SMPGD 2017

2017-01-12

St Mary’s Campus, Imperial College, London

Neil D. Lawrence

Amazon and University of Sheffield

@lawrennd inverseprobability.com

Mathematical Statistics

• ‘Founded’ by Karl Pearson (1857-1936)

‘Mathematical Data Science’

• ‘Founded’ by ? (?-?)

Background: Big Data

• 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.

“Embodiment Factors”

compute	~10 gigaflops	~ 1000 teraflops?
communicate	~1 gigbit/s	~ 100 bit/s
embodiment (compute/communicate)	10	~ 1013

Evolved Relationship

Effects

• This phenomenon has already revolutionised biology.

• Large scale data acquisition and distribution.

• Transcriptomics, genomics, epigenomics, ‘rich phenomics’.

• Great promise for personalized health.

Societal Effects

• 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.

Societal Effects

• 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

Challenges

1. Paradoxes of the Data Society

2. Quantifying the Value of Data

3. Privacy, loss of control, marginalization

Breadth vs Depth Paradox

• 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

What?

• 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’

Wood or Tree

• Can either see a wood or a tree.

Examples

• 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

Consistency

• Typically \(\boldsymbol{\theta} \in \Re^{\mathcal{O}(p)}\)

• Consistency reliant on large sample approximation of KL divergence

\[ \text{KL}(P(\mathbf{Y})|| p(\mathbf{Y}|\boldsymbol{\theta}))\]

• Minimization is equivalent to maximization of likelihood.

• A foundation stone of classical statistics.

Large \(p\)

• For large \(p\) the parameters are badly determined.

• Large \(p\) small \(n\) problem.

• Easily dealt with through definition.

Breadth vs Depth

• Modern Measurement deals with depth (many subjects) … or breadth lots of detail about subject.

• But what about
  • \(p\approx n\)?
  • Stratification of populations: batch effects etc.

• Multi-task learning (Natasha Jaques)

Does \(p\) Even Exist?

• Massively missing data.

• Classical bias towards tables.

• Streaming data.

\[ \mathbf{Y} = \begin{bmatrix} y_{1, 1} & y_{1, 2} &\dots & y_{1,p}\\ y_{2, 1} & y_{2, 2} &\dots & y_{2,p}\\ \vdots & \vdots &\dots & \vdots\\ y_{n, 1} & y_{n, 2} &\dots & y_{n,p} \end{bmatrix} \in \Re^{n\times p} \]

General index on \(y\)

\[y_\mathbf{x}\]

where \(\mathbf{x}\) might include time, spatial location …

Streaming data. Joint model of past, \(\mathbf{y}\) and future \(\mathbf{y}_*\)

\[p(\mathbf{y}, \mathbf{y}_*)\]

Prediction through:

\[p(\mathbf{y}_*|\mathbf{y})\]

Kolmogorov Consistency — Exchangeability

• From the sum rule of probability we have \begin{align*} p(\mathbf{y}|n^*) = \int p(\mathbf{y}, \mathbf{y}^*) \text{d}\mathbf{y}^* \end{align*}

  \(n^*\) is length of \(\mathbf{y}^*\).

• Consistent if \(p(\mathbf{y}|n^*) = p(\mathbf{y})\)

• Prediction then given by product rule \begin{align*} p(\mathbf{y}^*|\mathbf{y}) = \frac{p(\mathbf{y}, \mathbf{y}^*)}{p(\mathbf{y})} \end{align*}

\(p(\mathbf{y}^*|\mathbf{y})\)

Parametric Models

• Kolmogorov consistency trivial in parametric model. \begin{align*} p(\mathbf{y}, \mathbf{y}^*) = \int \prod_{i=1}^n p(y_{i} | \boldsymbol{\theta})\prod_{i=1}^{n^*}p(y^*_i|\boldsymbol{\theta}) p(\boldsymbol{\theta}) \text{d}\boldsymbol{\theta} \end{align*}
• Marginalizing \begin{align*} p(\mathbf{y}) = \int \prod_{i=1}^n p(y_{i} | \boldsymbol{\theta})\prod_{i=1}^{n^*} \int p(y^*_i|\boldsymbol{\theta}) \text{d}y^*_i p(\boldsymbol{\theta}) \text{d}\boldsymbol{\theta} \end{align*}

Parametric Bottleneck

• Bayesian methods suggest a prior over \(\boldsymbol{\theta}\) and use posterior, \(p(\boldsymbol{\theta}|\mathbf{y})\) for making predictions. \begin{align*} p(\mathbf{y}^*|\mathbf{y}) = \int \prod_i p(y_i^* | \boldsymbol{\theta}) p(\boldsymbol{\theta}|\mathbf{y})\text{d}\boldsymbol{\theta} \end{align*}

• Design time problem: parametric bottleneck. \[p(\boldsymbol{\theta} | \mathbf{y})\]

• Streaming data could turn out to be more complex than we imagine.

Finite Storage

• Despite our large interconnected brains, we only have finite storage.

• Similar for digital computers. So we need to assume that we can only store a finite number of things about the data \(\mathbf{y}\).

• This pushes us back towards parametric models.

• Our solutions: Gaussian processes + inducing variable approximations

Also need

• More classical statistics!
  • Like the ‘paperless office’

• A better characterization of human (see later)

• Larger studies (100,000 genome)
  • Combined with complex models: algorithmic challenges

Quantifying the Value of Data

There’s a sea of data, but most of it is undrinkable

We require data-desalination before it can be consumed!

Data — Quotes from NIPS Workshop on ML for Healthcare

• 90% of our time is spent on validation and integration (Leo Anthony Celi)
• “The Dirty Work We Don’t Want to Think About” (Eric Xing)
• “Voodoo to get it decompressed” (Francisco Giminez)
• In health care clinicians collect the data and often control the direction of research through guardianship of data.

Value

• How do we measure value in the data economy?
• How do we encourage data workers: curation and management
• Incentivization for sharing and production.
• Quantifying the value in the contribution of each actor.

Credit Allocation

• 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)

Solutions

• Encourage greater interaction between application domains and data scientists

• Encourage visualization of data

• Implications for incentivization schemes

• Adoption of data readiness levels

Privacy, Loss of Control and Marginalization

• Society is becoming harder to monitor

• Individual is becoming easier to monitor

Modelling

Modelling

Hate Speech or Political Dissent?

• social media monitoring for ‘hate speech’ can be easily turned to political dissent monitoring

Marketing

• can become more sinister when the target of the marketing is well understood and the (digital) environment of the target is also so well controlled

Free Will

• What does it mean if a computer can predict our individual behavior better than we ourselves can?

Discrimination

• 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

Marginalization

• Credit scoring, insurance, medical treatment
• What if certain sectors of society are under-represented in our aanalysis?
• What if Silicon Valley develops everything for us?

Digital Revolution and Inequality?

Amelioration

• Work to ensure individual retains control of their own data
• We accept privacy in our real lives, need to accept it in our digital

• Control of persona and ability to project

• Need better technological solutions: trust and algorithms.

Awareness

• Need to increase awareness of the pitfalls among researchers
• Need to ensure that technological solutions are being delivered not merely for few (#FirstWorldProblems)
• Address a wider set of challenges that the greater part of the world’s population is facing

Conclusion

• Data science offers a great deal of promise for personalized health
• There are challenges and pitfalls
• It is incumbent on us to avoid them
• Need new ways of thinking!
• Mathematical Data Science

Many solutions rely on education and awareness

Thanks!

• twitter: @lawrennd
• blog: http://inverseprobability.com