bayesian nonparametric modeling of suicide attempts

Introduction Indian Buffet Process Observation Model Inference Experiments Conclusions References

Bayesian Nonparametric Modeling of SuicideAttempts

Francisco J. R. Ruiz, Isabel Valera, Carlos Blanco,Fernando Perez-Cruz

[3]

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Outline

1 Introduction

2 Indian Buffet Process

3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Introduction

Every year, more than 34,000 suicides occur and over 370,000individuals are treated for self-inflicted injuries in emergency roomsin the U.S.

First step to suicide prevention: Identification of the factors thatincrease the risk of attempting suicide [1].

Challenging and complex task.

We apply Bayesian nonparametric tools to find out the latentfeatures that increase the risk of attempting suicide.

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Goals

Find out and analyze the latent causes of suicides attempts.

NESARC database (National Epidemiologic Survey on Alcohol andRelated Conditions):

Samples the U.S. population.Nearly 3,000 questions regarding the way of life, medical conditions,depression and other mental disorders.Mainly yes-or-no questions, and some multiple-choice and questionswith ordinal answers.

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Indian Buffet Process [2]

The IBP places a prior distribution over binary matrices where thenumber of columns (features) K →∞.

Matrix ZN×K ∼ IBP(α) (α: concentration parameter).

Each element znk ∈ {0, 1} indicates whether the k th featurecontributes to the nth data point.

For a finite number of data points N, the number of non-zerocolumns K+ is finite.

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Culinary Metaphor

Generative Process:

N customers enter sequentially a restaurant with an infinitely longbuffet of dishes.

The first customer takes a serving from a Poisson(α) number ofdishes.

The i-th customer:1 Moves along the buffet, sampling dishes in proportion to their

popularity, i.e., with probability proportional to mk =∑

j<i zjk .2 Having reached the end of all previous sampled dishes, the i-th

customer then tries a Poisson(αi

) number of new dishes.

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Inference via Gibbs Sampling

N × D observation matrix X, where the nth row contains aD-dimensional observation vector xn·.

The algorithm iteratively samples the value of each element znk :

p(znk = 1|X,Z¬nk) ∝ p(X|Z)p(znk = 1|Z¬nk).

Exchangeability property of the IBP:

p(znk = 1|Z¬nk) =m−n,kN

,

where m−n,k =∑

i 6=n zik .

Then, the number of new columns is drawn from a distributionwhere the prior is Poisson(αN ).

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Observation Model (1/2)

Discrete observations: xnd ∈ {1, . . . ,R} ({‘Yes’, ‘No’, ‘Blank’,‘Unkown’, . . . } in the NESARC database).

Multiple-logistic function:

p(xnd = ‘yes’|zn·,Bd) ∝ exp (zn·bd·yes),

p(xnd = ‘no’|zn·,Bd) ∝ exp (zn·bd·no),

. . .

p(xnd = r |zn·,Bd) =exp (zn·b

d·r )

R∑r ′=1

exp (zn·bd·r ′)

, r = 1, . . . ,R,

where bd·r ∼ N (0,Σb = σ2B I) is the r−th column of matrix Bd

K×R .

The matrices Bd weight differently the contribution of every latentfeature for every component d (the probability of every response toa question).

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Observation Model (2/2)

Given the latent feature matrix Z and the weighting matrices Bd , theelements in X are independent:

p(X|Z,B1, . . . ,BD) =N∏

n=1

D∏d=1

p(xnd |zn·,Bd).

Graphical model:

XBd�B

� Z

d = 1, . . . , D

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Inference

To apply the Gibbs sampling algorithm, we need to integrate outBd , for which an approximation is required.

The posterior of B1, . . . ,BD factorizes as

Non Gauss︷︸︸︷p(B1, . . . ,BD |X,Z) =

D∏d=1

p(Bd |x·d ,Z) =D∏

d=1

Non Gauss︷︸︸︷p(x·d |Bd ,Z)

Gauss︷︸︸︷p(Bd)

p(x·d |Z).

We define the function:

ψ(βd) = log p(x·d |βd ,Z) + log p(βd),

where βd = Bd(:).

The likelihood term we want to compute is

p(x·d |Z) =

∫exp

(ψ(βd)

)dβd .

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Laplace Approximation [4]

Our objective is to approximate the integral

p(x·d |Z) =

∫exp

(ψ(βd)

)dβd .

Approximate ψ(βd) by its second-order Taylor series expansion ≡Approximate p(βd |X,Z) by a Gaussian distribution.

ψ(βd) is a strictly concave function of βd and therefore it has aunique maximum, which coincides with the maximum of p(βd |X,Z).

Then, p(βd |X,Z) = N (βdMAP ,−∇∇ψ|βd

MAP).

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Toy Example

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(a) (b)

(c) (d)

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

Iteration

Num

ber o

f Fea

ture

s (K

+)

(e)

0 20 40 60 80 100 120 140 160 180 200−4400

−4200

−4000

−3800

−3600

Iteration

log

p(X|

Z)

(f)

Figure 1: Experimental results of the infinite binary multinomial-logistic model over the image dataset. (a) The four base images used to generate the 200 observations. (b) Probability of each pixelbeing white, when a single feature is active (ordered to match the images on the left), computedusing Bd

MAP . (c) Four data points generated as described in the text. The numbers above eachfigure indicate which features are present in that image. (d) Probabilities of each pixel being whiteafter 200 iterations of the Gibbs sampler inferred for the four data points on (c). The numbers aboveeach figure show the inferred value of zn· for these data points. (e) The number of latent featuresK+ and (f) the log of p(X|Z) over the 200 iterations of the Gibbs sampler.

4.2 NESARC

The National Epidemiologic Survey on Alcohol and Related Conditions was designed to determinethe magnitude of alcohol use disorders and their associated disabilities. Two waves of interviewshave been fielded for this survey (first wave in 2001-2002 and second wave in 2004-2005). For thefollowing experimental results, we only use the data from the first wave, for which 43,093 peoplewere selected to represent the U.S. population 18 years of age and older. Public use data are currentlyavailable for this wave of data collection.

Through 2,991 entries, the NESARC collects data on the background of participants, alcohol andother drug consumption and abuse, medicine use, medical treatment, mental disorders, phobias,family history, etc. The survey includes a question about having attempted suicide as well as otherrelated questions such as ‘felt like wanted to die’ and ‘thought a lot about own death’. In the presentpaper, we use the IBP with discrete observations for a preliminary study in seeking the latent causeswhich lead to committing suicide. Most of the questions in the survey (over 2,500) are yes-or-noquestions, which have four possible outcomes: ‘blank’ (B), ‘unknown’ (U), ‘yes’ (Y) and ‘no’ (N).If a question is left blank the question was not asked1. If a question is said to be unknown either itwas not answered or was unknown to the respondent.

In our ongoing study, we want to find a latent model that describes this database and can be usedto infer patterns of behavior and, specifically, be able to predict suicide. In this paper, we buildan unsupervised model with the 20 variables that present the highest mutual information with thesuicide attempt question, which are shown in Table 1 together with their code in the questionnaire.

We run the Gibbs sampler over 500 randomly chosen subjects out of the 13,670 that have answeredaffirmatively to having had a period of low mood. In this study, we use another 9,500 as test casesand have left the remaining samples for further validation. We have initialized the sampler with an

1In a questionnaire of this size some questions are not asked when a previous question was answered in apredetermined way to reduce the burden of taking the survey. For example, if a person has never had a periodof low mood, the attempt suicide question is not asked.

6

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NESARC database (1/3)

The National Epidemiologic Survey on Alcohol and RelatedConditions was designed to determine the magnitude of alcohol usedisorders.

43, 093 subjects representing the U.S. population.

2, 991 questions about alcohol and other drug consumption andabuse, medicine use, medical treatment, mental disorders, phobias,family history, etc.

One question about having attempted suicide.

We select the 20 questions that present the highest mutualinformation with the ‘suicide attempt’ question.

Goal: Search for the latent variables related to the suicide attemptrisk.

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Figure: Probability of answering ‘blank’ (B), ‘unknown’ (U), ‘yes’ (Y) and ‘no’(N) to each of the 20 selected questions, when only one of the seven latentfeatures is active.

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Suicide attempt probability in the whole database: ∼ 8%.

Latent featuresSuicide attempt probability Number of cases

Train Test Train Test

1 - - - - - - 6.74% 5.55% 430 8072- 1 - - - - - 10.56% 11.16% 322 6083- - 1 - - - - 3.72% 4.60% 457 8632- - - 1 - - - 25.23% 22.25% 111 2355- - - - 1 - - 8.64% 9.69% 301 5782- - - - - 1 - 6.90% 7.18% 464 8928- - - - - - 1 14.29% 14.18% 91 1664

- - 0 0 - - - 30.77% 28.55% 26 571- - 0 1 - - - 82.35% 61.95% 17 297- - 1 0 - - - 0.83% 0.87% 363 6574- - 1 1 - - - 14.89% 16.52% 94 2058

- - 0 1 - - 1 100.00% 69.41% 4 850 - 0 1 - - - 80.00% 66.10% 5 1181 - 1 0 - 1 0 0.00% 0.25% 252 4739- - 1 0 - - 0 0.33% 0.63% 299 55431 - 1 0 - - - 0.32% 0.41% 317 5807

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Conclusions and Future Work

Conclusions

We have developed a likelihood model for the IBP that allowsdealing with discrete observations.

For that purpose, we need to resort to Laplace approximation.

We found that 7 latent features can model the factors that increasethe risk of attempting suicide.

Future work

Adding a constant term to the likelihood function, i.e.,p(xnd = r |zn·,Bd) ∝ exp (zn·bd·r + bd0r ).

An inference algorithm that allows working with more data points,e.g., variational inference or sequential Monte Carlo.

Prior that enforces sparsity in the weighting matrices Bd .

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Outline

1 Introduction


3 Observation Model

4 Inference

5 Experiments

6 Conclusions

7 References

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Summary of national strategy for suicide prevention: Goals and objectivesfor action, 2007.

Available at: http://www.sprc.org/library/nssp.pdf.

T. L. Griffiths and Z. Ghahramani.

The Indian Buffet Process: An introduction and review.

Journal of Machine Learning Research, 12:1185–1224, 2011.

F. J. R. Ruiz, I. Valera, C. Blanco, and F. Perez-Cruz.

Bayesian nonparametric modeling of suicide attempts.

Advances in Neural Information Processing Systems (NIPS), 2012.

C. K. I. Williams and D. Barber.

Bayesian classification with Gaussian Processes.

IEEE Transactions on Pattern Analysis and Machine Intelligence,20:1342–1351, 1998.

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bayesian nonparametric modeling of suicide attempts

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