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Bayesian inference is a statistical tool that can be applied to motor learning, specifically to adaptation. adaptation is a short-term learning process involving gradual improvement in performance in response to a change in sensory information. Bayesian inference is used to describe the way the nervous system combines this sensory information with prior knowledge to estimate the position or other characteristics of something in the environment. Bayesian inference can also be used to show how information from multiple senses (e.g. visual and proprioception) can be combine for the same purpose. In either case, Bayesian inference dictates that the estimate is most influenced by whichever information is most certain.

Statistical Overview
Bayes' Theorem states


 * $$P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}. \,$$

In the language of Bayesian statistics, $$P(A|B)$$, or probability of A given B, is called the posterior, while $$P(B|A)$$ and $$P(A)$$ are the likelihood and the prior probabilities, respectively. $$P(B)$$ is a constant scaling factor which allows the posterior to be between zero and one). Translating this into the language of motor learning, the prior represents previous knowledge about the physical state of the thing being observed, the likelihood is sensory information used to update the prior, and the posterior is the nervous system's estimate of the physical state. Therefore, for adaptation, Bayes' Theorem can be expressed as

estimate = previous knowledge × sensory information/scaling factor

The 3 terms in the equation above are all probability distributions. To find the estimate in non-probabilistic terms, a weighted sum can be used.


 * $$E = \frac{W_p S + W_s P}{W_p + W_s}.

$$

where $$E$$ is the estimate, $$S$$ is sensory information, $$P$$ is previous knowledge, and the weighting factors $$Wp$$ and $$Ws$$ are the variances of $$S$$ and $$P$$, respectively. Variance is a measure of uncertainty in a variable, so the above equation indicates that higher uncertainty in sensory information causes previous knowledge to have more influence on the estimate and vice versa.

A more rigorous mathematical description of Bayesian inference is available here and here.

Integration of Prior Knowledge and Sensory Information
In this case, a person uses Bayesian inference to create an estimate that is a weighted combination of his current sensory information and the previous knowledge. Consider Bayesian inference as applied to tennis. If you are playing tennis against a familiar opponent who likes to serve such that the ball strikes on the sideline, your prior would lead you to place your racket above the sideline to return the serve. However, when you see the ball moving toward you, it may appear that it will land closer to the middle of the court. Rather than completely following this sensory information or completely following the prior, you would move to a location between the sideline (suggested by the prior) and the point where your eyes indicate the ball will land. Another major point in Bayesian inference is that the estimate will be closer to the physical state suggested by sensory information if it has a high degree of certainty and will be closer to the state of the prior if the sensory information is more uncertain than the prior. Extending this to the tennis example, a player facing an opponent for the first time would have little certainty in his/her previous knowledge of the opponent and would therefore have an estimate weighted more heavily on visual information concerning ball position. Alternatively, if one were familiar with one's opponent but were playing in foggy or dark conditions that would hamper sight, sensory information would be less certain and one's estimate would rely more heavily on previous knowledge.

Reaching
Adaptation to visual shifts in position during reaching follows Bayesian inference. In a study in 2004, subjects moved their right hands toward a target, but were blocked from seeing their actual hand movement. Instead, they saw their hand and the target represented on a screen, in a similar manner to seeing a cursor move toward a target on a computer screen when moving the mouse. However, the hand position represented on the computer screen is shifted horizontally a small distance from the actual hand position. The subject began to adapt to this shift and was soon able to move the hand such that the cursor on the screen was over the target. The subjects did 1,000 reaches with a 1-cm shift, establishing the 1-cm shift as their previous knowledge. Next, the subjects did 1,000 more reaches with the position of the cursor shift 2-cm from hand position. however, instead of being one small dot, the cursor was sometimes represented by different numbers of dots, creating different levels of uncertainty in the current visual (sensory) information (see figure). For lower levels of uncertainty (i.e. one dot) the subjects adapted by moving their hands almost 2-cm to counteract the shift, and the previous knowledge of a 1-cm shift had little influence. For higher levels of uncertainty, the hand's adaptation was closer to 1-cm, indicating that the previous knowledge of a 1-cm shift had more influence on the adaptation. This supports the Bayesian idea that sensory information with more certainty will have greater influence on a person's adaptation to shifted sensory feedback.

Bayesian inference also explains how we adapt the force exerted during reaching. Will add more on this

Recent studies of reaching... Wei article(s) mentioned here

Posture
More recently, Bayesian inference has been found to play a part in adaptation of posture control. In one study, for example, subjects use a Wii Balance Board to do a surfing task in which they must move a cursor representing their center of pressure on a screen. Summarize Dokka and Stevenson articles here. example of how to insert a jpeg

Pic and ref examples from collective animal behavior article. Influence of a predator on the optimal foraging behavior of sticklebacks. Nature 275, pp642-644. 1978. This theory is based on the idea that it becomes difficult for predators to pick out individual prey from groups because the many moving targets create a sensory overload of the predator's visual channel.

presumably provide a higher level of vigilance, it could also allow more time for individual feeding.

Integrating Multiple Senses
Explain other van Beers' papers here as well

Possible Contradictions to Bayesian Inference?
Scheidt paper. Should I even include this?