,
    Tuusula, June, 2007.
  • Bayesian Robust Policies Find optimal policy
    relative to a prior distribution on the
    uncertainty class.
  • Pal, R., Datta, A., and E. R. Dougherty,
    Bayesian Robustness in the Control of Gene
    Regulatory Networks, IEEE Statistical Signal
    Processing Workshop, Madison, August, 2007.

78
Collaborators
  • Translational Genomics Research Institute
    (Arizona)
  • University of Texas M. D. Anderson Cancer
    Institute
  • Tampere University of Technology (Finland)
  • University of Sao Paulo (Brazil)
  • Strathclyde University (Scotland)
  • Columbia University
  • Acknowledgements
  • National Science Foundation
  • National Cancer Institute
  • National Human Genome Research Institute
  • W. M. Keck Foundation

79
References
  • Genomic Signal Processing and Statistics, eds. E.
    R. Dougherty, I. Shmulevich, J. Chen, J. Wang,
    Hindawi Press, 2005.
  • Genomic Signal Processing, I. Shmulevich and E.
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