Noise, Fake News, and Tenacious Bayesians

Abstract

A modeling framework, based on the theory of signal processing, for characterizing the dynamics of systems driven by the unraveling of information is outlined, and is applied to describe the process of decision making. The model input of this approach is the specification of the flow of information. This enables the representation of (i) reliable information, (ii) noise, and (iii) disinformation, in a unified framework. Because the approach is designed to characterize the dynamics of the behavior of people, it is possible to quantify the impact of information control, including those resulting from the dissemination of disinformation. It is shown that if a decision maker assigns an exceptionally high weight on one of the alternative realities, then under the Bayesian logic their perception hardly changes in time even if evidences presented indicate that this alternative corresponds to a false reality. Thus, confirmation bias need not be incompatible with Bayesian updating. By observing the role played by noise in other areas of natural sciences, where noise is used to excite the system away from false attractors, a new approach to tackle the dark forces of fake news is proposed.

Keywords: communication theory; confirmation bias; disinformation; electoral competition; marketing; noise; signal processing.

Figure 1

Impact of fake news. The two alternatives are represented by the values X = 0 and X = 1, but at time zero the uncertainty is at its maximum (the opinion is equally split between the two alternatives). Then noisy observation begins, and the prior opinion is updated in time (expressed in years). Plotted here are sample paths for posterior probabilities that X = 1. In all simulations the correct choice is (secretly) preselected to be the one corresponding to X = 1. Depending on how the Brownian noise develops the best inference develops differently, as shown here on the left panel, but by waiting sufficiently long, ultimately enough truth is learned and all decisions on the left panel will converge to the correct one, typically by time t≈10. The impact of disinformation, released at a random point in time (at t = 0.6 here), is shown on the right panel. The disinformation is released at a constant rate in time, intended to mislead the decision maker to select the choice corresponding to X = 0. The rate of release is taken to be sufficiently strong that if decision makers are unaware of the disinformation then they will be led to making the incorrect decision (X = 0), even though the simulator had preselected the correct decision to be that corresponding to X = 1.

Figure 2

Probability of winning a future election. The winning probability of a candidate in a two-candidate electoral competition, to take place in 1 year time, is plotted. On the left panel, the probabilities are shown as a function of today's support rating S for two different values of the information-flow rate: σ = 0.15 (purple) and σ = 0.95 (magenta). If today's poll S were an indicator for the winning probability, then it would be given by a straight line (brown), but in reality the probability of winning a future election of a candidate, whose current support rate is S>50%, is always greater than S. On the right panel, the winning probability is shown as a function of the information-flow rate σ of a candidate whose support rate today is S = 52%. If the candidate is leading the poll, then the best strategy is to reveal as little information relevant to the election as possible.

Figure 3

Tenacious Bayesian behavior in a binary decision making. The two alternatives are represented by the values X = 0 and X = 1, and in all simulations, the “correct” decision is preselected to correspond to the choice X = 1. On the left panel, four sample paths for the a posteriori probability that X = 1 are shown when the a priori probability for the incorrect decision X = 0 is given by 99%. Although the evidences presented by the observed time series consistently indicate that X = 0 is not the correct choice, the decision makers in these simulations have hardly changed their views even after 2 years, in spite of following the Bayesian logic. In contrast, if the prior probability for the incorrect choice X = 0 is reduced to, say, 80%, then with the same amount of information-revelation rate (σ = 1 in both cases) there will be more variabilities, as shown on the right panel.

Figure 4

Separation distance under Bayesian updating. The polarity, or distance δ of two decision makers, when they are provided with an identical set of noisy information, has a tendency to increase under the Bayesian updating, even though on average it decreases in time. Five sample paths are shown here for two different choices of σ. On the left panel the information flow rate (signal to noise ratio) is taken to be σ = 0.2. Simulation studies (not shown here) indicate that in this case the upward trend persists for some 40 years in about 50% of the sample paths, and the separation distance is typically reduced to half of the initial value after about 100 years. When the information flow rate is increased eleven-fold to σ = 2.2, polarized Bayesian learners are forced to converge a lot quicker, as shown on the right panel, where the separation is reduced to half of its initial value typically within 2 years.

Figure 5

Simulating alternative fact. Three alternatives are represented by the values X = 1, X = 2, and X = 3. Plotted here are sample paths for the mean values of X subject to information process ξ_t = σXt+ϵ_t, where σ = 2 and {ϵ_t} denotes Brownian noise. In all simulations, the simulator has chosen the alternative X = 2 to be the correct one. On the left panel, all decision makers start with the prior belief that the probability of X = 2 is only 10%, whereas the two other alternatives are equally likely realized at 45% each. Hence the initial mean of X equals 2. Initially, their views tend to converge either to X = 1 or X = 3; but over time, sufficient facts are revealed that they all converge to the correct choice made by the simulator. But what if they do not believe in the “truth” at all? On the right panel, the simulator has again chosen X = 2 to represent the true value, but the decision makers assume that this is impossible and that the two other alternatives are equally likely realized at 50% each. Hence again the initial mean of X equals 2. In this case, the decision makers' views tend to converge quickly to one of the two “false” alternatives X = 1 or X = 3. These two beliefs are, however, only quasi-stable; the views will never converge indefinitely. Instead, their views tend to flip back and forth between the alternatives X = 1 and X = 3, but never converging to either one, and certainly never come close to the correct alternative X = 2.

Figure 6

Tenacious Bayesian binary decision with enhanced noise. What happens to the tenacious Bayesian behavior of the left panel in Figure 3 if the noise level is enhanced in such a way that decision makers are unaware of it? As in the example of Figure 3, the decision makers here have their priors set at 99% for the choice X = 0, but the simulator has chosen X = 1 to be the correct choice. Plotted here are sample paths for the a posteriori probability that X = 1. In the left panel the noise level is doubled as compared to that of the left panel in Figure 3; whereas in the right panel it is quadrupled. In both cases, the information flow rates are the same as that chosen in Figure 3, so no more reliable information is provided here for the Bayesian decision makers. Nevertheless, the introduction of unknown noise enhances the chance of arriving at the correct decision considerably sooner, with positive probability. In particular, if the noise level is quadrupled, then there is about 15% chance that the noise will assist such an escape from a false reality after two years.