Research Centre for Vocational Education

1
Research Centre for Vocational Education
Bayesian Modeling in Educational Research
Petri Nokelainen petri.nokelainen_at_uta.fi
v2.2 This lecture is available at
http//www.uta.fi/aktkk/lectures/bayes_en
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Outline

• Research Overview
• Introduction to Bayesian Modeling
• Investigating Non-linearities with Bayesian
  Networks
• Bayesian Classification Modeling
• Bayesian Dependency Modeling
• Bayesian Unsupervised Model-based Visualization

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Research Overview
VOCATIONAL EDUCATION
COMPUTER SCIENCE
Professional growth and learning
Applied research
Pekka Ruohotie Petri Nokelainen
Online pedagogy
Authentic use of educational technology
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Research Overview
RCVE Data Server - On-line Questionnaire
BayMiner - Bayesian Supervised and Unsupervised
Visualization
2008
1997
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Research Overview
1. BAYESIAN MODELING -gt Linear vs. non-linear
dependencies -gt Dependency and
classification modeling
(B-COURSE) -gt Unsupervised model-based
visualisation (BAYMINER)
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Linear and Non-linear Dependencies
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Bayesian Classification Modeling
The classification accuracy of the best model
found is 83.48 (58.57).
COMMON FACTORS PUB_T CC_PR CC_HE PA C_SHO C_FAIL
CC_AB CC_ES
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Bayesian Dependency Modeling
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Bayesian Unsupervised Model-based Visualization
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Current Research

• 1. Bayesian track
• Normalized Maximum Likelihood (NML) test
• Bayesian dependency modeling with latent
  variables
• Visualization of Bayesian networks (Banex)
• 2. Educational technology track
• Proactive search to help learners to manage
  information online
• Empirical validation of the technical and
  pedagogical usability criteria of digital
  learning material

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Outline

• Research Overview
• Introduction to Bayesian Modeling
• Investigating Non-linearities with Bayesian
  Networks
• Bayesian Classification modeling
• Bayesian Dependency modeling
• Bayesian Unsupervised Model-based Visualization

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Putting Bayesian Techniques on the Map
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Introduction to Bayesian Modeling

• Probability is a mathematical construct that
  behaves in accordance with certain rules and can
  be used to represent uncertainty.
• Bayesian inference uses conditional probabilities
  to represent uncertainty.
• P(M D,I) - the probability of unknown things
  (M) given the data (D) and background information
  (I).

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Introduction to Bayesian Modeling

• The essence of Bayesian inference is in the rule,
  known as Bayes' theorem, that tells us how to
  update our initial probabilities P(M I) if we
  see data D, in order to find out P(M D,I).

• A priori probability
• Conditional probability
• Posteriori probability

P(DM) P(M) P(MD) P(DM)P(M
) P(DM) P(M)
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Introduction to Bayesian Modeling

• Bayesian inference comprises the following three
  principal steps
• (1) Obtain the initial probabilities P(M I) for
  the unknown things. (Prior distribution.)
• (2) Calculate the probabilities of the data D
  given different values for the unknown things,
  i.e., P(D M, I). (Likelihood.)
• (3) Finally the probability distribution of
  interest, P(M D, I), is calculated using
  Bayes' theorem. (Posterior distribution.)
• Bayes' theorem can be used sequentially.

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Introduction to Bayesian Modeling

• Bayesian method
• (1) is parameter-free and the user input is not
  required, instead, prior distributions of the
  model offer a theoretically justifiable method
  for affecting the model construction,
• (2) works with probabilities and can hence be
  expected to produce robust results with discrete
  data containing nominal and ordinal attributes,
• (3) has no limit for minimum sample size,
• (4) is able to analyze both linear and
  non-linear dependencies,
• (5) assumes no multivariate normal model.

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C_Example 1 Applying Bayes Theorem

• Company A is employing workers on short term
  jobs that are well paid.
• The job sets certain prerequisites to applicants
  linguistic abilities and looks.
• Earlier all the applicants were interviewed, but
  nowadays it has become an impossible task as both
  the number of open vacancies and applicants has
  increased enormously.
• Personnel department of the company was ordered
  to develop a questionnaire to preselect the most
  suitable applicants for the interview.

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C_Example 1 Applying Bayes Theorem

• Psychometrician who developed the instrument
  estimates that it would work out right on 90 out
  of 100 applicants, if they are honest.
• We know on the basis of earlier interviews that
  the terms (linguistic abilities, looks) are valid
  for one per 100 person living in the target
  population.
• The question is If an applicant gets enough
  points to participate in the interview, is he or
  she hired for the job (after an interview)?

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C_Example 1 Applying Bayes Theorem

• A priori probability P(H) is described by the
  number of those people in the target population
  that really are able to meet the requirements of
  the task (1 out of 100 .01).
• Counter assumption of the a priori is P(H) that
  equals to 1-P(H), thus it is .99.
• Psychometricians beliefs about how the instrument
  works is called conditional probability P(EH)
  .9.
• Instruments failure to indicate non-valid
  applicants, i.e., those that are not able to
  succeed in the following interview, is stated as
  P(EH) that equals to .1.
• These values need not to sum to one!

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C_Example 1 Applying Bayes Theorem

• A priori probability
• Conditional probability
• Posterior probability

• P(EH) P(H)
• P(HE)
• P(EH) P(H) P(EH) P(H)

(.9) (.01) P(HE) (.9)
(.01) (.1) (.99)
.08
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C_Example 1 Applying Bayes Theorem
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C_Example 1 Applying Bayes Theorem

• What if the measurement error of the
  psychometricians instrument would have been 20
  per cent?
• P(EH)0.8 P(EH)0.2

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C_Example 1 Applying Bayes Theorem
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C_Example 1 Applying Bayes Theorem

• What if the measurement error of the
  psychometricians instrument would have been only
  one per cent?
• P(EH)0.99 P(EH)0.01

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C_Example 1 Applying Bayes Theorem
26
C_Example 1 Applying Bayes Theorem

• Quite often people tend to estimate the
  probabilities to be too high or low, as they are
  not able to update their beliefs even in simple
  decision making tasks when situations change
  dynamically (Anderson, 1995, p. 326).

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• One of the most important rules educational
  science scientific journals apply to judge the
  scientific merits of any submitted manuscript is
  that all the reported results should be based on
  so called null hypothesis significance testing
  procedure (NHSTP) and its featured product,
  p-value.
• Gigerenzer, Krauss and Vitouch (2004, p. 392)
  describe the null ritual as follows
• 1) Set up a statistical null hypothesis of no
  mean difference or zero correlation. Dont
  specify the predictions of your research or of
  any alternative substantive hypotheses
• 2) Use 5 per cent as a convention for rejecting
  the null. If significant, accept your research
  hypothesis
• 3) Always perform this procedure.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• A p-value is the probability of the observed data
  (or of more extreme data points), given that the
  null hypothesis H0 is true, P(DH0) (id.).
• The first common misunderstanding is that the
  p-value of, say t-test, would describe how
  probable it is to have the same result if the
  study is repeated many times (Thompson, 1994).
• Gerd Gigerenzer and his colleagues (id., p. 393)
  call this replication fallacy as P(DH0) is
  confused with 1P(D).

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• The second misunderstanding, shared by both
  applied statistics teachers and the students, is
  that the p-value would prove or disprove H0.
  However, a significance test can only provide
  probabilities, not prove or disprove null
  hypothesis.
• Gigerenzer (id., p. 393) calls this fallacy an
  illusion of certainty Despite wishful thinking,
  p(DH0) is not the same as P(H0D), and a
  significance test does not and cannot provide a
  probability for a hypothesis.
• A Bayesian statistics provide a way of
  calculating a probability of a hypothesis
  (discussed later in this section).

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• I have been teaching elementary level statistics
  for educational science students for the last 12
  years.
• My latest statistics course grades (Autumn 2006,
  n 12) ranged from one to five as follows 1) n
  3 2) n 2 3) n 4 4) n 2 5) n 1,
  showing that the lowest grade frequency (1) from
  the course is three (25.0).
• Previous data from the same course (2000-2005)
  shows that only five students out of 107 (4.7)
  had the lowest grade.
• Next, I will use the classical statistical
  approach (the likelihood principle) and Bayesian
  statistics to calculate if the number of the
  lowest course grades is exceptionally high on my
  latest course when compared to my earlier stat
  courses.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• There are numerous possible reasons behind such
  development, for example, I have become more
  critical on my assessment or the students are
  less motivated in learning quantitative
  techniques.
• However, I believe that the most important
  difference between the last and preceding courses
  is that the assessment was based on a computer
  exercise with statistical computations.
• The preceding courses were assessed only with
  essay answers.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• I assume that the 12 students earned their grade
  independently (independent observations) of each
  other as the computer exercise was conducted
  under my or my assistants supervision.
• I further assume that the chance of getting the
  lowest grade (?), is the same for each student.
• Therefore X, the number of lowest grades (1) in
  the scale from 1 to 5 among the 12 students in
  the latest stat course, has a binomial (12, ?)
  distribution X Bin(12, ?).
• For any integer r between 0 and 12,

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• The expected number of lowest grades is 12(5/107)
  0.561.
• Theta is obtained by dividing the expected number
  of lowest grades with the number of students
  0.561 / 12 ? 0.05.
• The null hypothesis is formulated as follows H0
  ? 0.05, stating that the rate of the lowest
  grades from the current stat course is not a big
  thing and compares to the previous courses rates.
  
• Three alternative hypotheses are formulated to
  address the concern of the increased number of
  lowest grades H1 ? 0.06 H2 ? 0.07 H3 ?
  0.08.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• To compare the hypotheses, we calculate binomial
  distributions for each value of ?.
• For example, the null hypothesis (H0) calculation
  yields

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• The results for the alternative hypotheses are as
  follows
• PH1(3.06, 12) ? .027
• PH2(3.07, 12) ? .039
• PH3(3.08, 12) ? .053.
• The ratio of the hypotheses is roughly 1223
  and could be verbally interpreted with statements
  like the second and third hypothesis explain the
  data about equally well, or the fourth
  hypothesis explains the data about three times as
  well as the first hypothesis.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• Lavine (1999) reminds that P(r?, n), as a
  function of r (3) and ? .05 .06 .07 .08,
  describes only how well each hypotheses explains
  the data no value of r other than 3 is relevant.
  
• For example, P(4.05, 12) is irrelevant as it
  does not describe how well any hypothesis
  explains the data.
• This likelihood principle, that is, to base
  statistical inference only on the observed data
  and not on a data that might have been observed,
  is an essential feature of Bayesian approach.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• The Fisherian, so called classical approach to
  test the null hypothesis (H0 ? .05) against
  the alternative hypothesis (H1 ? gt .05) is to
  calculate the p-value that defines the
  probability under H0 of observing an outcome at
  least as extreme as the outcome actually
  observed

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• After calculations, the p-value becomes ? 0.02
  and would suggest H0 rejection, if the rejection
  level of significance is set at 5 per cent.
• Another problem with the p-value is that it
  violates the likelihood principle by using P(r?,
  n) for values of r other than the observed value
  of r 3 (Lavine, 1999)
• The summands of P(4.05, 12), P(5.05, 12), ,
  P(12.05, 12) do not describe how well any
  hypothesis explains observed data.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• A Bayesian approach will continue from the same
  point as the classical approach, namely
  probabilities given by the binomial
  distributions, but also make use of other
  relevant sources of a priori information.
• In this domain, it is plausible to think that the
  computerized test would make the number of total
  failures more probable than in the previous times
  when the evaluation was based solely on the
  essays.
• On the other hand, the computer test has only 40
  per cent weight in the equation that defines the
  final stat course grade .3(Essay_1)
  .3(Essay_2) .4(Computer test)/3 Final grade.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• Another aspect is to consider the nature of the
  aforementioned tasks, as the essays are distance
  work assignments while the computer test is to be
  performed under observation.
• Perhaps the course grades of my earlier stat
  courses have a narrower dispersion due to
  violence of the independent observation
  assumption?
• For example, some students may have copy-pasted
  text from other sources or collaborated without a
  permission.
• As we see, there are many sources of a priori
  information that I judge to be inconclusive and,
  thus, define that null hypothesis is as likely to
  be true or false.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• This a priori judgment is expressed
  mathematically as P(H0) ? 1/2 ? P(H1) P(H2)
  P(H3). If I further assume that the alternative
  hypotheses H1, H2 or H3 share the same likelihood
  P(H1) ? P(H2) ? P(H3) ? 1/6.
• These prior distributions summarize the knowledge
  about ? prior to incorporating the information
  from my course grades.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• An application of Bayes' theorem yields

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• Similar calculations for the alternative
  hypotheses yields P(H1r3) ? .16 P(H2r3) ?
  .29 P(H3r3) ? .31.
• These posterior distributions summarize the
  knowledge about ? after incorporating the grade
  information.
• The four hypotheses seem to be about equally
  likely (.30 vs. .16, .29, .31).
• The odds are about 2 to 1 (.30 vs. .70) that the
  latest stat course had higher rate of lowest
  grades than 0.05.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• The difference between the classical and Bayesian
  statistics would be only philosophical
  (probability vs. inverse probability) if they
  would always lead to similar conclusions.
• However, in this case the p-value would suggest
  rejection of H0 (p .02), but the Bayesian
  analysis indicate not very strong evidence
  against ? .05, only about 2 to 1.

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C_Example 2 Comparison of Traditional
Frequentistic and Bayesian Approach

• What if the number of the lowest grades is two?
• The classical approach would not anymore suggest
  H0 rejection (p .12).
• Bayesian result would stay quite the same (.39
  vs. .17, .20, .24), saying that there is not much
  evidence against the H0.

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Outline

• Research Overview
• Introduction to Bayesian Modeling
• Investigating Non-linearities with Bayesian
  Networks
• Bayesian Classification Modeling
• Bayesian Dependency Modeling
• Bayesian Unsupervised Model-based Visualization

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Investigating the Number of Non-linear and
Multi-modal Relationships Between Observed
Variables Measuring A Growth-oriented Atmosphere
A_Example 1

• Petri Nokelainen
• Pekka Ruohotie
• Research Centre for Vocational Education (RCVE)
• University of Tampere, Finland
• Tomi Silander
• Complex Systems Computation Group (CoSCo)
• Helsinki University of Technology, Finland
• Henry Tirri
• Nokia Research Center, Finland

See printed version!
(Nokelainen, Silander, Ruohotie Tirri, 2007,
2003)
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Introduction
A_Example 1

• In the social science researchers point of view,
  the requirements of traditional frequentistic
  statistical analysis are very challenging.
• For example, the assumption of normality of both
  the phenomena under investigation and the data is
  prerequisite for traditional parametric
  frequentistic calculations.

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Introduction
A_Example 1

• In situations where a latent construct cannot be
  appropriately represented as a continuous
  variable, or where ordinal or discrete indicators
  do not reflect underlying continuous variables,
  or where the latent variables cannot be assumed
  to be normally distributed, traditional Gaussian
  modeling is clearly not appropriate.
• In addition, normal distribution analysis sets
  minimum requirements for the number of
  observations, and the measurement level of
  variables to be continuous.

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Introduction
A_Example 1

• Bayesian modeling approach is a good alternative
  to traditional frequentistic statistics as it is
  capable of handling small discrete, non-normal
  samples as well as large-scale continuous data
  sets.
• The purpose of this paper is to investigate the
  number of non-linear and multi-modal
  relationships between variables in various
  real-world empirical Growth-oriented Atmosphere
  data in order to find how much they weaken the
  robustness of linear statistical methods.

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Research Questions
A_Example 1

• What kind of non-linearities and how many are
  captured by discrete Bayesian networks?
• Is there difference between the results of linear
  bivariate correlations and Bayesian networks?
• Does an empirical sample containing pure linear
  dependencies have better overall fit indices in
  CFA than sample containing less linear
  dependencies?
• Does an empirical sample containing pure linear
  dependencies have higher CFA parameter estimates
  than sample containing less linear dependencies?
• Are discrete Bayesian networks viable way to
  pre-model data before CFA?

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Types of Non-linearities Studied
A_Example 1

• We study two different kinds of
  "non-linearities",
• Non-linear relationships between continuous
  variables and
• Multi-modal relationships between continuous
  variables.
• Further, we only study simple non-linear
  relationships between two variables
• The dependency between variables X and Y is
  considered non-linear if the mean of the
  conditional distribution of Y is not a monotonous
  (i.e., increasing or decreasing) function of X.
• Similarly, the dependency between variables X and
  Y is considered multi-modal if the mode of the
  conditional distribution of Y is not a monotonous
  function of X.

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Bayesian Dependency Models
A_Example 1

• This study resembles to some extent the work by
  Hofmann and Tresp in which they use method of
  Parzen windows to allow non-linear dependencies
  between continuous variables.
• The emphasis in their work was to demonstrate the
  possibility to build Bayesian networks that can
  capture non-linear relationships.
• By using discretized variables this possibility
  comes trivially, but our objective is to find out
  to what extent this possibility is used i.e. how
  many and what kind of non-linearities are
  captured by discrete Bayesian networks.

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Bayesian Dependency Models
A_Example 1

• Given the identically and independently
  distributed multivariate data set D over
  variables V and the prior probability
  distribution ? over Bayesian networks, Bayesian
  probability theory allows us to calculate the
  probability P(G  D, ?) of any Bayesian network
  G.
• Different networks can then be compared by their
  probability.
• Finding the most probable Bayesian network for
  any given data is known to be NP-hard, which
  practically ruins the hopes for the automatic
  discovery of the most probable network.
• However, stochastic search methods have proven to
  be successful in finding high probability
  networks Once the network G has been constructed
  using data D, we can use it to calculate
  predictive joint distributions P(V  G, D).

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Bayesian Dependency Models
A_Example 1

• Bayesian network structure can be used to
  effectively calculate conditional marginals of
  the predictive joint distribution for single
  variables i.e. P(Vi  A, G, D), where A is any
  subset of variables of V.
• In this paper we only study the marginals, where
  A is a singleton Vj and there is either an
  arrow from Vi to Vj or an arrow from Vj to Vi (we
  say that Vi and Vj are adjacent in G).

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Linear and Non-linear Dependencies
A_Example 1

• Frequentistic parametric statistical techniques
  are designed for normally distributed (both
  theoretically and empirically) indicators that
  have linear dependencies.
• Univariate normality
• Multivariate normality
• Bivariate linearity

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Linear and Non-linear Dependencies
A_Example 1
rP -1.00
58
Linear and Non-linear Dependencies
A_Example 1

• Sometimes univariate/multivariate assumption is
  true, but bivariate linearity is violated.

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Linear and Non-linear Dependencies
A_Example 1
rP -.39
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Linear and Non-linear Dependencies
A_Example 1

• In some cases, univariate normality is violated,
  resulting in linear dependency.

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Linear and Non-linear Dependencies
A_Example 1
rP.77
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Linear and Non-linear Dependencies
A_Example 1

• In some cases, univariate normality is violated,
  resulting in non-linear dependency.

63
Linear and Non-linear Dependencies
A_Example 1
rP.59
64
Linear and Non-linear Dependencies
A_Example 1
rP.59
65
Measuring Non-linearities
A_Example 1

• To measure non-linear dependencies captured by
  Bayesian networks, we tested every variable in
  each network by conditioning it one by one with
  its immediate neighbors in the network.
• We then observed whether the modes and means of
  the conditional distributions were "linear" and
  whether the conditional distributions were
  "unimodal".
• Linearity of modes and means was tested by
  recording whether the means and modes were
  increasing or decreasing functions of
  conditioning variable.
• Even clear departures from line like behavior
  were accepted as linear as long as the direction
  of correlation (positive, negative) did not
  change.

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Measuring Non-linearities
A_Example 1

• In these experiments, linear means relationship
  that can be more or less adequately modeled by a
  line describing how central tendency of the
  dependent variable varies as a function of the
  independent variable.
• In measuring unimodality of conditional
  distributions, we judged the dependency to be
  unimodal if (and only if) none of the conditional
  distributions P(YX) were clearly multimodal.

67
Results
A_Example 1

• What kind of non-linearities and how many are
  captured by discrete Bayesian networks?
• Investigation of two independent empirical data
  (n 2430, n 762) showed that only 39 per
  cent of all dependencies between variables were
  purely linear (linear mode, linear mean,
  unimodal).
• Nine per cent of dependencies were purely
  non-linear (non-linear mode, non-linear mean,
  multimodal).
• Multimodality was the most common reason for
  violation of linearity in both datasets.

68
Results
A_Example 1

• We continued investigations with two distinct
  samples of the latter data, namely D21 (n 447)
  and D23 (n 208).
• Those two data were selected for the following
  reasons
• First, the sample sizes are more close to each
  other when compared to the D22 data (n 71).
• Second, the two samples are collected with the
  same self-rated five-point Likert-scale
  questionnaire.
• Third, the D21 sample represents in this study
  linear empirical data with 23.9 per cent of
  pure linear and 15.0 per cent of pure non-linear
  dependencies, and the D23 sample represents
  non-linear data with only 16.2 per cent of pure
  linear dependencies and 18.3 per cent of pure
  non-linear dependencies.

69
Results
A_Example 1

• Our first goal was to compare subject domain
  interpretations of linear correlational analysis
  and non-linear Bayesian dependency models in
  order to investigate if the models would differ
  in terms of interpretation according to the
  Growth-oriented Atmosphere model.
• Is there difference between the results of linear
  bivariate correlations and Bayesian networks?
• The results showed that in general Bayesian
  network models were congruent with the
  correlation matrixes as both methods found the
  same variables independent of all the other
  variables.
• However, non-linear modeling found with both
  samples greater number of strong dependencies
  between growth-oriented atmosphere factors.

70
Results
A_Example 1

• Our second goal was to investigate the following
  four aspects of the growth-oriented atmosphere
  theory
• support and rewards from the management,
• the incentive value of the job,
• operational capacity of the team, and
• work-related stress.
• Does an empirical sample containing pure linear
  dependencies have better overall fit indices in
  CFA than sample containing less linear
  dependencies?

71
Results
A_Example 1

• First, when comparing CFA and Bayesian modeling,
  we learned that latter is unable to find support
  for the second aspect under investigation, namely
  the relationship between incentive value of the
  job, know-how developing and valuation of the
  job.
• Second, we found no major differences in results
  between linear and non-linear samples.
• However, the linear data has higher parameter
  estimates in all four aspects under
  investigation.

72
Results
A_Example 1

• Next we investigated if theoretically justifiable
  dependencies between factors found by Bayesian
  models are also present in the CFA models.
• Does an empirical sample containing pure linear
  dependencies have higher CFA parameter estimates
  than sample containing less linear dependencies?
• We conducted confirmatory factor analysis with
  the growth-oriented atmosphere model and examined
  the differences between the linear (D21) and
  non-linear (D23) factor covariance matrixes.
• The results showed that the CFA model performed
  better with the linear sample.

73
Results
A_Example 1

• Is there difference between substantive
  interpretations of the results of CFA and BDM
  with linear and non-linear samples?
• Both Bayesian dependency models did not support
  the second theoretical assumption about the
  relationship between Incentive value of the job
  (INV) and Know-how developing (DEV) and Valuation
  of the job (VAL).

74
Results
A_Example 1
75
Results
A_Example 1

• Is there difference between substantive
  interpretations of the results of CFA and BDM
  with linear and non-linear samples?
• The second observation is that the fourth
  theoretical assumption about the negative
  influence of Psychic stress (PSY) on all the
  other factors is only partially supported in both
  Bayesian models.

76
Results
A_Example 1
77
Results
A_Example 1

• Is there difference between substantive
  interpretations of the results of CFA and BDM
  with linear and non-linear samples?
• Finally, the linear sample (D21) has in most
  cases higher CFA parameter estimates than the
  non-linear sample.

78
Conclusions
A_Example 1

• This study investigated the number of non-linear
  and multi-modal relationships between variables
  in various real-world empirical Growth-oriented
  Atmosphere data.
• Investigation of two independent empirical data
  (n 2430 and n 762) showed that only 39 per
  cent of all dependencies between variables were
  purely linear (linear mode, linear mean,
  unimodal).
• Nine per cent of dependencies were purely
  non-linear (non-linear mode, non-linear mean,
  multimodal).
• Multimodality was the most common reason for
  violation of linearity in both datasets.

79
Conclusions
A_Example 1

• Two subgroups of the latter data were identified
  as linear (D21, n 447) and non-linear (D23,
  n 208).
• Both correlational analysis and Bayesian
  dependency modeling were applied to these data in
  order to investigate relationships between
  fourteen factors of the growth-oriented
  atmosphere model.
• Our conclusion that is based on the preliminary
  analysis of two relatively small empirical
  samples is that descriptive power of traditional
  linear models (e.g., correlational analysis) is
  sufficient with non-linear data (pure linear
  dependencies vary between 16.2 - 23.9 ).

80
Outline

• Research Overview
• Introduction to Bayesian Modeling
• Investigating Non-linearities with Bayesian
  Networks
• Bayesian Classification Modeling
• Bayesian Dependency Modeling
• Bayesian Unsupervised Model-based Visualization

81
Bayesian Classification Modeling

• Which variables are the best predictors for
  different group memberships (e.g., A or C group,
  gender, productivity, level of giftedness).
• In the classification process, the automatic
  search is looking for the best set of variables
  to predict the class variable for each data item.

82
Bayesian Classification Modeling

• The search procedure resembles the traditional
  linear discriminant analysis (LDA, Huberty, 1994,
  118-126), but the implementation is totally
  different.
• For example, a variable selection problem that is
  addressed with forward, backward or stepwise
  selection procedure in LDA is replaced with a
  genetic algorithm approach (e.g., Hilario,
  Kalousisa, Pradosa Binzb, 2004 Hsu, 2004) in
  the Bayesian classification modeling.

83
Bayesian Classification Modeling

• The genetic algorithm approach means that
  variable selection is not limited to one (or two
  or three) specific approach instead many
  approaches and their combinations are exploited.
• One possible approach is to begin with the
  presumption that the models (i.e., possible
  predictor variable combinations) that resemble
  each other a lot (i.e., have almost same
  variables and discretizations) are likely to be
  almost equally good.
• This leads to a search strategy in which models
  that resemble the current best model are selected
  for comparison, instead of picking models
  randomly.

84
Bayesian Classification Modeling

• Another approach is to abandon the habit of
  always rejecting the weakest model and instead
  collect a set of relatively good models.
• The next step is to combine the best parts of
  these models so that the resulting combined model
  is better than any of the original models.
• B-Course is capable of mobilizing many more
  viable approaches, for example, rejecting the
  better model (algorithms like hill climbing,
  simulated annealing) or trying to avoid picking
  similar model twice (tabu search).

85
C_Example 3 Motivational Predictors for Study
Group Membership

• Researcher picked from a student population (N
  240) a sub sample (one study group) (n 23) for
  closer investigation.
• His goal was to study how students self-reported
  learning motivation was related to academic
  success of group works.
• The sub sample consisted of five groups that
  studens had formed by themselves in the early
  stages of their studies.

86
C_Example 3 Motivational Predictors for Study
Group Membership

• All the participants (n 23) filled the
  Abilities for Professional Learning Questionnaire
  (Ruohotie, 2002 Nokelainen Ruohotie, 2002)
  that measures their motivational level.
• See research article 3 for item descriptions.
• Researcher interviewed the participants to
  profile the groups.

87
C_Example 3 Motivational Predictors for Study
Group Membership

• Classification analysis was conducted using class
  membership as class variable.
• Aim of the analysis was to learn which of the
  APLQ items (i.e., learning motivation dimensions)
  would best predict differencies between the
  groups.
• Naïve Bayes Network that is produced in the
  analysis is used to examine special features of
  the groups.
• In addition, it allows groupwise comparison.

88
C_Example 3 Motivational Predictors for Study
Group Membership
Sample size n 23
Classification accuracy 60.87.
Common components V2, V3, V4, V6, V7, V8, V10,
V11, V12, V13, V14, V16, V17, V20, V21, V22, V23,
V25, V26, V28
89
C_Example 3 Motivational Predictors for Study
Group Membership
90
C_Example 4 Mobile Learning Components
Predicting the Use of Three Types of Computers

• The study investigated how components of mobile
  learning predict the use of different computer
  devices (Syvänen, Nokelainen, Ahonen Turunen,
  2003).
• Sample (n 87) consisted of 5th and 6th grade
  Finnish elementary school students.

91
C_Example 4 Mobile Learning Components
Predicting the Use of Three Types of Computers

• Classification variable was device that has
  three values
• 1 Handheld computer2 Portable computer3
  Desktop computer
• Fourteen questions were asked from the students
  to measure their mobile learning experineces.

92
C_Example 4 Mobile Learning Components
Predicting the Use of Three Types of Computers

• Variable Description
• DEEP Deep approach
• HELPSEE Help-seeking
• MANAGEM Learning management
• CREATIV Creativity in problem solving
• EFFECTI Perceived effectiveness
• SELFEFF Self-efficacy
• SEARCH Knowledge seeking
• SHARE Knowledge sharing
• DUALISM Conception of knowledge
• SURFACE Surface approach
• CONFIDENCE Computer confidence
• PEERLEARN Peer learning
• EASINESS Perceived easiness of use
• CONSTRUC Knowledge construction

93
C_Example 4 Mobile Learning Components
Predicting the Use of Three Types of Computers
Sample size n 87
Classification accuracy 62.32.
Common components DUALISM, SURFACE, CONFIDENCE,
PEERLEARN, EASINESS, CONSTRUC
94
C_Example 4 Mobile Learning Components
Predicting the Use of Three Types of Computers
95
Investigating the Influence of Attribution Styles
on the Development of Mathematical Talent
A_Example 2

• Petri Nokelainen
• Research Centre for Vocational Education
• University of Tampere, Finland
• Kirsi Tirri
• Department of Practical Theology
• University of Helsinki, Finland
• Hanna-Leena Merenti-Välimäki
• Espoo-Vantaa Institute of Technology, Finland

See printed version!
(Nokelainen, Tirri Merenti-Välimäki, 2007.)
96
Outline

• Research Overview
• Introduction to Bayesian Modeling
• Investigating Non-linearities with Bayesian
  Networks
• Bayesian Classification Modeling
• Bayesian Dependency Modeling
• Bayesian Unsupervised Model-based Visualization

97
Bayesian Dependency Modeling

• Bayesian dependency modeling (BDM) is applied to
  examine dependencies between variables by both
  their visual representation and probability ratio
  of each dependency
• Graphical visualization of Bayesian network
  contains two components
• 1) Observed variables visualized as ellipses.
• 2) Dependences visualized as lines between nodes.

98
C_Example 5 Calculation of Bayesian Score

• Next, I will present how Bayesian score (BS),
  that is, the probability of the model P(MD), is
  firstly calculated and secondly compared for the
  two models presented in the figure

Figure 9. An Example of Two Competing Bayesian
Network Structures
(Nokelainen, 2008, p. 121)
99
C_Example 5 Calculation of Bayesian Score

• Let us assume that we have the following data
• x1 x2
• 1 1
• 1 1
• 2 2
• 1 2
• 1 1
• Model 1 (M1) represents the two variables, x1 and
  x2 respectively, without statistical dependency,
  and the model 2 (M2) represents the two variables
  with a dependency (i.e., with a connecting arc).
• The binomial data might be a result of an
  experiment, where the five participants have
  solved a job related task before (x1) and after
  (x2) a vocational training period.

100
C_Example 5 Calculation of Bayesian Score

• In order to calculate P(M1,2D), we need to solve
  P(DM1,2) for the two models M1 and M2.
• Probability of the data given the model is solved
  by using the following marginal likelihood
  equation (Congdon, 2001, p. 473 Myllymäki,
  Silander, Tirri, Uronen, 2001 Myllymäki
  Tirri, 1998, p. 63)

101
C_Example 5 Calculation of Bayesian Score

• In the Equation 4, following symbols are used
• n is the number of variables (i indexes variables
  from 1 to n)
• r1 is the number of values in ith variable (k
  indexes these values from 1 to ri
• qi is the number of possible configurations of
  parents of ith variable
• Nij describes the number of rows in the data that
  have jth configuration for parents of ith
  variable
• Nijk describes how many rows in the data have
  kth value for the ith variable also have jth
  configuration for parents of Ith variable
• N is the equivalent sample size set to be the
  average number of values divided by two.
• The marginal likelihood equation produces a
  Bayesian Dirichlet score that allows model
  comparison (Heckerman et al., 1995 Tirri, 1997
  Neapolitan Morris, 2004).

102
C_Example 5 Calculation of Bayesian Score

• First, I will calculate P(DM1) given the values
  of variable x1

(2/2)/1
(2/2)/21
x1 x2 1 1 1 1 2 2 1 2 1 1
103
C_Example 5 Calculation of Bayesian Score

• Second, the values for the x2 are calculated

104
C_Example 5 Calculation of Bayesian Score

• The BS, probability for the first model P(M1D),
  is 0.027 0.012 ? 0.000324.

105
C_Example 5 Calculation of Bayesian Score

• Third, P(DM2) is calculated given the values of
  variable x1

106
C_Example 5 Calculation of Bayesian Score

• Fourth, the values for the first parent
  configuration (x1 1) are calculated

107
C_Example 5 Calculation of Bayesian Score

• Fifth, the values for the second parent
  configuration (x1 2) are calculated

108
C_Example 5 Calculation of Bayesian Score

• The BS, probability for the second model P(M2D),
  is 0.027 0.027 0.500 ? 0.000365.

109
C_Example 5 Calculation of Bayesian Score

• Bayes theorem enables the calculation of the
  ratio of the two models, M1 and M2.
• As both models share the same a priori
  probability, P(M1) P(M2), both probabilities
  are canceled out.
• Also the probability of the data P(D) is canceled
  out in the following equation as it appears in
  both formulas in the same position

110
C_Example 5 Calculation of Bayesian Score

• The result of model comparison shows that since
  the ratio is less than 1, the M2 is more probable
  than M1.
• This result becomes explicit when we investigate
  the sample data more closely.
• Even a sample this small (n 5) shows that there
  is a clear tendency between the values of x1 and
  x2 (four out of five value pairs are identical).

x1 x2 1 1 1 1 2 2 1 2 1 1
111
C_Example 6 Modeling of Prerequisites for
Organizational Learning

• Staff of five Finnish police organization (n
  281) filled out Growth Oriented Atmosphere
  Questionnaire (Luoma, Nokelainen Ruohotie,
  2002).
• BDM was applied to study how the theoretical
  model is represented in the Bayesian Network.

112
C_Example 6 Modeling of Prerequisites for
Organizational Learning
113
C_Example 6 Modeling of Prerequisites for
Organizational Learning
114
Investigating Subordinates' Evaluations on their
Superiors Emotional Leadership
A_Example 3

• Petri Nokelainen
• Pekka Ruohotie
• Research Centre for Vocational Education
• University of Tampere, Finland

See printed version!
(Nokelainen Ruohotie, 2005.)
115
Conceptual Modeling of Self-rated
Intelligence-profile
A_Example 4

• Kirsi Tirri and Erkki Komulainen
• University of Helsinki, Finland
• Petri Nokelainen and Henry Tirri
• Helsinki University of Technology, Finland

See printed version!
(Tirri, K., Komulainen, Nokelainen Tirri, H.,
2002.)
116
Outline

• Research Overview
• Introduction to Bayesian Modeling
• Investigating Non-linearities with Bayesian
  Networks
• Bayesian Classification Modeling
• Bayesian Dependency Modeling
• Bayesian Unsupervised Model-based Visualization

117
Bayesian Unsupervised Model-based Visualization

• Dispersion of single data vectors in
  three-dimensional space in order to find how the
  factors are interrelated in individual level.
• The data is mapped into different set of
  dimensions according to the optimized solution
  from which Bayesian algorithm produced one
  optimal model.
• The three-dimensional model is plotted into
  series of two-dimensional figures each presenting
  one dimension at the time.
• BayMiner http//www.bayminer.com

118
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119
Investigating Growth Prerequisites in a Finnish
Polytechnic Institute of Higher Education
A_Example 5

• Petri Nokelainen
• Pekka Ruohotie
• Research Centre for Vocational Education
• University of Tampere, Finland

See printed version!
(Nokelainen Ruohotie, in press, to appear in
the Journal of Workplace Learning.)
120
Links

• Research Centre for Vocational Education ltURL
  http//www.uta.fi/aktkk gt
• Complex Systems Computation Group ltURL
  http//cosco.hiit.fi gt
• EDUTECH ltURL http//cosco.hiit.fi/edutech gt
• B-COURSE ltURL http//b-course.hiit.fi gt
• BayMiner ltURL http//www.bayminer.com gt

121
References

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• Fisher, R. (1935). The design of experiments.
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• Gigerenzer, G. (2000). Adaptive thinking. New
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• Gill, J. (2002). Bayesian methods. A Social and
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  (pp. 391-408). Thousand Oaks Sage.
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• Heckerman, D., Geiger, D., Chickering, D.
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• Luoma, M., Nokelainen, P., Ruohotie, P. (2003,
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• Nokelainen, P. (2008). Modeling of Professional
  Growth and Learning Bayesian Approach. Tampere
  Tampere University Press.
• Nokelainen, P., Ruohotie, P. (2005).
  Investigating the Construct Validity of the
  Leadership Competence and Characteristics Scale.
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  Work and Learning 2005 Conference, Sydney,
  Australia.
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  51(1), 64-81.
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  Turunen, H. (2003, August). Approaches to
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