Ishanu Chattopadhyay

Modeling Opinion Dynamics

with individual-level predictive capability

University of Chicago 

Nov 18 2022

cognet 0.0.193 lates version 

Learning dependency structure between opinions from data

Using Learned structure for prediction

Estimate worldview from incomplete information

Learning dependency structure between opinions from data

Using Learned structure for prediction

Estimate worldview from incomplete information

Learning polarization dynamics

\Phi_i:\prod_{j \neq i} \Sigma_j \rightarrow \mathcal{D}(\Sigma_i)

1. Qnets

Data-inferred hidden dependencies between beliefs

  • Recursive forest of conditional inference trees, constructed from survey responses.
  • No prior assumption of dependency structure

Recursive expansion

Three fundamental equations:

\Phi_i:\prod_{j \neq i} \Sigma_j \rightarrow \mathcal{D}(\Sigma_i)

1. Qnets

\theta(x,y) \triangleq \mathbf{E}_i \left ( \mathbb{J}^{\frac{1}{2}} \left (\Phi_i^P(x_{-i}) , \Phi_i^Q(y_{-i})\right ) \right )

2. Q-distance

\mathbb{D}^P(x,i)\triangleq 1 - \Phi^P_i(x_{-i})\vert_{x_i}

3. Dissonance

Data-inferred hidden dependencies between beliefs

Canonical distance between belief vectors

Quantifying the notion of dissonance

Belief vectors define a metric space;  close beliefs are ones that can spontaneously change to or jump across

Left and Right  Polar Vectors

GSS variable,R-pole,L-pole

-----------------------------------------------------
abany,no,yes
abdefctw,always wrong,not wrong at all
abdefect,no,yes
abhlth,no,yes
abnomore,no,yes
abpoor,no,yes
abpoorw,always wrong,not wrong at all
abrape,no,yes
absingle,no,yes
bible,inspired word,book of fables
colcom,fired,not fired
colmil,not fired,not allowed
comfort,strongly agree,strongly disagree
conlabor,hardly any,a great deal
godchnge,"believe now, always have","don't believe now, never have"
grass,not legal,legal
gunlaw,oppose,favor
intmil,very interested,not at all interested
libcom,remove,not remove
libmil,not remove,remove
maboygrl,true,false
owngun,yes,no
pillok, agree,strongly agree
pilloky,strongly disagree,strongly agree
polabuse,no,yes
pray,several times a day,never
prayer,disapprove,approve
prayfreq,several times a day,never
religcon,strongly disagree,strongly agree
religint,strongly disagree,strongly agree
reliten,strong,no religion
rowngun,yes,no
shotgun,yes,no
spkcom,not allowed,allowed
spkmil,allowed,not allowed
taxrich,about right,much too low
viruses,definitely true,definitely not true

37 dimensional polar baseline

Extreme Left

  • questions on key social issues
  • questions that are unambiguously tied to liberal/conservative ideologies
GSS variable actual (masked) Reconstructed
spkcom allowed allowed
colcom not fired not fired
spkmil allowed allowed
colmil allowed not allowed
libmil not remove not remove
libhomo not remove not remove
reliten strong no religion
pray once a day once a day
bible inspired word word of god
abhlth yes yes
abpoor no no
pillok agree agree
intmil very interested very interested
abpoorw always wrong not wrong at all
godchnge believe now, always have believe now, always have
prayfreq several times a week several times a week
religcon strong disagree disagree
religint disagree disagree
comfort strongly agree neither agree nor disagree

Reconstruction

Example 1

GSS variable actual (masked) Reconstructed
spkcom allowed allowed
colcom not fired not fired
libmil not remove not remove
libhomo not remove not remove
gunlaw favor favor
reliten no religion no religion
prayer approve approve
bible book of fables inspired word
abnomore yes yes
abhlth yes yes
abpoor yes yes
abany yes yes
owngun no no
intmil moderately interested moderately interested
abpoorw not wrong at all not wrong at all
godchnge believe now, didn't used to believe now, always have
prayfreq several times a week several times a week
religcon strongly agree agree
religint strongly agree not agree/dsagre

Reconstruction

Example 2

Out-of-sample validation

GSS data

Out-of-sample validation

Eurobarometer data

Modeling societal polarization

Primary outcome:

Estimate worldview from partial knowledge

 

Secondary outcome:

Validate geometric theory of belief shift

Future Work: Validation in Survey Experiments in the wild

Sep 2 2021

YouGov

Primary outcome:

1. Estimate worldview from partial knowledge

 

Secondary outcome:

1. Predict belief shift

2. Validate mechanism (emedding geometry is more important to nature of interactions)

highly consequential !

ZeD Lab

TruthNet: ML For Survey Data Validation

&

Identification of Adversarial Responses

University of Chicago 

Algorithmic Lie Detector

QNet: Interrogating Structure of Survey Responses

  • Each response vector is an element of the "response space".
  • Is there a natural metric on the response space? What would such a metric mean intuitively
  • Can we determine if a response vector is "valid"?
  • Can we distinguish algorithmically between actual/honest responses vs random/adversarial responses?

Nodes Hyperlinked to Trees

click on nodes to change trees

"Mass tells space-time how to curve, and space-time tells mass how to move."

 

-- John Wheeler

I think here "mass" (concentration of people) modulates the geometry,

which then drives the belief shifts.

Future Work:  The "why" question

Symmetry breaking in opinion formation: Towards the physics of opinion change

Symmetry breaking in opinion formation: Towards the physics of opinion change

  1. We develop an intrinsic metric between opinion / belief vectors (the q-distance)
  2. Allows us to consider the space of opinions as a metric space
  3. Local metric embedding gives us a manifold
  4. We can reconstruct missing beliefs
  5. We hypothesize and validate that belief shifts occur along opinion density gradients : a mechanistic theory of belief propagation that can yield falsifiable predictions from field data.
  6. "Popular beliefs" act as "gravity" wells
\displaystyle \eta(s) = \lim_{\epsilon \rightarrow 0+} \frac{1}{\epsilon} \left \vert \big \{ s' \in \Omega : \theta(s,s') \leqq \epsilon \big \}\right \vert

"mass" distribution:

 

Z-axis: local density of samples

A

B

C

D

Pr\{s \rightarrow s'\} \propto \langle s'-s, \nabla\eta(s)\rangle

Opinion mass creates "gravity" wells

perturbation causes opinions to shift towards its closest "gravity" well

Explains Chris Bail's observation

Dissonance more at ideological boundaries

Red and gray represent high and low dissonance on the GSS variable "bible"

Red and blue represent conservative-leaning and liberal-leaning populations

Predicting worldviews

from partial observation

reconstruct

response 

random

mask

  1. We develop an intrinsic metric between opinion / belief vectors (the q-distance)
  2. Allows us to consider the space of opinions as a metric space
  3. Local metric embedding gives us a manifold
  4. We can reconstruct missing beliefs
  5. We hypothesize and validate that belief shifts occur along opinion density gradients : a mechanistic theory of belief propagation that can yield falsifiable predictions from field data.
  6. "Popular beliefs" act as "gravity" wells

collect data

construct qnet

\theta

reconstruct worldview

predict belief shifts

Summary

Towards A Mechanistic Theory of Belief Shift Dynamics: Peaks Attract

Towards A Mechanistic Theory of Belief Shift Dynamics: Peaks Attract

Pr\{s \rightarrow s'\} \propto \langle s'-s, \nabla\eta(s)\rangle

A

B

C

D

Pr\{s \rightarrow s'\} \propto \langle s'-s, \nabla\eta(s)\rangle

A

B

C

D

current opinion

Shifted belief

local density gradient

embedded versions

Pr\{s \rightarrow s'\} \propto e^{-\theta(s,s')}

A

B

C

D

current opinion

Shifted belief

Pr\{s \rightarrow s'\} \propto \langle s'-s, \nabla\eta(s)\rangle

Proven property of q-distance

Theorem (conjectured)

Eurobarometer is a series of public opinion surveys conducted regularly on behalf of the European Commission and other EU Institutions since 1973. These surveys address a wide variety of topical issues relating to the European Union throughout its member states.

Reconstruction Error Distribution

Eurobarometer 4

GSS

Eurobarometer 4

EC Membership good, bad, neither?

Future:

 

1. Survey expt

2. Statistical Physics of Opinion Change

\omega_y e^{\frac{\sqrt{8}N^2}{1-\alpha}\theta(x,y)} \geqq Pr(x \in P \rightarrow y \in Q) \geqq \omega_y e^{-\frac{\sqrt{8}N^2}{1-\alpha}\theta(x,y)}\\
\bigg \lvert \log \frac{ Pr(x \in P \rightarrow y \in Q) }{\omega_y } \bigg \rvert \leqq -\frac{\sqrt{8}N^2}{1-\alpha}\theta(x,y)

theorem

\textrm{as } \theta \rightarrow 0, \\ Pr(x \in P \rightarrow y \in Q) \rightarrow \omega_y

symmetry breaking

The q-distance Metric

Collection of all such conditional inference trees is  the recursive forest, answering the following question:

\textrm{If we have $n$ questions } X_1, \cdots , X_n, \\ \textrm{ and we have a subject responding with}\\ {\color{green} x_1, \cdots, x_{i-1},x_{i+1},\cdots, x_{n-1}, }\\ \textrm{ then the distribution of responses to question $X_i$ is given by } \\ {\color{green}\Phi_i:\prod_{j \neq i} \Sigma_j \rightarrow \mathcal{D}(\Sigma_i)}\\ \textrm{ where } \mathcal{D}(\Sigma_i) \textrm{ is the set of all possible distributions}\\ \textrm{over the set of all possible responses $\Sigma_i$ }

The q-distance Metric

\textrm{where $P,Q$ are possibly two distinct populations}\\ \textrm{with distinct qnets, such that }\\ x \in P, y \in Q \textrm{ and }\\ J \textrm{ is the Jensen-Shannon divergence }
{\theta(x,y) \triangleq \mathbf{E}_i \left ( \mathbb{J}^{\frac{1}{2}} \left (\Phi_i^P(x_{-i}) , \Phi_i^Q(y_{-i})\right ) \right )}\\
\textrm{For two opinion vectors $x,y$}
\textrm{Intrinsic metric between opinion vectors}

The q-distance Metric: Why Is  This a Natural Metric?

\textrm{items } X_1, X_2, \cdots , X_{i-1},X_{i+1}, \cdots, X_N
a
b
c
d
e
X_i

Similar opinion vectors can spontaneously switch:

intrinsic metric quantifies the odds of this spontaneous switch

Theorem: q-distance is "natural"

\textrm{With $N$ distinct questions, at a significance level $\alpha$, we have }\\ \omega_y e^{\frac{\sqrt{8}N^2}{1-\alpha}\theta(x,y)} \geqq Pr(x \in P \rightarrow y \in Q) \geqq \omega_y e^{-\frac{\sqrt{8}N^2}{1-\alpha}\theta(x,y)}\\ \textrm{ where } \omega_y \textrm{ is the probability $y \in P$ }

Theorem 1.

Sanov's Theorem & Pinsker's Inequality

Assume that one question $$X_i$$ is unanswered.

\textrm{items } X_1, X_2, \cdots , X_i, \cdots, X_N
a
b
c
d
e

Distribution of responses to this item given remaining responses

X_i

Given this distribution the probability that "b" is the answer

Pr(x \in P \rightarrow y \in Q) = \prod_{i=1}^N\Phi_i^P(x_{-i}) \vert_{y_i}
\theta(x,y) \triangleq \mathbf{E}_i \left ( \mathbb{J}^{\frac{1}{2}} \left (\Phi_i^P(x_{-i}) , \Phi_i^Q(y_{-i})\right ) \right )\\
\omega_y e^{\frac{\sqrt{8}N^2}{1-\alpha}{\color{green}\theta(x,y)}} \geqq {\color{red} Pr(x \in P \rightarrow y \in Q) } \geqq \omega_y e^{-\frac{\sqrt{8}N^2}{1-\alpha}{\color{green}\theta(x,y)}}

From first principles:

Distance metric such that log-likelihood of jump scales as the distance

theorem

Summarizing

Three fundamental equations:

\Phi_i:\prod_{j \neq i} \Sigma_j \rightarrow \mathcal{D}(\Sigma_i)

1. Qnets

\theta(x,y) \triangleq \mathbf{E}_i \left ( \mathbb{J}^{\frac{1}{2}} \left (\Phi_i^P(x_{-i}) , \Phi_i^Q(y_{-i})\right ) \right )

2. Q-distance

\mathbb{D}^P(x,i)\triangleq 1 - \Phi^P_i(x_{-i})\vert_{x_i}

3. Dissonance

Qnet

  1. Distance Metric
  2. Dissonance calculation

masterDarpa_W_Corvey_Nov2022_ishanu

By Ishanu Chattopadhyay

masterDarpa_W_Corvey_Nov2022_ishanu

Opinion dynamics and belief shift using recursive decision forests

  • 143

More from Ishanu Chattopadhyay