Uncertainty, Ambiguity, and Noise in Perception

Why probability?

Let’s start with a broad question: what makes probability theory the right tool to model perception? To put it simply, the world is much fuzzier and less certain than it seems. Given the chance to design a system that functions in an uncertain world, the optimal thing to do would be to have it explicitly reason about probable and improbable things. But if the brain is not designed per se, then is there more than human ego to make us think that it would function in a similarly optimal way? In a future post, I will outline the kinds of evidence there are for this, but for now let’s just assume (or let’s hypothesize) that the brain is at least trying to do the right thing, and is getting pretty close. This is known as the Bayesian Brain Hypothesis,[1] and its roots go at least as far back as Hermann von Helmholtz, who in 1867 described perception as a process of “unconscious [probabilistic] inference”.[2]

A similar story can be told for cognition: where are the crisp boundaries between concepts like ‘cup’ and ‘mug’? What makes something ‘art’? When is thought precise? Instead, the mind works largely by induction, generalization, simulation, and reasoning, all of which are naturally formalized in probabilistic terms.

It should be clarified that a probabilistic brain is not necessarily a random brain. If you are predicting the outcomes of a weighted coin flip that is heads 55% of the time and tails 45% of the time, the ideal probabilistic response would be to guess heads every time. What’s important is that a probabilistic system is not very confident in that guess. Later, we will see examples of ‘sampling’ algorithms where generating random numbers is a tool for reasoning probabilistically, but other algorithms achieve probabilistic answers with no randomness at all!

Whence comes the uncertainty

Even in early sensory processing, the brain faces substantial uncertainty about the ‘true’ state of the world. I like to break this problem down into two parts that I call ambiguity and noise. Using visual terms,

  1. ambiguity arises when many different ‘world states’ can give rise to exactly the same image.
  2. noise refers to the fact that subtly different images might evoke the same activity in the brain.

"illusion" in which an object casts a shadow on a chess board. The light squares in shadow are the exact same color as the dark squares not in shadow, but are perceived differently. The chessboard image shown here is a classic example of how ambiguity arises and how the brain resolves it. White squares in the cylinder’s shadow are exactly the same color as dark squares outside the shadow, yet they are perceived differently. To put it another way, exactly the same image (a gray square) was created from different world states (light square + shadow or dark square + no shadow). It is easy to think that this is a trivial problem since we so quickly and effortlessly perceive the true nature of the scene, but these same mechanisms make us susceptible to other kinds of illusions.

Noise occurs partly because neurons are imperfect, so the same image on the retina evokes different activity in visual cortex at different times. Noise also occurs when irrelevant parts of a scene are changing, like dust moving on a camera lens, or small movements of an object while trying to discern what it is. These are called “internal noise” and “external noise” respectively. Strictly speaking, noise as described here is not by itself an issue; the same input may map to many different patterns of neural activity in the brain, but as long as we can invert the mapping there is no problem! The only time we cannot do this inversion is when the noise results in overlapping neural patterns for different states in the world.1 That is, noise is only an issue if it results in ambiguity!

Cartoon of mappings from world states "A" "B" and "C" to brain states. All three mappings contain "noise" so that a single world state maps to multiple brain states. This is only a problem when the noise causes two world states to be mapped to the same brain state.
A, B, and C refer to different world states (different images) on the left, and different brain states (patterns of neural activity) on the right. Despite noise (gray ovals), C can be distinguished from both A and B. Where A and B overlap, the mapping from the world to the brain cannot be inverted, so there would be uncertainty in whether A or B was the cause.

Noise explains why there is a limit to our ability to make extremely fine visual distinctions, like the difference between a vertical line and a line tilted off of vertical by a small fraction of a degree – similar enough inputs will have indistinguishable patterns of neural activity.

Finally, it is important to note that an information bottleneck in the visual system also indirectly implies a kind of noise.2 Information bottlenecks arise whenever a ‘channel’ can take on fewer states than the messages sent across it; a classic example is that the optic nerve has too few axons to transmit the richness of all retinal patterns.

Visualization of how a complicated scene passed through an "information bottleneck" results in a loss of detail.
Think of an information bottleneck as losing detail about a scene. The logic may seem backwards, but the lack of detail in the right implies that some scenes on the left are forced to become indistinguishable after passing through the bottleneck.

The fact that an information bottleneck implies uncertainty is counter-intuitive at first since it makes no statement about how any particular image or scene is affected.3 Perhaps it is more intuitive to think of an information bottleneck as a kind of continuous many-to-one mapping, where different inputs are forced to map to similar neural states. As seen in the overlap between states “A” and “B” in the second illustration above, a many-to-one mapping cannot be inverted, so there must be uncertainty about the true scene.

Wrapping up

Wherever there is uncertainty, the optimal thing to do is to play the odds and think probabilistically. As I hope to have conveyed in this post, uncertainty about the ‘true’ state of the world is a ubiquitous problem for perception, though it may not seem so introspectively. Future posts will elaborate on the process of inference, which resolves such uncertainty and settles on the most likely interpretation(s) of an image or scene.


Footnotes

1. advanced readers may recognize this as the logic behind information-limiting correlations.[3]

2. for those familiar with information theory, this is because an upper bound on the mutual information between input (an image) and evoked neural activity implies a lower bound on the conditional entropy of the neural activity given the input. For those unfamiliar with information theory, stay tuned for a future post =)

3. rate distortion theory allows one to make more concrete statements about this mapping, but only having assumed a loss function that quantifies how bad it is to lose some details relative to others

References

[1] Knill, D. C., & Pouget, A. (2004). The Bayesian brain: the role of uncertainty in neural coding and computation. Trends in Neurosciences, 27(12), 712–9. http://doi.org/10.1016/j.tins.2004.10.007

[2] von Helmholtz, Hermann. (1867). Handbuch der physiologischen Optik.

[3] Moreno-Bote, R., Beck, J. M., Kanitscheider, I., Pitkow, X., Latham, P., & Pouget, A. (2014). Information-limiting correlations. Nature Neuroscience, 17(10), 1410–1417. http://doi.org/10.1038/nn.3807