Bayes for Beginners



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Bayes for Beginners

  • Reverend Thomas Bayes (1702-61)
  • Stuart Rosen° and Elena Rusconi§
  • °Department of Phonetics and Linguistics, UCL,
  • §Institute of Cognitive Neuroscience, UCL
  • MfD
  • 15–II-2006

Overview of SPM Analysis

  • Motion
  • Correction
  • Smoothing
  • Spatial
  • Normalisation
  • General Linear Model
  • Statistical Parametric Map
  • fMRI time-series
  • Design matrix
  • Anatomical Reference

Spatial Normalisation - Overfitting

  • Template
  • image
  • Affine registration.
  • (2 = 472.1)
  • Non-linear
  • registration
  • without
  • Bayes constraints.
  • (2 = 287.3)
  • Non-linear
  • registration
  • using
  • Bayes.
  • (2 = 302.7)
  • Without Bayesian constraints, the non-linear spatial normalisation can introduce unnecessary warps.
  • Bayes and image segmentation
  • Want to automatically classify regions into grey matter, white matter, CSF and non-brain tissue.
  • How do we use prior information? (probabilities of each voxel being of a particular type derived from a database of 152 scans)
  • Bayes!

A big advantage of a Bayesian approach

  • Allows a principled approach to the exploitation of all available data …
  • with an emphasis on continually updating one’s models as data accumulate
  • as seen in the consideration of what is learned from a positive mammogram

Bayesian Reasoning

  • - Casscells, Schoenberger & Grayboys, 1978
  • - Eddy, 1982
  • Gigerenzer & Hoffrage, 1995, 1999
  • Butterworth, 2001
  • Hoffrage, Lindsey, Hertwig & Gigerenzer, 2001
  • When PREVALENCE, SENSITIVITY, and FALSE POSITIVE RATES are given, most physicians misinterpret the information in a way that could be potentially disastrous for the patient.

Bayesian Reasoning

  • ASSUMPTIONS
  • 1% of women aged forty who participate in a routine screening have breast cancer
  • 80% of women with breast cancer will get positive tests
  • 9.6% of women without breast cancer will also get positive tests
  • EVIDENCE
  • A woman in this age group had a positive test in a routine screening
  • PROBLEM
  • What’s the probability that she has breast cancer?

Bayesian Reasoning

  • ASSUMPTIONS
  • 10 out of 1000 women aged forty who participate in a routine screening have breast cancer
  • 800 out of 1000 of women with breast cancer will get positive tests
  • 95 out of 1000 women without breast cancer will also get positive tests
  • PROBLEM
  • If 1000 women in this age group undergo a routine screening, about what fraction of women with positive tests will actually have breast cancer?

Bayesian Reasoning

  • ASSUMPTIONS
  • 100 out of 10,000 women aged forty who participate in a routine screening have breast cancer
  • 80 of every 100 women with breast cancer will get positive tests
  • 950 out of 9,900 women without breast cancer will also get positive tests
  • PROBLEM
  • If 10,000 women in this age group undergo a routine screening, about what fraction of women with positive tests will actually have breast cancer?

Bayesian Reasoning

  • Before the screening:
  • 100 women with breast cancer
  • 9,900 women without breast cancer
  • After the screening:
  • A = 80 women with breast cancer and positive test
  • B = 20 women with breast cancer and negative test
  • C = 950 women without breast cancer and positive test
  • D = 8,950 women without breast cancer and negative test
  • Proportion of cancer patients with positive results, within the group of ALL patients with positive results:
  • A/(A+C) = 80/(80+950) = 80/1030 = 0.078 = 7.8%

Compact Formulation

  • C = cancer present, T = positive test
  • p(A|B) = probability of A, given B, ~ = not
  • PRIOR PROBABILITY
  • p(C) = 1%
  • CONDITIONAL PROBABILITIES
  • p(T|C) = 80%
  • p(T|~C) = 9.6%
  • POSTERIOR PROBABILITY (or REVISED PROBABILITY)
  • p(C|T) = ?
  • PRIORS

Bayesian Reasoning

  • Before the screening:
  • 100 women with breast cancer
  • 9,900 women without breast cancer
  • After the screening:
  • A = 80 women with breast cancer and positive test
  • B = 20 women with breast cancer and negative test
  • C = 950 women without breast cancer and positive test
  • D = 8,950 women without breast cancer and negative test
  • Proportion of cancer patients with positive results, within the group of ALL patients with positive results:
  • A/(A+C) = 80/(80+950) = 80/1030 = 0.078 = 7.8%

Bayesian Reasoning

  • Prior Probabilities:
  • 100/10,000 = 1/100 = 1% = p(C)
  • 9,900/10,000 = 99/100 = 99% = p(~C)
  • Conditional Probabilities:
  • A = 80/10,000 = (80/100)*(1/100) = p(T|C)*p(C) = 0.008
  • B = 20/10,000 = (20/100)*(1/100) = p(~T|C)*p(C) = 0.002
  • C = 950/10,000 = (9.6/100)*(99/100) = p(T|~C)*p(~C) = 0.095
  • D = 8,950/10,000 = (90.4/100)*(99/100) = p(~T|~C) *p(~C) = 0.895
  • Rate of cancer patients with positive results, within the group of ALL patients with positive results:
  • A/(A+C) = 0.008/(0.008+0.095) = 0.008/0.103 = 0.078 = 7.8%

-----> Bayes’ theorem

  • p(T|C)*p(C)
  • p(C|T) = ______________________
  • P(T|C)*p(C) + p(T|~C)*p(~C)
  • A
  • A + C

Comments

  • Common mistake: to ignore the prior probability
  • The conditional probability slides the revised probability in its direction but doesn’t replace the prior probability
  • A NATURAL FREQUENCIES presentation is one in which the information about the prior probability is embedded in the conditional probabilities (the proportion of people using Bayesian reasoning rises to around half).
  • Test sensitivity issue (or: “if two conditional probabilities are equal, the revised probability equals the prior probability”)
  • Where do the priors come from?

-----> Bayes’ theorem

  • p(X|A)*p(A)
  • p(A|X) = ______________________
  • P(X|A)*p(A) + p(X|~A)*p(~A)
  • Given some phenomenon A that we want to investigate, and an observation X that is evidence about A, we can update the original probability of A, given the new evidence X.
  • It relates the conditional density of a parameter (posterior probability) with its unconditional density (prior, since depends on information present before the experiment).
  • The likelihood is the probability of the data given the parameter and represents the data now available.
  • Bayes’ Theorem for a given parameter 
  • p (data) = p (data) p () / p (data)
  • 1/P (data) is basically
  • a normalizing constant
  • The prior is the probability of the parameter and represents what was thought before seeing the data.
  • The posterior represents what is thought given both prior information and the data just seen.
  • Likelihood: p(y|) = (Md, d-1)
  • Prior: p() = (Mp, p-1)
  • Posterior: p(y) ∝ p(y|)*p() = (Mpost, post-1)
  • Mp
  • p-1
  • Mpost
  • post-1
  • d-1
  • Md
  • post = d + p Mpost = d Md + p Mp
  •  post
  • Posterior Probability Distribution
  • precision  = 1/2

Activations in fMRI….

  • Classical
    • ‘What is the likelihood of getting these data given no activation occurred?’
  • Bayesian option (SPM2)
    • ‘What is the chance of getting these parameters, given these data?

What use is Bayes in deciding what brain regions are active in a particular study?

  • Problems with classical frequentist approach
    • All inferences relate to disproving the null hypothesis
    • Never fully reject H0, only say that the effect you see is unlikely to occur by chance
    • Corrections for multiple comparisons
      • significance depends on the number of contrasts you look at
    • Very small effects can be declared significant with enough data
  • Bayesian Inference offers a solution through Posterior Probability Maps (PPMs)

SPMs and PPMs

  • SPMs: show voxels with
  • non-zero activations

PPMs

  • Advantages
  • Disadvantages
  • One can infer a cause
  • DID NOT elicit a response
  • SPMs conflate effect-size
  • and effect-variability
  • Computationally
  • demanding (priors are determined empirically)
  • Utility of Bayesian
  • approach is yet
  • to be established

Frequentist vs. Bayesian by Berry

  • 1. Probabilities of data vs. probabilities of parameters (& also data).
  • 2. Evidence used:
    • Frequentist measures specific to experiment.
    • Posterior distribution depends on all available information.
  • Makes Bayesian approach appealing, but assembling, assessing, & quantifying information is work.

3. Depend on probabilities of results that could occur vs. did occur:

  • 3. Depend on probabilities of results that could occur vs. did occur:
    • Frequentist measures (e.g., p values, confidence intervals) incorporate probabilities of data that were possible but did not occur.
    • Posterior depends on data only through the likelihood, which is calculated from observed data.
  • 4. Flexibility:
    • Frequentist measures depend on design; require that design be followed.
    • Bayesian view: update continually as data accumulate (only requirement is honesty). Sample size: need not choose in advance. Weigh costs/benefits; decide whether to start experiment. After experiment starts, decide whether to continue—stop at any time, for any reason.

5. Decision making

  • 5. Decision making
    • Frequentist: historically avoided.
    • Bayesian: tailored to decision analysis; losses and gains considered explicitly.

Statistics: A Bayesian Perspective D. Berry, 1996, Duxbury Press.

  • Statistics: A Bayesian Perspective D. Berry, 1996, Duxbury Press.
  • http://ftp.isds.duke.edu/WorkingPapers/97-21.ps
    • “Using a Bayesian Approach in Medical Device Development”, also by Berry
  • http://www.pnl.gov/bayesian/Berry/
    • a powerpoint presentation by Berry
  • http://yudkowsky.net/bayes/bayes.html
    • Extremely clear presentation of the mammography example; highly polemical and fun too!
  • http://www.stat.ucla.edu/history/essay.pdf
    • Bayes’ original essay
  • Jaynes, E. T., 1956, `Probability Theory in Science and Engineering,' (No. 4 in `Colloquium Lectures in Pure and Applied Science,' Socony-Mobil Oil Co. USA. http://bayes.wustl.edu/etj/articles/mobil.pdf
    • A physicist’s take on Bayesian approaches. Proposes an interesting metric of probability using decibels (yes, the unit used for sound levels!).
  • http://www.sportsci.org/resource/stats/
    • a skeptical account of Bayesian approaches. The rest of the site is very informative and sensible about basic statistical issues.
  • References

Bayes’ ending



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