Bayes Nets Meetup Sept 29th 2016 - Bayesian Network Modelling by Marco Scutari

Bayesian Network Modelling
with Examples in Genetics and Systems Biology
Marco Scutari
scutari@stats.ox.ac.uk
Department of Statistics
University of Oxford
September 29, 2016

What Are Bayesian Networks?
Marco Scutari University of Oxford

A Graph and a Probability Distribution
Bayesian networks (BNs) are deﬁned by:
ˆ a network structure, a directed acyclic graph G = (V, A), in which
each node vi ∈ V corresponds to a random variable Xi;
ˆ a global probability distribution X with parameters Θ, which can
be factorised into smaller local probability distributions according to
the arcs aij ∈ A present in the graph.
The main role of the network structure is to express the conditional
independence relationships among the variables in the model through
graphical separation, thus specifying the factorisation of the global
distribution:
P(X) =
p
i=1
P(Xi | ΠXi ; ΘXi ) where ΠXi = {parents of Xi}

Key Books to Reference
(Best perused as ebooks, the Koller & Friedman is ≈ 21/2 inches thick.)

How the DAG Maps to the Probability Distribution
C
A B
D
E
F
DAG
Graphical
separation
Probabilistic
independence
Formally, the DAG is an independence map of the probability
distribution of X, with graphical separation (⊥⊥G) implying probabilistic
independence (⊥⊥P ).

Graphical Separation in DAGs (Fundamental Connections)
separation (undirected graphs)
d-separation (directed acyclic graphs)
C
A B
C
A B
C
A B
C
A B

Graphical Separation in DAGs (General Case)
Now, in the general case we can extend the patterns from the
fundamental connections and apply them to every possible path between
A and B for a given C; this is how d-separation is deﬁned.
If A, B and C are three disjoint subsets of nodes in a directed
acyclic graph G, then C is said to d-separate A from B,
denoted A ⊥⊥G B | C, if along every path between a node in
A and a node in B there is a node v satisfying one of the
following two conditions:
1. v has converging edges (i.e. there are two edges pointing
to v from the adjacent nodes in the path) and none of v
or its descendants (i.e. the nodes that can be reached
from v) are in C.
2. v is in C and does not have converging edges.
This deﬁnition clearly does not provide a computationally feasible
approach to assess d-separation; but there are other ways.

A Simple Algorithm to Check D-Separation (I)
C
A B
D
E
F
C
A B
D
E
F
Say we want to check whether A and E are d-separated by B. First, we
can drop all the nodes that are not ancestors (i.e. parents, parents’
parents, etc.) of A, E and B since each node only depends on its
parents.

A Simple Algorithm to Check D-Separation (II)
C
A B
E
C
A B
E
Transform the subgraph into its moral graph by
1. connecting all nodes that have one parent in common; and
2. removing all arc directions to obtain an undirected graph.
This transformation has the double eﬀect of making the dependence
between parents explicit by “marrying” them and of allowing us to use
the classic deﬁnition of graphical separation.

A Simple Algorithm to Check D-Separation (III)
C
A B
E
Finally, we can just perform e.g. a depth-first or breadth-first search and
see if we can find an open path between A and B, that is, a path that is
not blocked by C.

Completely D-Separating: Markov Blankets
Parents Children
Children's other parents
(Spouses)
Markov blanket of A
A
FI
H E
D
C
B
G
We can easily use the DAG to solve
the feature selection problem. The
set of nodes that graphically
isolates a target node from the rest
of the DAG is called its Markov
blanket and includes:
ˆ its parents;
ˆ its children;
ˆ other nodes sharing a child.
Since ⊥⊥G implies ⊥⊥P , we can
restrict ourselves to the Markov
blanket to perform any kind of
inference on the target node, and
disregard the rest.

Different DAGs, Same Distribution: Topological Ordering
A DAG uniquely identifies a factorisation of P(X); the converse is not
true. Consider again the DAG on the left:
P(X) = P(A) P(B) P(C | A, B) P(D | C) P(E | C) P(F | D).
We can rearrange the dependencies using Bayes theorem to obtain:
P(X) = P(A | B, C) P(B | C) P(C | D) P(D | F) P(E | C) P(F),
which gives the DAG on the right, with a different topological ordering.
C
A B
D
E
F
C
A B
D
E
F

Different DAGs, Same Distribution: Equivalence Classes
On a smaller scale, even keeping the same underlying undirected graph
we can reverse a number of arcs without changing the dependence
structure of X. Since the triplets A → B → C and A ← B → C are
probabilistically equivalent, we can reverse the directions of their arcs as
we like as long as we do not create any new v-structure (A → B ← C,
with no arc between A and C).
This means that we can group DAGs into equivalence classes that are
uniquely identified by the underlying undirected graph and the
v-structures. The directions of other arcs can be either:
ˆ uniquely identifiable because one of the directions would introduce
cycles or new v-structures in the graph (compelled arcs);
ˆ completely undetermined.

Completed Partially Directed Acyclic Graphs (CPDAGs)
C
A B
D
E
F
C
A B
D
E
F
C
A B
D
E
F
D
E
F
C
BA
DAG CPDAG

What About the Probability Distributions?
The second component of a BN is the probability distribution P(X).
The choice should such that the BN:
ˆ can be learned eﬃciently from data;
ˆ is ﬂexible (distributional assumptions should not be too strict);
ˆ is easy to query to perform inference.
The three most common choices in the literature (by far), are:
ˆ discrete BNs (DBNs), in which X and the Xi | ΠXi are
multinomial;
ˆ Gaussian BNs (GBNs), in which X is multivariate normal and the
Xi | ΠXi are univariate normal;
ˆ conditional linear Gaussian BNs (CLGBNs), in which X is a
mixture of multivariate normals and the Xi | ΠXi are either
multinomial, univariate normal or mixtures of normals.
It has been proved in the literature that exact inference is possible in
these three cases, hence their popularity.

Discrete Bayesian Networks
visit to Asia? smoking?
tuberculosis? lung cancer? bronchitis?
either tuberculosis
or lung cancer?
positive X-ray?
dyspnoea?
A classic example of DBN is
the ASIA network from
Lauritzen & Spiegelhalter
(1988), which includes a
collection of binary variables.
It describes a simple
diagnostic problem for
tuberculosis and lung cancer.
Total parameters of X :
28 − 1 = 255

Conditional Probability Tables (CPTs)
visit to Asia?
tuberculosis?
smoking?
lung cancer?
smoking?
bronchitis?
tuberculosis? lung cancer?
either tuberculosis
or lung cancer?
either tuberculosis
or lung cancer?
positive X-ray?
bronchitis?either tuberculosis
or lung cancer?
dyspnoea?
visit to Asia? smoking?
The local distributions
Xi | ΠXi take the form
of conditional
probability tables for
each node given all the
conﬁgurations of the
values of its parents.
Overall parameters of
the Xi | ΠXi : 18

Gaussian Bayesian Networks
mechanics analysis
vectors statistics
algebra
A classic example of GBN is
the MARKS networks from
Mardia, Kent & Bibby JM
(1979), which describes the
relationships between the
marks on 5 math-related
topics.
Assuming X ∼ N(µ, Σ), we can compute Ω = Σ−1. Then Ωij = 0
implies Xi ⊥⊥P Xj | X {X,Xj}. The absence of an arc Xi → Xj in
the DAG implies Xi ⊥⊥G Xj | X {X,Xj}, which in turn implies
Xi ⊥⊥P Xj | X {X,Xj}.
Total parameters of X : 5 + 15 = 20

Partial Correlations and Linear Regressions
The local distributions Xi | ΠXi take the form of linear regression
models with the ΠXi acting as regressors and with independent error
terms.
ALG = 50.60 + εALG ∼ N(0, 112.8)
ANL = −3.57 + 0.99ALG + εANL ∼ N(0, 110.25)
MECH = −12.36 + 0.54ALG + 0.46VECT + εMECH ∼ N(0, 195.2)
STAT = −11.19 + 0.76ALG + 0.31ANL + εSTAT ∼ N(0, 158.8)
VECT = 12.41 + 0.75ALG + εVECT ∼ N(0, 109.8)
(That is because Ωij ∝ βj for Xi, so βj > 0 if and only if Ωij > 0. Also
Ωij ∝ ρij, the partial correlation between Xi and Xj, so we are
implicitly assuming all probabilistic dependencies are linear.)
Overall parameters of the Xi | ΠXi : 11 + 5 = 16

Conditional Linear Gaussian Bayesian Networks
CLGBNs contain both discrete and continuous nodes, and combine
DBNs and GBNs as follows to obtain a mixture-of-Gaussians network:
ˆ continuous nodes cannot be parents of discrete nodes;
ˆ the local distribution of each discrete node is a CPT;
ˆ the local distribution of each continuous node is a set of linear
regression models, one for each conﬁgurations of the discrete
parents, with the continuous parents acting as regressors.
sexdrug
weight loss
(week 1)
weight loss
(week 2)
One of the classic examples is
the RATS’ WEIGHTS network
from Edwards (1995), which
describes weight loss in a drug
trial performed on rats.

Mixtures of Linear Regressions
The resulting local distribution for the ﬁrst weight loss for drugs D1, D2
and D3 is:
W1,D1 = 7 + εD1 ∼ N(0, 2.5)
W1,D2 = 7.50 + εD2 ∼ N(0, 2)
W1,D3 = 14.75 + εD3 ∼ N(0, 11)
with just the intercepts since the node has no continuous parents. The
local distribution for the second loss is:
W2,D1 = 1.02 + 0.89βW1 + εD1 ∼ N(0, 3.2)
W2,D2 = −1.68 + 1.35βW1 + εD2 ∼ N(0, 4)
W2,D3 = −1.83 + 0.82βW1 + εD3 ∼ N(0, 1.9)
Overall, they look like random eﬀect models with random intercepts and
random slopes.

Case Study:
A Protein Signalling Network

Case Study: A Protein Signalling Network
Source and Overview of the Data
Causal Protein-Signalling Networks Derived from
Multiparameter Single Cell Data
Karen Sachs, et al., Science, 308, 523 (2005);
DOI: 10.1126/science.1105809
That’s a landmark paper in applying BNs because it highlights the use of
interventional data; and because results are validated using existing literature.
The data consist in the 5400 simultaneous measurements of 11 phosphorylated
proteins and phospholypids:
ˆ 1800 data subject only to general stimolatory cues, so that the protein
signalling paths are active;
ˆ 600 data with with speciﬁc stimolatory/inhibitory cues for each of the
following 4 proteins: Mek, PIP2, Akt, PKA;
ˆ 1200 data with speciﬁc cues for PKA.
The goal of the analysis is to learn what relationships link these 11 proteins,
that is, the signalling pathways they are part of.

Analysis and Validated Network
Akt
Erk
Jnk
Mek
P38
PIP2
PIP3
PKA
PKC
Plcg
Raf
1. Outliers were removed and the data
were discretised, since it was
impossible to model them with a
GBN.
2. A large number of DAGs were
learned and averaged to produce a
more robust model. The averaged
DAG was created using the arcs
present in at least 85% of the DAGs.
3. The validity of the averaged BN was
evaluated against established
signalling pathways from literature.

The Hill-Climbing Algorithm
The most common score-based structure learning algorithm, in which
we are looking for the DAG that maximises a score such as the posterior
P(G | D) or BIC, is a greedy search such as hill-climbing:
1. Choose an initial DAG G, usually (but not necessarily) empty.
2. Compute the score of G, denoted as ScoreG = Score(G).
3. Set maxscore = ScoreG.
4. Repeat the following steps as long as maxscore increases:
4.1 for every possible arc addition, deletion or reversal not introducing
cycles:
4.1.1 compute the score of the modiﬁed DAG G∗
, ScoreG∗ = Score(G∗
):
4.1.2 if ScoreG∗ > ScoreG, set G = G∗
and ScoreG = ScoreG∗ .
4.2 update maxscore with the new value of ScoreG.
5. Return the DAG G.
Only one local distribution changes in each step, which makes algorithm
computationally eﬃcient and easy to speed up with caching.

Learning Multiple DAGs from the Data
Searching from diﬀerent starting points increases our coverage of the
space of the possible DAGs; the frequency with which an arc appears is
a measure of the strength of the dependence.

Model Averaging for DAGs0.00.20.40.60.81.0
arc strength
ECDF(arcstrength)
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0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
Arcs with signiﬁcant strength can be identiﬁed using a threshold estimated
from the data by minimising the distance from the observed ECDF and the
ideal, asymptotic one (the blue area in the right panel).

Combining Observational and Interventional Data
model without interventions
Akt
Erk
Jnk
Mek
P38
PIP2
PIP3
PKA
PKC
Plcg
Raf
model with interventions
Akt
Erk
Jnk
Mek
P38
PIP2
PIP3
PKA
PKC
Plcg
Raf
Observations must be scored taking into account the eﬀects of the
interventions, which break biological pathways; the overall network score is a
mixture of scores adjusted for each experiment.

Using The Protein Network to Plan Experiments
This idea goes by the name of hypothesis generation: using a statistical
model to decide which follow-up experiments to perform. BNs are
especially easy to use for this because they automate the computation
of arbitrary events.
P(Akt)
probability
Akt
LOW
AVG
HIGH
0.0 0.2 0.4 0.6
without intervention
with intervention
P(PKA)
probability
PKA
LOW
AVG
HIGH
0.2 0.4 0.6
without intervention
with intervention

Conditional Probability Queries
DBNs, GBNs, CLGBNs all have exact inference algorithms to compute
conditional probabilities for an event given some evidence. However,
approximate inference scales better and the same algorithms can be
used for all kinds of BNs. An example is likelihood weighting below.
Input: a BN B = (G, Θ), an query Q = q and a set of evidence E
Output: an approximate estimate of P(Q | E, G, Θ).
1. Order the Xi according to the topological ordering in G.
2. Set wE = 0 and wE,q = 0.
3. For a suitably large number of samples x = (x1, . . . , xp):
3.1 generate x(i), i = 1, . . . , p from X(i) | ΠX(i)
using the values
e1, . . . , ek speciﬁed by the hard evidence E for Xi1
, . . . , Xik
.
3.2 compute the weight wx = P(Xi∗ = e∗ | ΠXi∗ )
3.3 set wE = wE + wx;
3.4 if x includes Q = q , set wE,q = wE,q + wx.
4. Estimate P(Q | E, G, Θ) with wE,q/wE.

Case Study:
Genome-Wide Predictions

Case Study: Genome-Wide Predictions
MAGIC Populations: Wheat and Rice
Sequence data (e.g. SNP markers) is routinely used in statistical genetics to
understand the genetic basis of human diseases, and to breed traits of
commercial interest in plants and animals. Multiparent (MAGIC) populations
are ideal for the latter. Here we consider two:
ˆ A winter wheat population: 721 varieties, 16K markers, 7 traits.
ˆ An indica rice population: 1087 varieties, 4K markers, 10 traits.
Phenotypic traits include ﬂowering time, height, yield, a number of disease
scores; and physical and quality traits for grains in rice.
The goal of the analysis is to ﬁnd key markers controlling the traits; the causal
relationships between them; keep a good predictive accuracy.
Multiple Quantitative Trait Analysis Using Bayesian
Networks
Marco Scutari, et al., Genetics, 198, 129–137 (2014);
DOI: 10.1534/genetics.114.165704

Bayesian Networks for Selection and Association Studies
If we have a set of traits and markers for each variety, all we need are
the Markov blankets of the traits; most markers are discarded in the
process. Using common sense, we can make some assumptions:
ˆ traits can depend on markers, but not vice versa;
ˆ dependencies between traits should follow the order of the
respective measurements (e.g. longitudinal traits, traits measured
before and after harvest, etc.);
ˆ dependencies in multiple kinds of genetic data (e.g. SNP + gene
expression or SNPs + methylation) should follow the central
dogma of molecular biology.
Assumptions on the direction of the dependencies allow to reduce
Markov blankets learning to learning the parents and the children of
each trait, which is a much simpler task.

Parametric Assumptions
In the spirit of classic additive genetics models, we use a Gaussian BN.
Then the local distribution of each trait Ti is a linear regression model
Ti = µTi
+ ΠTi
βTi
+ εTi
= µTi
+ TjβTj
+ . . . + TkβTk
traits
+ GlβGl
+ . . . + GmβGm
markers
+εTi
and the local distribution of each marker Gi is likewise
Gi = µGi
+ ΠGi
βGi
+ εGi
=
= µGi
+ GlβGl
+ . . . + GmβGm
markers
+εGi
in which the regressors (ΠTi or ΠGi ) are treated as fixed effects. ΠTi
can be interpreted as causal effects for the traits, ΠGi as markers being
in linkage disequilibrium with each other.

Learning the Bayesian Network (I)
1. Feature Selection.
1.1 Independently learn the parents and the children of each trait with
the SI-HITON-PC algorithm; children can only be other traits,
parents are mostly markers, spouses can be either. Both are selected
using the exact Student’s t test for partial correlations.
1.2 Drop all the markers that are not parents of any trait.
Parents and children of T1 Parents and children of T2 Parents and children of T3 Parents and children of T4
Redundant markers that are not in the
Markov blanket of any trait

Constraint-Based Structure Learning Algorithms
C
A B
D
E
F
CPDAG
Graphical
separation
Conditional
independence tests
The mapping between edges and conditional independence relationships
lies at the core of BNs; therefore, one way to learn the structure of a
BN is to check which such relationships hold using a suitable conditional
independence test. Such an approach results in a set of conditional
independence constraints that identify a single equivalence class.

The Semi-Interleaved HITON-PC Algorithm
Input: each trait Ti in turn, other traits (Tj) and all markers (Gl), a
signiﬁcance threshold α.
Output: the set CPC parents and children of Ti in the BN.
1. Perform a marginal independence test between Ti and each Tj (Ti ⊥⊥ Tj)
and Gl (Ti ⊥⊥ Gl) in turn.
2. Discard all Tj and Gl whose p-values are greater than α.
3. Set CPC = {∅}.
4. For each the Tj and Gl in order of increasing p-value:
4.1 Perform a conditional independence test between Ti and Tj/Gl
conditional on all possible subsets Z of the current CPC
(Ti ⊥⊥ Tj | Z ⊆ CPC or Ti ⊥⊥ Gl | Z ⊆ CPC).
4.2 If the p-value is smaller than α for all subsets then
CPC = CPC ∪ {Tj} or CPC = CPC ∪ {Gl}.
NOTE: the algorithm is deﬁned for a generic independence test, you can plug
in any test that is appropriate for the data.

Learning the Bayesian Network (II)
2. Structure Learning. Learn the structure of the network from the nodes
selected in the previous step, setting the directions of the arcs according
to the assumptions above. The optimal structure can be identiﬁed with a
suitable goodness-of-ﬁt criterion such as BIC. This follows the spirit of
other hybrid approaches that have shown to be well-performing in
literature.
Empty network Learned network

Learning the Bayesian Network (III)
3. Parameter Learning. Learn the parameters: each local distribution is a
linear regression and the global distribution is a hierarchical linear model.
Typically least squares works well because SI-HITON-PC selects sets of
weakly correlated parents; ridge regression can be used otherwise.
Learned network Local distributions

Model Averaging and Assessing Predictive Accuracy
We perform all the above in 10 runs of 10-fold cross-validation to
ˆ assess predictive accuracy with e.g. predictive correlation;
ˆ obtain a set of DAGs to produce an averaged, de-noised consensus DAG
with model averaging.

WHEAT: a Bayesian Network (44 nodes, 66 arcs)
YR.GLASS
HT
YR.FIELDMIL
FT
G418
G311
G1217
G800
G866
G795
G2570
G260
G2920
G832
G1896
G2953
G266
G847
G942
G200
G257
G2208
G1373
G599
G261
G383
G1853
G1033
G1945
G1338
G1276
G1263
G1789
G2318
G1294
G1800
YLD
FUS
G1750
G524
G775
G2835
G43
PHYSICAL TRAITS
OF THE PLANT
DISEASES

RICE: a Bayesian Network (64 nodes, 102 arcs)
HT
FT
AMY
GL
GW
GTEMP
G4432
G2670
G4744
G3092
G3486
G1533
G2639
G3823
G1378
G3105
G3102
G3098
G317
G5167
G1778
G3872
G4529
G1440
G5794
G4668
G2764
G457
G3862
G3964
G4109
G2089
G3219
G3209
G3222
G3212
G4573
G1311
G2949
G6003
G6010 G6123
G2815
G3049
YLD
CHALK
BROWN
SUB
G1888
G1958
G4156
G4382
G3136
G4145
G3106
G1266
G3927
G5997
G4553
G2179
G5006
G3992
G678
G3925
PHYSICAL TRAITS
OF THE PLANT
DISEASES
ABIOTIC
STRESS
PHYSICAL AND QUALITY
TRAITS OF THE GRAINS

Predicting Traits for New Individuals
We can predict the traits:
1. from the averaged consensus
network;
2. from each of the 10 × 10 networks
we learn during cross-validation,
and average the predictions for
each new individual and trait.
Option 2. almost always provides better
accuracy than option 1., especially for
polygenic traits; 10 × 10 networks can
cover the genome much better, and we
have to learn them anyway.
So: averaged network for interpretation,
ensemble of networks for predictions.
cross−validatedcorrelation
0.2
0.4
0.6
0.8
YR.GLASS YLD HT YR.FIELD FUS MIL FT
AVERAGED NETWORK(α = 0.05, ρC)
AVERAGED PREDICTIONS(α = 0.05, ρC)
AVERAGED NETWORK(α = 0.05, ρG)
AVERAGED PREDICTIONS(α = 0.05, ρG)
cross−validatedcorrelation
0.2
0.4
0.6
0.8
YLD HT FT AMY GL GW CHALKGTEMP SUB BROWN
AVERAGED NETWORK(α = 0.05, ρC)
AVERAGED PREDICTIONS(α = 0.05, ρC)
AVERAGED NETWORK(α = 0.05, ρG)
AVERAGED PREDICTIONS(α = 0.05, ρG)

Causal Relationships Between Traits
One of the key properties of BNs is their ability of
capturing the direction of the causal relationships
among traits in the absence of latent confounders
(the experimental design behind the data collection
should take care of a number of them).
It works because each trait will have at least one
incoming arc from the markers, say Gl → Tj, and
then (Gl →) Tj ← Tk and (Gl →) Tj → Tk are not
probabilistically equivalent. So the network can
ˆ suggest the direction of novel relationships;
ˆ conﬁrm the direction of known relationships,
troubleshooting the experimental design and
data collection.
HT
YR.FIELD
FT
G418
1217
0
G260
G2920
832
896
G2953
G266
G847
G942
G257
G2208
G
G1338
G
G1294
G1800
YLD
G2835
YR.GLASS
HT
YRMIL
G418
G1217
0
G2570
G832
G1896
G2953
G9
G257
G2208
G1373G1945
G1338
G1800
YLD
FUS
G1750
G2835
(WHEAT)
(WHEAT)

Spotting Confounding Effects
HT
G2570
G832
G1896
G2953
YLD
FUS
G2835
(WHEAT) Traits can interact in complex ways that may not
be obvious when they are studied individually,
but that can be explained by considering
neighbouring variables in the network.
An example: in the WHEAT data, the difference
in the mean YLD between the bottom and top
quartiles of the FUS disease scores is +0.08.
So apparently FUS is associated with increased YLD! What we are actually
measuring is the confounding effect of HT (FUS ← HT → YLD); conditional
on each quartile of HT, FUS has a negative effect on YLD ranging from -0.04
to -0.06. This is reassuring since it is known that susceptibility to fusarium is
positively related to HT, which in turn affects YLD.

Disentangling Pleiotropic Effects
When a marker is shown to be associated to
multiple traits, we should separate its direct
and indirect effects on each of the traits.
(Especially if the traits themselves are linked!)
Take for example G1533 in the RICE data set:
it is putative causal for YLD, HT and FT.
HT
FT
G4432
G1533
G4109
YLD
(RICE)
ˆ The difference in mean between the two homozygotes is +4.5cm in HT, +2.28 weeks in
FT and +0.28 t/ha in YLD.
ˆ Controlling for YLD and FT, the difference for HT halves (+2.1cm);
ˆ Controlling for YLD and HT, the difference for FT is about the same (+2.3 weeks);
ˆ Controlling for HT and FT the difference for YLD halves (+0.16 t/ha).
So, the model suggests the marker is causal for FT and that the effect on the
other traits is partly indirect. This agrees from the p-values from an
independent association study (FT: 5.87e-28 < YLD: 4.18e-10, HT:1e-11).

Identifying Causal (Sets of) Markers
Compared to additive regression models, BNs make it trivial to compute:
ˆ posterior probability of association for a marker and a trait after all the
other markers and traits are taken into account to rule out linkage
disequilibrium, confounding, pleiotropy, etc.;
ˆ expected allele counts nLOW and nHIGH for a marker and low/high values
of a set of traits (nLOW − nHIGH should be large if the marker tags a
causal variant and thus should show which allele is favourable).
G1778 G3872 G4529 G1440 G5794
small grains 0.00 0.78 0.29 0.16 0.74
large grains 2.00 0.47 0.63 0.35 0.12
G4668 G2764 G3927 G3992 G4432
small grains 0.24 0.29 0.18 0.09 0.00
large grains 0.17 0.00 0.62 0.29 0.82
small grains = bottom 10% GL, bottom 10% GW in the RICE data.
large grains = top 10% GL, top 10% GW.

Conclusions

Conclusions
Conclusions and Remarks
ˆ BNs provide an intuitive representation of the relationships linking
heterogeneous sets of variables, which we can use for qualitative and
causal reasoning.
ˆ We can learn BNs from data while including prior knowledge as
needed to improve the quality of the model.
ˆ BNs subsume a large number of probabilistic models and thus can
readily incorporate other techniques from statistics and computer
science (via information theory).
ˆ For most tasks we can start just reusing state-of-the-art, general
purpose algorithms.
ˆ Once learned, BNs provide a ﬂexible tool for inference, while
maintaining good predictive power.
ˆ Markov blankets are a valuable tool for feature selection, and make it
possible to learn just part of the BN depending on the goals of the
analysis.

That’s all, Thanks!

Bayes Nets Meetup Sept 29th 2016 - Bayesian Network Modelling by Marco Scutari

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Bayes Nets Meetup Sept 29th 2016 - Bayesian Network Modelling by Marco Scutari