This document discusses applications of Bayesian nonparametric methods to various domains including toxicology, ecology, marketing, human fertility, and more. It provides examples of using rounded Gaussian mixtures and Dirichlet process mixtures to model count data from developmental toxicity studies and animal abundance data. Applications to modeling multivariate mobile phone usage data and basal body temperature curves are also described. The document emphasizes that Bayesian nonparametric approaches allow inclusion of prior information and flexible modeling of complex data structures.

1. Bayesian Nonparametrics
Applications to biology, ecology, and marketing
Antonio Canale
Universit`a di Torino &
Collegio Carlo Alberto
StaTalk
19 February, 2016

2. Outline
Applications to
1 Toxicology
2 Ecology
3 Marketing
4 Human fertility
5 More applications

3. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Developmental toxicity studies
• Developmental toxicity is any alteration which interferes with
normal growth caused by environmental factors
• environmental factors include drugs, lifestyle factors such as
alcohol, smoke, and environmental toxic chemicals or physical
factors
• typical settings involve animals experiments

4. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Ethylene glycol
• Ethylene glycol is used in many industrial processes as e.g. an
antifreeze, an industrial humectant, a solvent in paint and plastic
industry.
• we consider data from a developmental toxicity study of ethylene
glycol in mice conducted by the National Toxicology Program
(Price et al. 1985)
• Pregnant mice were assigned to dose groups of 0, 750, 1500, or
3000 mg/kg/day, with the number of implants measured for each
mouse at the end of the experiment.
• The scientific interest lies in studying a dose-response trend in the
distribution of the number of implants

5. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Ethylene glycol data (control group, mean 13.32,
variance 4.89)
5 10 15 20
0123456
frequencies

6. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Let’s go nonparametric!
Clearly we cannot try to estimate the pmf of the number of implants
with
yi ∼ Pois(λ)
λ ∼ Ga(a, b)
since the sampling model is too restrictive.
Hence we have a good reason to be nonparametric

7. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Simple approach
• A draw form the DP process produce an almost sure discrete
distribution.
• We may think to assume
yi ∼ P
P ∼ DP(α, P0)

8. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Simple approach
• The posterior is in closed form, i.e.
(P | yn
) ∼ DP (α + n) αP0 +
i
δyi ,
• which is actually quite unappealing in not allowing borrowing of
information about local deviations from P0.

9. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Simple approach
5 10 15 20
0.000.050.100.150.20
pmf

10. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Mixture of Poisson
• An alternative is
Pr(Y = j) = Poi(j; λ)dP(λ), P ∼ DP(αP0),
• DPM of Poisson seems extremely flexible and to provide a natural
modification of the DPM of Gaussians;
• the resulting prior on the count distribution is actually quite
inflexible;
• distributions that are under-dispersed cannot be approximated;

11. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Round a continous distribution
• Take a continuous density
• Define a0 = 0, a1 = 1, . . .
• Calculate p(j) =
aj+1
aj
f (x)dx
• Obtain the discrete count
distribution
0 1 2 3 4 5
0.00.20.40.6
x
f(x)

12. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Round a continous distribution
• Take a continuous density
• Define a0 = 0, a1 = 1, . . .
• Calculate p(j) =
aj+1
aj
f (x)dx
• Obtain the discrete count
distribution
0 1 2 3 4 5
0.00.20.40.6
x
f(x)

13. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Round a continous distribution
• Take a continuous density
• Define a0 = 0, a1 = 1, . . .
• Calculate p(j) =
aj+1
aj
f (x)dx
• Obtain the discrete count
distribution
0 1 2 3 4 5
0.00.20.40.6
x
f(x)

14. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Round a continous distribution
• Take a continuous density
• Define a0 = 0, a1 = 1, . . .
• Calculate p(j) =
aj+1
aj
f (x)dx
• Obtain the discrete count
distribution
0 1 2 3 4
0.00.10.20.30.40.50.6
y
p(y)

15. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Rounded Gaussian Mixture (Canale and Dunson,
2011)
p(·; P) = RG(·; µ, τ−1
)dP(µ, τ−1
),
P ∼ DP(αP0),

16. Toxicology Ecology Marketing Human fertility More applications
Developmental toxicity studies
Rounded Gaussian Mixture
5 10 15 20
0.000.050.100.150.20
estimated pmf (blue) and empirical pmf (black)
pmf
0.0 0.2 0.4 0.6 0.8 1.0
−6−4−202
quantile
changein#implants

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Animal abundance

18. Toxicology Ecology Marketing Human fertility More applications
Animal abundance
0 50 100 150 200 250
012345
animal abundance

19. Toxicology Ecology Marketing Human fertility More applications
Another reason to avoid Poisson mixtures
• We compare
p(·; P) = RG(·; µ, τ−1
)dP(µ, τ−1
),
p(·; P) = Poi(·; λ)dP(λ)
• the DP is highly sensitive to the prior specifications of α which has
a major impact in the total number of clusters
• a more general NP prior can lead to more accurate estimates,
especially for the number of mixture components. (Ishwaran and
James, 2001 and Lijoi et al. 2005, 2007)
P ∼ PY (θ, σ, P0)

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Improving rounded mixtures (Canale and Pr¨unster,
2016)
q
q
q
q
1020304050
σ
E(Kn|−)
0.00 0.25 0.50 0.75
q
q
q
q
q q q
q
q q q q
q
q
q
q
1020304050
σ
E(Kn|−)
0.00 0.25 0.50 0.75
q
q
q
qq q q
q
q
q
q
q
Figure: Posterior mean number of distinct clusters E[Kn|−] for the Okaloosa
darters dataset: Poisson mixture and RG mixture for different
σ = 0, 0.25, 0.5, 0.75 and prior expected number of components E(Kn).

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Marketing application
• we focus on data from 2, 050 SIM cards from customers having a
prepayed contract in a single period;
• yi = (yi1, . . . , yi5) with the number of outgoing calls to fixed
numbers (yi1), to mobile numbers of competing operators (yi2)
and to mobile numbers of the same operator (yi3), the total
number of MMS (yi4) and SMS (yi5) sent;

22. Toxicology Ecology Marketing Human fertility More applications
• the RK method can be adapted in the multivariate context
• it is able to characterize the entire joint distribution;
• the use of underlying Gaussian mixtures allows the joint modeling
of variables on different measurement scales (continuous,
categorical, binary and counts). See also Canale and Dunson
(2015)
• we can do inference on different objects: the whole multivariate
density, the marginals, the conditionals.
• there are not so many alternatives to model a multivariate count
distribution!

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Each concepts of before can be generalized into its multivariate
counterpart.
Pr(y = J) = RKp(J; Θ)dP(Θ),
P ∼ DP(αP0)
with J ∈ Np and
RK(J; Θ) =
AJ
K(y∗
; Θ)dy∗
where AJ = {y∗ : a1,J1 ≤ y∗
1 < a1,J1+1, . . . , ap,Jp ≤ y∗
p < ap,Jp+1}
defines a disjoint partition of the sample space.

24. Toxicology Ecology Marketing Human fertility More applications
Marketing application
• we focused on the forecast of yi1, using data on yi2, . . . , yi5
• we split the dataset in a training and test subset;
• the approach is compared with prediction under a generalized
additive model (GAM) with spline smoothing function;
• Smaller out-of-sample MAD (8.08 vs 8.76)
• side prediction automatically accomodate - e.g. pr(y1 = 0) or
pr(y1 > T)

25. Toxicology Ecology Marketing Human fertility More applications
Human reproductive functioning
• we focus now on female reproductive functioning
• data refer to the basal body temperature (bbt), across the
menstrual cycle.
• bbt curves follow a characteristic trajectory: during the follicular
phase of the cycle leading up to ovulation, the bbt values tend to
be low, while after ovulation bbt rises progressively before dropping
prior to the next cycle.

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bbt curves

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bbt curves model
• we model the data assuming
fij (t) = ηij (t) + ijt,
where fij (t) is the cycle j of woman i at day t and η is the
underling bbt curve. The curve is observed with random noise ijt.

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bbt curves: abnormal cycles

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bbt curves mixture model
• we use the mixture model
p(ηij ) = P, P =
∞
h=1
πhηh,
with a stick-breaking prior on the weights π and a suitable base
measures ηh ∼ P0 (note that the atoms, here are curves)
• but the regular shape of a healthy woman is a well known fact a
priori
• we are Bayesians, we can include this prior information!

30. Toxicology Ecology Marketing Human fertility More applications
bbt curves mixture model
• we use the mixture model
p(ηij ) = P, P =
∞
h=1
πhηh,
with a stick-breaking prior on the weights π and a suitable base
measures ηh ∼ P0 (note that the atoms, here are curves)
• but the regular shape of a healthy woman is a well known fact a
priori
• we are Bayesians, we can include this prior information!

31. Toxicology Ecology Marketing Human fertility More applications
bbt curves mixture model
• we use the mixture model
p(ηij ) = P, P =
∞
h=1
πhηh,
with a stick-breaking prior on the weights π and a suitable base
measures ηh ∼ P0 (note that the atoms, here are curves)
• but the regular shape of a healthy woman is a well known fact a
priori
• we are Bayesians, we can include this prior information!

32. Toxicology Ecology Marketing Human fertility More applications
Atomic base measure
• it is sufficient to assume that
P = wδη0 + (1 − w)
∞
h=1
πhηh,
with η0 representing the S-shape trajectory known a priori.
• there are technical challenges in assuming an atomic base measure
that we are trying to solve (Canale, Nipoti, Lijoi and Pruenster,
20??)

33. Toxicology Ecology Marketing Human fertility More applications
Atomic base measure
• it is sufficient to assume that
P = wδη0 + (1 − w)
∞
h=1
πhηh,
with η0 representing the S-shape trajectory known a priori.
• there are technical challenges in assuming an atomic base measure
that we are trying to solve (Canale, Nipoti, Lijoi and Pruenster,
20??)

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Image reconstruction
(Wang, Canale, and Dunson 2016)

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Brain-network data analysis
(Durante, Canale, and Dunson 201?)

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Demand-supply model
(Canale and Ruggiero 2016)

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To conclude
• BNP provides a set of useful tools for challenging applications
• the Bayesian approach allow us to include prior informations
• the nonparametric approach let the data speak
• today there are thousands of situations generating interesting data
with complex structures
• “The best thing about being a statistician is that you get to play in
everyone else’s backyard.” (John Tukey)

38. Toxicology Ecology Marketing Human fertility More applications
To conclude
• BNP provides a set of useful tools for challenging applications
• the Bayesian approach allow us to include prior informations
• the nonparametric approach let the data speak
• today there are thousands of situations generating interesting data
with complex structures
• “The best thing about being a statistician is that you get to play in
everyone else’s backyard.” (John Tukey)

39. Toxicology Ecology Marketing Human fertility More applications
To conclude
• BNP provides a set of useful tools for challenging applications
• the Bayesian approach allow us to include prior informations
• the nonparametric approach let the data speak
• today there are thousands of situations generating interesting data
with complex structures
• “The best thing about being a statistician is that you get to play in
everyone else’s backyard.” (John Tukey)

40. Toxicology Ecology Marketing Human fertility More applications
To conclude
• BNP provides a set of useful tools for challenging applications
• the Bayesian approach allow us to include prior informations
• the nonparametric approach let the data speak
• today there are thousands of situations generating interesting data
with complex structures
• “The best thing about being a statistician is that you get to play in
everyone else’s backyard.” (John Tukey)

41. Toxicology Ecology Marketing Human fertility More applications
To conclude
• BNP provides a set of useful tools for challenging applications
• the Bayesian approach allow us to include prior informations
• the nonparametric approach let the data speak
• today there are thousands of situations generating interesting data
with complex structures
• “The best thing about being a statistician is that you get to play in
everyone else’s backyard.” (John Tukey)

42. Toxicology Ecology Marketing Human fertility More applications
That’s all folks!
Thanks for your attention!