580.691 learning theory reza shadmehr bayesian estimation in everyday behaviors text to read:...
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580.691 Learning Theory
Reza Shadmehr
Bayesian estimation in everyday behaviors
Text to read: chapter 2
The influence of priorsWhen at the coffee shop the attendant hands you a cup full of tea, your brain needs to guess how heavy the drink is. This guess need to be accurate, otherwise you would have trouble grasping the cup (activating your finger muscles in a way that they does not let it slip out of your hand), and holding it steady (activating your arm muscles so the cup does not rise up in the air or fall down). The only cue that you have is the size information provided by vision. Fortunately, the other piece of information is the prior experience that you have had with cups of tea. The fact that most people have little trouble holding cups that are handed to them in coffee shops suggests that they are making accurate estimates of weight. How are they making these guesses?
x
p x
s
p s x
Suppose we label the weight of the cup of tea as
We have some prior belief about the distribution of these weights,
We have some prior belief about the relationship between visual property (size) of a tea cupand its weight
.
p s x p xp x s
p sThen our guess about weight of this particular cup of tea should be
based on the posterior distribution
The effect of prior beliefs during interactions with everyday objects. A) Volunteers were asked to use their fingers to lift up a small, medium, or a large box. The instrumented device measured grip and load forces. The three boxes were the same weight. B) People tended to produce the smaller grip and load forces for the smallest box, resulting in large lift velocities for the largest box. (From (Gordon et al., 1991)).
Left. The ball starts with a non-zero velocity from a given height and falls in 0g or 1g gravity. In the 0g scenario (i.e., in space), the recently arrived astronaut will use a 1g internal model to predict the ball’s trajectory, expecting it to arrive earlier (dashed line). Right. EMG activity from arm muscles of an astronaut in 0g and 1g. In 0g, the arm muscles activate sooner, suggesting that the astronaut expected the ball to arrive sooner than in reality. (From McIntyre et al. (2001))
i e eE
i e
e i ii e
e
p pp
p
Guess a person’s life expectancy:Insurance agencies employ actuaries to make predictions about people’s life spans (the age at which they will die). If you were assessing an insurance case for an 18 year old man, what would you predict for his life span? How about a 39 year old man? 61 year old man? 83 year old man? 96 year old man?
Survey results from about 150 undergraduates. From Griffiths and Tenenbaum (2006) Psych. Sci.
0 20 40 60 80 100
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
p t x p xp x t
p t
Life span
Current age
2
0
1 if , 0 otherwise
, , 75, 15 years
t
p t x x tx
p x N
p t p t x p x dx p t x p x dx
This is the prob. of meeting someone who is t years old, given that their life span is x. We assume that for a given life span, we can meet that person with equal probability at all times.
This is the life span distribution for an American male.
Probability of meeting someone who is t years old.
p t
(years)t
0 20 40 60 80 1000
0.005
0.01
0.015
0.02
0.025
30p x t
0 20 40 60 80 1000
0.005
0.01
0.015
0.02
0.025
40p x t
0 20 40 60 80 1000
0.005
0.01
0.015
0.02
0.025 50p x t
0 20 40 60 80 1000
0.01
0.02
0.03
0.04
me
dia
n
me
dia
n
70p x t
0 20 40 60 80 1000
0.02
0.04
0.06 80p x t
0 20 40 60 80 1000
0.02
0.04
0.06
0.08
0.1 90p x t
Pro
babi
lity
of li
fesp
an,
give
n cu
rren
t ag
e
p x t
20 40 60 80 100
20
40
60
80
100
If we have the posterior probability of lifespan for a person of a given age, how do we make a guess about their lifespan? We should pick the median of this probability distribution, because this is the value for which we are going to be right half the time.
0
1if then is median of
2
m
p x t dx m p x t
Current age (yrs)
Med
ian
of p
ost.
pro
b. (
yrs)
