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Home / Week 10 Exercises / Repeated Readings

Repeated Readings

The questions below are due on Sunday April 21, 2019; 11:00:00 PM.
 
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They say goldfish lack the ability to perform Bayesian updates

A single measurement can often have a lot of uncertainty associated with it. We've already seen evidence of this in the DNA sequencing problem or in everyday life when we think that guy we just saw in front of the CVS in Central Square might have been Nicolas Cage, but then we go and look again (take a second measurement) and realize it is just some random guy. We'll investigate this phenomenon below.

A color sensor for object detection is used to determine if an object is red, green, or blue.

The output of the sensor is "R", "G", or "B"; however the sensor is not perfect. An engineer generated a conditional probability table estimated from lab experiments where an object of each of the color was placed in front of the sensor and many independent measurements were made. The measurements were subsequently pooled together to generate output probabilities given each color. These probabilities are shown in the table below. For example, the sensor has a 0.10 probability of printing a "B" when it is given a red object to read.,

P(Obs="R"|Object=x)P(Obs="G"|Object=x)P(Obs="B"|Object=x)
Object=Red0.600.300.10
Object=Green0.250.500.25
Object=Blue0.050.250.70

  • Assume blue, red, and green objects occur with equal probability.
  • Objects have fixed colors (either red, green, or blue) that do not change in time.

Part 0: Zeroth Reading

Before making any reading, what is the probability of each of three readings? Fill in the probability for each reading to six significant digits. The solution boxes will accept Python numerical expressions so feel free to enter your computations into them. (for example either 0.1666667 or 1.0/6 is valid)

Pr(Obs1="R")
Pr(Obs1="G")
Pr(Obs1="B")

Part 1: First Reading

The sensor is provided an object and reads a "B". Given this new information, fill in the probability for each object color to six significant digits. The solution boxes will accept Python numerical expressions so feel free to enter your computations into them. (for example either 0.1666667 or 1.0/6 is valid)

Pr(Obj=Red | Obs="B")
Pr(Obj=Green | Obs="B")
Pr(Obj=Blue | Obs="B")

Part 2: Second Reading

The sensor slightly repositions itself and again measures the same object as before. It returns a "B". Given this new information, fill in the probability for each object color to six significant digits. The solution boxes will accept Python numerical expressions so feel free to enter your computations into them. (for example either 0.1666667 or 1.0/6 is valid)

Pr(Obj=Red | Obs="BB")
Pr(Obj=Green | Obs="BB")
Pr(Obj=Blue | Obs="BB")

Part 3: Third Reading

The sensor slightly repositions itself and again measures the same object as before. It returns a "B" for the third time. Given this new information, fill in the probability for each object color to six significant digits. The solution boxes will accept Python numerical expressions so feel free to enter your computations into them. (for example either 0.1666667 or 1.0/6 is valid)

Pr(Obj=Red | Obs="BBB")
Pr(Obj=Green | Obs="BBB")
Pr(Obj = Blue | Obs="BBB")

Part 4: Fourth Reading

Based on all the readings you have made so far, what is the probability of seeing each of the following readings the fourth time we observe the object? (Answer to six significant digits.) The solution boxes will accept Python numerical expressions so feel free to enter your computations into them. (for example either 0.1666667 or 1.0/6 is valid)

Pr(Obs4="R" | Obs="BBB")
Pr(Obs4="G" | Obs="BBB")
Pr(Obs4="B" | OBs="BBB")