• The idea is that before we observe 20 flips of a coin
• and if we think of 17 heads and 3 tails as a possible outcome
• then exchangeablility means that we don't think of the positions that the 3 heads can occupy as being special

• An infinite sequence \(X_1,\dots,X_n,\dots\) of random variables is said to be exchangeable if for all \(n = 2,3,\dots\), \[X_1,\dots,X_n \overset{d}{=} X_{\pi(1)},\dots,X_{\pi(n)} \text{ for all } \pi \in S(n)\] where \(S(n)\) is the group of permutations of \(\{1,\dots,n\}\)
• For example, for a binary sequence, we may have: \[p(1,1,0,0,0,1,1,0) = p(1,0,1,0,1,0,0,1)\]
• If \(X_1,\dots,X_n,\dots\) are independent and identically distributed, they are exchangeable, but not conversely

• Consider an urn with \(b\) black balls and \(w\) white balls
• Draw a ball at random and note its colour
• Replace the ball together with \(a\) balls of the same colour
• Repeat the procedure ad infinitum
• Let \(X_i =1\) if the \(i\)th draw yields a black ball and \(X_i = 0\) otherwise
• The sequence \(X_1,\dots,X_n,\dots\) is exchangeable: \[ \begin{align} p(1,1,0,1) & = \frac{b}{b+w} \frac{b+a}{b+w+a} \frac{w}{b+w+2a} \frac{b+2a}{b+w+3a} \\ & = \frac{b}{b+w} \frac{w}{b+w+a} \frac{b+a}{b+w+2a} \frac{b+2a}{b+w+3a} = p(1,0,1,1) \end{align} \]
• \(X_1,\dots,X_n,\dots\) are not independent

• Suppose \(X_1,\dots,X_n\) are conditionally i.i.d. given some unknown parameter \(\theta\)
• Then for any permutation \(\pi\) of \(\{1,...,n\}\) and any set of values \((x_1,\dots,x_n)\)
• \(X_i\)'s are conditionally i.i.d. \[f(x_1,\dots,x_n) = \int f(x_1,\dots,x_n | \theta) p(\theta) d\theta = \int \left( \prod_{i=1}^n f(x_i|\theta) \right) p(\theta) d\theta\]
• product does not depend on order: \[= \int \left( \prod_{i=1}^n f(x_{\pi(i)} | \theta) \right) p(\theta) d\theta = f\left( x_{\pi(i)},\dots,x_{\pi(n)} \right)\]

• This means: \[ X_1,\dots,X_n \text{ are exchangeable for all $n$ } \Leftarrow \begin{cases} X_1,\dots,X_n \, \big| \, \theta \text{ i.i.d. } \\ \theta \sim p(\theta) \end{cases} \]

• In the other direction:
  • de Finetti showed in 1931 that all infinite exchangeable binary sequences are i.i.d. Bernoulli with beta prior \(\mathcal{B}(b/a,w/a)\) on the success parameter
  • Hewitt and Savage (1955) generalized it to any infinite exchangeable sequences
  • Diaconis and Freedman (1980) generalized it to finite exchangeable sequences

• Suppose we have text written in foreign language
• We are asked whether text is meaningful or meaningless
• We have very limitted partial dictionary: e.g. umbrella and rain
• Can we decide using this limited dictionary whether text is meaningful?
• Idea: Check whether conceputally related words appear in close proximity
• If text is meaningless, we don't expect this to happen very often
• If text is meaningful, we expect this to happen more often than by pure chance

Write bible as two-dimensional array of letters ignoring spaces

Source: Witztum, Rips, and Rosenberg (1994)

1. Define distance between two words \(w\) and \(w'\):
   Euclidean distance (considering letter matrix separated by spacing \(1\)) \(d(w,w') =\)
   squared distance between consecutive letters of \(w\) +
   squared distance between consecutive letters of \(w'\) +
   minimal distance between a letter from \(w\) and \(w'\)
2. Define a statistic: Average distance between a set of predefined word pairs calculated over the entire text
3. Choose word pair list: From other book (Encyclopedia of Great Men in Israel): 32 Rabbi names and their birthdays
4. Compute null distribution of average distance between all the word pairs using permutations

• They computed \(p\)-value using \(999\,999\) random name–birthday permutations (from the total of \(32!\) permutations to assign \(i\)th name of Rabbi to randomly permuted birthday)
• Rank the observed statistics among the permutations
• They found a \(p\)-value of \(0.00002\)
• Thus, we can say that equidistant letter sequences of Rabbi name and birthday is not due to chance
• For more details see: Witztum, Rips, and Rosenberg (1994). Equidistant Letter Sequences in the Book of Genesis

• We encouter two-sample problems in many neuroimaging studies
  • Compare two populations, such as people who are autistic and healthy controls
  • Find morphological, connectivity, and functional difference
  • Inform new therapies and cures
• We work on preprocessed neuroimaging data from the Autism Brain Imaging Data Exchange (ABIDE). The data is openly avaialbe on the ABIDE website
  • ABIDE is a collaboration of 16 international imaging sites
  • Openly sharing neuroimaging data from 539 individuals suffering from Autism Spectrum Disorder (ASD) and 573 typical controls
  • We will focus on a subset of 39 paricipants (all aquired at Stanford)

Source: multiple-sclerosis-research.blogspot.com

Source: Das, Avants, Grossman, and Gee (2009). Registration based cortical thickness measurement

• Analyze geometric difference through deformation
• Find deformations \(\varphi_k\) from template \(\boldsymbol{I}_0\) to participant images \(\boldsymbol{I}_k\)

• Parametrize \(\varphi_k(x,\boldsymbol{q})\) with B-Splines (coordinates \(x_i \in \boldsymbol{I}_0\), parameters \(\boldsymbol{q}\)) \[\sum_{i=1}^N \left( \boldsymbol{I}_k(x_i) \circ \varphi_k(x_i,\boldsymbol{q}) - \boldsymbol{I}_0(x_i) \right)^2 + \| \operatorname{Jac} \varphi_k(x_i,q) \|_{\ell_2}^2\]
• B-Splines are piecewise polynomial functions and can be used as basis functions to describe more complicated nonlinear functions
• minimize this with fixed \(\lambda\) \[\widehat{\boldsymbol{q}} = \operatorname{minimize}_\boldsymbol{q} \Big\{ \operatorname{Similarity}(\boldsymbol{I}_0,\boldsymbol{I}_k,\boldsymbol{q}) + \lambda \operatorname{Regularity}(\boldsymbol{q}) \Big\}\]

Split the data in two groups: autistic (1) and healthy participants (2)

PaInfo <- read.csv("Phenotypic_V1_0b_preprocessed1.csv", header=TRUE)
copiedFilesInd = PaInfo$FILE_ID %in% substr(fileNames,1,16)
copiedFilesGroup = PaInfo$DX_GROUP[copiedFilesInd]
ThicknessAutistic = brainMatrix[copiedFilesGroup==1,mask==1]
ThicknessHealthy = brainMatrix[copiedFilesGroup==2,mask==1]

Peform voxelwise nonparametric test Wilcoxon two-sample rank test

uncorrectedPValues = sapply(1:dim(ThicknessAutistic)[2], 
                            function(i) 
                              wilcox.test(ThicknessAutistic[,i],
                                          ThicknessHealthy[,i],
                                          alternative = "two.sided")$p.value)

• One could now proceed and correct for multiple comparisons using Gaussian random field theory or controlling the false discorvery rate
• An alternative options is to use permutations tests
• The idea (do the following \(N\) times):
  • Permute group assignments
  • Compute voxelwise \(p\)-value
  • Threshold \(p\)-value image at fixed primary threshold (e.g. \(0.001\))
  • Note largest contiguous voxel cluster size
• This forms our null distribution of largest contiguous voxel cluster sizes
• Declare unpermuted observed clusters as significant when larger than \(\operatorname{floor}(\alpha \times N + 1)\) in permutation distribution

• This approach comes at the price of reduced localization power
• The null hypotheses for voxels within a significant cluster are not tested, so individual voxels cannot be declared significant
• Only the omnibus null hypothesis for the cluster can be rejected
• Further, the choice of primary threshold dictates the power of the test in detecting different types of deviation from the omnibus null hypothesis
• With small primary threshold (e.g. \(0.001\)),
  we will foucs on many but small clusters
• With large primary threshold (e.g. \(0.01\)),
  we will foucs on few but large clusters

• Diaconis (1977). Finite Forms Of De Finetti's Theorem On Exchangeability
• Diaconis and Freedman (1980). Finite exchangeable sequences
• de Finetti (1972). Probability Induction and Statistics
• Hewitt and Savage (1955). Symmetric measures on Cartesian products
• Peter Hoff (2009). A First Course in Bayesian Statistical Methods
• Good (2005). Permutations, Parametric, and Boostrap Test of Hypothesis
• Witztum, Rips, and Rosenberg (1994). Equidistant Letter Sequences in the Book of Genesis
• Nichols and Holmes (2001). Nonparametric Permutation Tests For Functional Neuroimaging: A Primer with Examples
• Das, Avants, Grossman, and Gee (2009). Registration Based Cortical Thickness Measurement
• Steffen Lauritzen (2007). Lecture notes