Neil D. Lawrence
There are three types of lies: lies, damned lies and statistics
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There are three types of lies: lies, damned lies and statistics
Benjamin Disraeli
There are three types of lies: lies, damned lies and statistics
Benjamin Disraeli 1804-1881
There are three types of lies: lies, damned lies and ‘big data’
Neil Lawrence 1972-?
The pervasiveness of data brings forward particular challenges.
Those challenges are most sharply in focus for personalized health.
Particular opportunities, in challenging areas such as mental health.
compute | ~10 gigaflops | ~ 1000 teraflops? |
communicate | ~1 gigbit/s | ~ 100 bit/s |
embodiment (compute/communicate) |
10 | ~ 10^{13} |
This phenomenon has already revolutionised biology.
Large scale data acquisition and distribution.
Transcriptomics, genomics, epigenomics, ‘rich phenomics’.
Great promise for personalized health.
Automated decision making within the computer based only on the data.
A requirement to better understand our own subjective biases to ensure that the human to computer interface formulates the correct conclusions from the data.
Particularly important where treatments are being prescribed.
But what is a treatment in the modern era: interventions could be far more subtle.
Shift in dynamic from the direct pathway between human and data to indirect pathway between human and data via the computer
This change of dynamics gives us the modern and emerging domain of data science
Paradoxes of the Data Society
Quantifying the Value of Data
Privacy, loss of control, marginalization
Able to quantify to a greater and greater degree the actions of individuals
But less able to characterize society
As we measure more, we understand less
Perhaps greater preponderance of data is making society itself more complex
Therefore traditional approaches to measurement are failing
Curate’s egg of a society: it is only ‘measured in parts’
Election polls (UK 2015 elections, EU referendum, US 2016 elections)
Clinical trials vs personalized medicine: Obtaining statistical power where interventions are subtle. e.g. social media
A better characterization of human (see later)
There’s a sea of data, but most of it is undrinkable
We require data-desalination before it can be consumed!
Direct work on data generates an enormous amount of ‘value’ in the data economy but this is unaccounted in the economy
Hard because data is difficult to ‘embody’
Value of shared data: Wellcome Trust 2010 Joint Statement (from the “Foggy Bottom” meeting)
Encourage greater interaction between application domains and data scientists
Encourage visualization of data
Adoption of ‘data readiness levels’
Implications for incentivization schemes
Society is becoming harder to monitor
Individual is becoming easier to monitor
Potential for explicit and implicit discrimination on the basis of race, religion, sexuality, health status
All prohibited under European law, but can pass unawares, or be implicit
GDPR: General Data Protection Regulation
Potential for explicit and implicit discrimination on the basis of race, religion, sexuality, health status
All prohibited under European law, but can pass unawares, or be implicit
GDPR: Good Data Practice Rules
Covariance function, \(\mathbf{K}\)
Determines properties of samples.
Function of \({\bf X}\), \[k_{i,j} = k({\bf x}_i, {\bf x}_j)\]
Posterior mean
\[f_D({\bf x}_*) = \mathbf{k}({\bf x}_*, {\bf X}) \mathbf{K}^{-1} \mathbf{y}\]
Posterior covariance \[\mathbf{C}_* = \mathbf{K}_{*,*} - \mathbf{K}_{*,\mathbf{f}} \mathbf{K}^{-1} \mathbf{K}_{\mathbf{f}, *}\]
Posterior mean
\[f_D({\bf x}_*) = \mathbf{k}({\bf x}_*, {\bf X}) \boldsymbol{\alpha}\]
Posterior covariance \[\mathbf{C}_* = \mathbf{K}_{*,*} - \mathbf{K}_{*,\mathbf{f}} \mathbf{K}^{-1} \mathbf{K}_{\mathbf{f}, *}\]
We want to protect a user from a linkage attack…
…while still performing inference over the whole group.
Making a dataset private is more than just erasing names.
To achieve a level of privacy one needs to add randomness to the data.
This is a fundamental feature of differential privacy.
See The Algorithmic Foundations of Differential Privacy by Dwork and Roth for a rigorous introduction to the framework.
We have a dataset in which the inputs, \(\mathbf{X}\), are public. The outputs, \(\mathbf{y}\), we want to keep private.
Data consists of the heights and weights of 287 women from a census of the !Kung
Hall et al. (2013) showed that one can ensure that a version of \(f\), function \(\tilde{f}\) is \((\varepsilon, \delta)\)-differentially private by adding a scaled sample from a GP prior.
3 pages of maths ahead!
We applied this method to the GP posterior.
The covariance of the posterior only depends on the inputs, \(X\). So we can compute this without applying DP.
The mean function, \(f_D(\mathbf{x_*})\), does depend on \(\mathbf{y}\). \[f_D(\mathbf{x_*}) = \mathbf{k}(x_*, \mathbf{X}) \mathbf{K}^{-1} \mathbf{y}\]
We are interested in finding
\[|| f_D(\mathbf{x_*}) - f_{D^\prime}(\mathbf{x_*}) ||_H^2\]
…how much the mean function (in RKHS) can change due to a change in \(\mathbf{y}\).
Using the representer theorem, we can write \[|| f_D(\mathbf{x_*}) - f_{D^\prime}(\mathbf{x_*}) ||_H^2\]
as:
\[\Big|\Big|\sum_{i=1}^n k(\mathbf{x_*},\mathbf{x}_i) \left(\alpha_i - \alpha^\prime_i\right)\Big|\Big|_H^2\]
where \(\boldsymbol{\alpha} - \boldsymbol{\alpha}^\prime = \mathbf{K}^{-1} \left(\mathbf{y} - \mathbf{y}^\prime \right)\)
L2 Norm
\[\Big|\Big|\sum_{i=1}^n k(\mathbf{x_*},\mathbf{x}_i) \left(\alpha_i - \alpha^\prime_i\right)\Big|\Big|_H^2\]
where \(\boldsymbol{\alpha} - \boldsymbol{\alpha}^\prime = \mathbf{K}^{-1} \left(\mathbf{y} - \mathbf{y}^\prime \right)\)
We constrain the kernel: \(-1\leq k(\cdot,\cdot) \leq 1\) and we only allow one element of \(\mathbf{y}\) and \(\mathbf{y}'\) to differ (by at most \(d\)).
So only one column of \(\mathbf{K}^{-1}\) will be involved in the change of mean (which we are summing over).
The distance above can then be shown to be no greater than \(d\;||\mathbf{K}^{-1}||_\infty\)
This ‘works’ in that it allows DP predictions…but to avoid too much noise, the value of \(\varepsilon\) is too large (here it is 100)
EQ kernel, \(\ell = 25\) years, \(\Delta=100\)cm
Using sparse methods (i.e. inducing inputs) can help reduce the sensitivity a little. We’ll see more on this later.
So far we’ve made the whole posterior mean function private…
…what if we just concentrate on making particular predictions private?
Standard approach: sample the noise is from the GP’s prior.
Not necessarily the most ‘efficient’ covariance to use.
Left: Function change. Right: test point change
Left: Function change. Right: test point change
Left: Function change. Right: test point change
Left: Function change. Right: test point change
Left: Function change. Right: test point change
Left: Function change. Right: test point change
Hall et al. (2013) also presented a bound on vectors.
Find a bound (\(\Delta\)) on the scale of the output change, in term of its Mahalanobis distance (wrt the added noise covariance).
\[\sup_{D \sim {D'}} ||\mathbf{M}^{-1/2} (\mathbf{y}_* - \mathbf{y}_{*}')||_2 \leq \Delta\]
We use this to scale the noise we add:
\[\frac{\text{c}(\delta)\Delta}{\varepsilon} \mathcal{N}_d(0,\mathbf{M})\]
We get to pick \(\mathbf{M}\)
Intuitively we want to construct \(\mathbf{M}\) so that it has greatest covariance in those directions most affected by changes in training points, so that it will be most able to mask those changes.
The change in posterior mean predictions is,
\[\mathbf{y}_* - \mathbf{y}'_* = \mathbf{K}_{*f} \mathbf{K}^{-1} (\mathbf{y}-\mathbf{y}')\]
Effect of perturbing each training point on each test point is represented in the cloaking matrix,
\[\mathbf{C} = \mathbf{K}_{*f} \mathbf{K}^{-1}\]
We assume we are protecting only one training input’s change, by at most \(d\).
So \(\mathbf{y}-\mathbf{y}'\) will be all zeros except for one element, \(i\).
So the change in test points will be (at most)
\[\mathbf{y}_*' - \mathbf{y}_* = d \mathbf{C}_{:i}\]
We’re able to write the earlier bound as,
\[d^2 \sup_{i} \mathbf{c}_i^\top \mathbf{M}^{-1} \mathbf{c}_i \leq\Delta\]
where \(\mathbf{c}_i \triangleq \mathbf{C}_{:i}\)
Dealing with \(d\) elsewhere and setting \(\Delta = 1\) (thus \(0 \leq \mathbf{c}_i^\top \mathbf{M}^{-1} \mathbf{c}_i \leq 1\)) and minimise \(\log |\mathbf{M}|\) (minimises the partial entropy).
Using Lagrange multipliers and gradient descent, we find
\[\mathbf{M} = \sum_i{\lambda_i \mathbf{c}_i \mathbf{c}_i^\top}\]
The noise added by this method is now practical.
EQ kernel, \(l = 25\) years, \(\Delta=100\)cm, \(\varepsilon=1\)
It also has some interesting features;
Tested on 4D citibike dataset (predicting journey durations from start/finish station locations).
The method appears to achieve lower noise than binning alternatives (for reasonable \(\varepsilon\)).
lengthscale in degrees, values above, journey duration (in seconds)
Outliers poorly predicted.
Too much noise around data ‘edges’.
Use inducing inputs to reduce the sensitivity to these outliers.
For 1D !Kung, RMSE improved from \(15.0 \pm 2.0 \text{cm}\) to \(11.1 \pm 0.8 \text{cm}\)
Use Age and Weight to predict Height
For 2D !Kung, RMSE improved from \(22.8 \pm 1.9 \text{cm}\) to \(8.8 \pm 0.6 \text{cm}\)
Note that the uncertainty across cross-validation runs smaller. 2D version benefits from data’s 1D manifold.
Many solutions rely on education and awareness