Thinking 1&2 - Bayesian Inference

Question 1

What is thinking?

Answer 1

• Can be described as the flexible organization and manipulation of internal representations
• An important aspect is therefore to understand how representations are formed:
  a) Rationalism vs empiricism debate in 17th century (Descartes Vs Locke & Hume)
• Emphasis of constructivist nature (according to a priori existing concepts) of the human mind vs sensory driven
• Perception is NOT a one-to-one mapping of the physical world into the mind
• Instead, the brain uses ‘algorithms’ and assumption to actively construct an image of the world
• Perceptual illusions, Gestalt laws (law of good continuity, law of closure)

Question 2

Thinking: perceptual illusions

Answer 2

• Brain integrates specific observations with all kinds of contextual information:
  1. The expectancy of the occurrence of a particular ‘object’
• Depending on the narrative context, observers either see the young or the old women
  2. Other available information (e.g. the sound the object makes)
• Ventriloquist effect: the perceived location of a sound is shifted in space by a simultaneously occurring visual stimulus at incongruent location
• McGurk effect: the perceived sound by a spoken syllable is altered by an incongruent visual input of lip-movement

Question 3

Summary of thinking

Answer 3

• Perception, even w.r.t. elementary sensations, is highly ambiguous
• To resolve ambiguity the brain needs to integrate sensory signals from a given modality with contextual information
• The human mind ‘has evolved’ a battery of strategies to deal with this uncertainty – select and integrate information according to “set routines” – which may be prone to failure in certain situations: perceptual illusions!!

Question 4

2 Key people in thinking

Answer 4

1. Hermann von Helmholtz – early experimental and theoretical work on perception
2. Wilhelm Wundt – founder of the first academic instate for Psychology in history
   - Perception as ‘unconscious inference’

Question 5

What is Bayesian cognition?

Answer 5

• One possible way of implementing the contextual integration of specific observations is through Bayesian inference:
• Infer what is going on in the ‘outside world’ on the basis of observations and expectations, i.e. what is likely happening in the outside world?
• Takes the observations (current/new ‘data’) into account
• Takes the expectations into account
• Chooses the scenario / hypothesis that is maximally satisfying both at the same time
• All formulated in (mathematical) probabilities

Question 6

The bayes theorem: what is a probability?

Answer 6

• Kolmogorov axioms in simplified terms:
  1. Probabilities are non-negative (real) numbers between 0 and 1 {0 < p < 1}
  2. The probability of the certain event is 1.
  3. The probabilities of all separate events that comprise a set add up and they add up to p=1.
• E.g. set of balls from 1-7 (in different subsets: one from 1-5, the other from 6-7)
• Probability of any given ball having one of the numbers from 1-5, or from 6-7
• Ball has number form 1-5 = 5/7, ball has number from 6-7 = 2/7
• The probability that any given ball has a number from 1-7 is p = 1

Question 7

What is conditional probability?

Answer 7

• P(B|A): reads “the probability of B given A”, or “the conditional probability of B given A
• P(A|B): reads “the probability of A given B”, or “the conditional probability of A given B”
• A conditional probability is the probability of a particular event within the set of another event

Question 8

What is Bayesian inference?

Answer 8

• Infer from observations to the probability of the hypothesis
• We want to know what the probability of a certain hypothesis is, given the observations we have made
• Inference of ‘ground truth’ (the state of the world) on the basis of (limited) data always comes with uncertainty, since the likelihoods, e.g. p(A|B) are virtually never 0 or 1. The data alone rarely ever tell us with 100% certainty whether Hypothesis 1 or 2 is correct
• In order to make the best judgement / inference under uncertainty, the Bayes theorem is crucial
• For judging which hypothesis is the most “likely”, consideration of the prior probability is often crucial. In other words, individual observations are rarely enough, but we need contextual information, e.g. prior probabilities, to make judgements better

Question 9

Bayes theorem: key facts

Answer 9

• The relative “impact” the prior probability has on the posterior probability also depends on the “strength of the observations
  For instance, if the likelihoods were deterministic then the prior probability is entirely irrelevant
• With more balanced ‘prior probabilities’, their importance is relatively reduced and will depend greatly on the “strength” of the data
• Decisions under uncertainty (e.g. in perception / perceptual decisions, medical diagnosis) require us to not only to look at the data supporting each of the hypotheses, but also at the prior probability of the hypotheses
• Ignoring the prior probability of the hypotheses can lead to seriously wrong decisions (choice of wrong options/hypotheses)
• The Bayes-theorem (we have seen its simplest form today) gives the equation how to calculate the (posterior) probability of a hypothesis given particular observations/data

Question 10

Bayesian cognition: two opposing camps

Answer 10

1. A wide range of studies and computational propositions. Human cognition is based on Bayesian algorithms and this is what brains have evolved to do
2. Daniel Kahneman: humans fail spectacularly in taking prior probabilities into account

Question 11

What is Bayesian perception?

Answer 11

• Use of prior/ contextual information will almost certainly considerably influence the perceived level of threat from one ad the same person
• Bayesian perceptual inference:
• Calculate the ‘posterior probability’ of perceptual hypotheses given the sensory evidence and prior probability of the causes
• Sensory and prior information weighted according to their ‘precision’ to compute posterior belief (subjective percept)
• One way to increase the certainty about objects in the real world from our sensory systems is to combine the signals from different sensory modalities
• The Bayes theorem allows to derive how these are ideally combined to come to a common judgement, in the same way as individual observations and prior probability are “combined”

Question 12

How do we test such a hypothesis?

Answer 12

• Visual and haptic 3D - virtual reality
• Task is to compare the height of two ‘bars’ – presented in VR visually and haptically
• Can measure the accuracy of judgement in visual domain and somatosensory domain (presented separately)
• Can estimate a person’s accuracy in each modality (separately)
• for vision also under various levels of noise
• In combined presentations, measure how much their judgement depends on visual, how much on haptic perception
  1. Probability density function (pdf) – normal distribution
  2. Cumulative density function (cdf) – normal distribution
• The standard deviation is here expressed by the slope: wider pdf expression, flatter slope in cdf

Question 13

The psychometric function

Answer 13

• The psychometric function is often modelled as a ‘cumulative Gaussian function’
• Slope of the psychometric function indicates the ‘performance’ of an observer:
• The better, the steeper
• This is captured by the standard deviation – or variance!
• Measure the psychometric function
• under haptic stimulation only
• under visual stimulation only
• at diff different noise levels!!
  = get different psychometric functions with different slopes!
• Now presenting the stimuli both visual AND haptically, often when the visual stimulus was different from the haptic stimulus, allows to estimate the empirical weight with which each sensory modality contributed to the joint estimate and this followed much exactly the optimal weight, as derived from the Bayes theorem
• Strong evidence that the human brain encodes sensory information probabilistically and applies Bayes-optimal computations for multi-sensory integration to optimally judge stimulus properties

Question 14

Noise characteristics and prior expectations in human visual speed perception

Answer 14

• Presented moving (‘drifting’) Gabor patches composed of different contrast-levels
• It is known that motion perception of low contrast stimuli is less accurate than at high contrasts
• It is known that there is a general bias to underestimate motion speeds, and this is thought to be due to the fact that lower speeds are more present in natural environments
• Hypotheses: low contrast stimuli will be perceived as more ambiguous
• Pp’s will be biased to perceive motions as slower than they are
• Therefore, the perceived motion of low contrast gratings (ambiguous) will be slower than those of high contrast
• Hypotheses were confirmed.
  A Bayesian observer model (solid lines in the right figure) could explain the behavioural data (filled dots) for different motion speeds and contrast levels sufficiently well and better than other models
• Showed clear and strong evidence that human observers are capable of integrating vision and touch signals as expected from a “Bayes-optimal observer”
• Showed that a Bayesian model can account for the data of human observers in a speed judgement task

Question 15

Criticisms of Bayesian perception

Answer 15

• There is criticism in that Ernst and Banks did not investigate whether participants integrate sensory signals and prior probability of the causes
• Stocker and Simoncelli could not manipulate the prior probability distribution – but had to assume these / estimate from the data. They only show that a Bayesian model can explain the data very well (and better than other models), yet there may be better models out there
• In general, this question (of perception reflecting Bayes-optimal inference) remains under intense investigation
• There are definitely a range of spectacularly positive results, in particular in the domain of multisensory integration, also studies that addressed the integration of different sensory modalities and prior probability
• Multisensory illusions (e.g. ventriloquist and McGurk effect) can be explained by Bayesian integration of the senses
• But there are also methodological shortcomings in studies investigating this for the combination of true prior probabilities and sensory information (i.e. the narrower sense of Bayesian perception). The question remains open