Difference between revisions of "Bayesian decision theory"

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'''Bayesian decision theory''' refers to a [[decision theory]] which is informed by [[Bayesian probability]]. It is a statistical system that tries to quantify the tradeoff between various decisions, making use of probabilities and costs. An agent operating under such a decision theory uses the concepts of Bayesian statistics to estimate the [[expected value]] of its actions, and update its expectations based on new information. These agents can and are usually referred to as estimators.
 
'''Bayesian decision theory''' refers to a [[decision theory]] which is informed by [[Bayesian probability]]. It is a statistical system that tries to quantify the tradeoff between various decisions, making use of probabilities and costs. An agent operating under such a decision theory uses the concepts of Bayesian statistics to estimate the [[expected value]] of its actions, and update its expectations based on new information. These agents can and are usually referred to as estimators.
  
Consider any kind of probability distribution - such as the weather for tomorrow (encompassing several variables such as humidity, rain or temperature). From a Bayesian perspective, that represents a [[priors|prior]] distribution. That is, it represents how we believe ''today'' the weather is going to be tomorrow. This contrasts with frequentist inference, the classical probability interpretation, where conclusions about an experiment are drawn from a set of repetitions of such experience, each producing statistically independent results. For a frequentist, a probability function would be a simple distribution function with no special meaning. A Bayesian decision rule is one that consistently tries to make decisions which minimize the risk of the probability distribution. Such risk can be seen as the difference between the prior beliefs and the real outcomes - the prediction and the actual weather tomorrow.
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From the perspective of Bayesian decision theory,  any kind of probability distribution - such as the distribution for tomorrow's weather - represents a [[priors|prior]] distribution. That is, it represents how we believe ''today'' the weather is going to be tomorrow. This contrasts with frequentist inference, the classical probability interpretation, where conclusions about an experiment are drawn from a set of repetitions of such experience, each producing statistically independent results. For a frequentist, a probability function would be a simple distribution function with no special meaning.
  
Computer algorithms such as those studied in the subject of [[Machine learning]] can also use Bayesian methods. Besides these explicit implementations, it also has been observed that naturally evolved [http://en.wikipedia.org/wiki/Bayesian_brain nervous systems] mirror these probabilistic methods when they adapt to an uncertain environment. Such systems, like the human brain, seem to construct Bayesian models of their environment and then use these models to make decisions. Such models and distributions are constantly being updated and reconfigured according to feedback from the environment.
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Suppose we know that we're going to meet a friend tomorrow, and that there's a .5 chance for it raining. If we are choosing between various options of how to spend the next day, with the pleasantness of some of the options (such as going to the park) being affected by the possibility of rain, we can [http://lesswrong.com/lw/8uj/compressing_reality_to_math/ assign values to the different options with or with rain]. We can then pick the option whose expected value is the highest, given the probability of rain.
  
==Bayesian reasoning in everyday life==
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One definition of [[Rationality rationality]], used both on Less Wrong and in economics and psychology, is behavior which obeys the rules of Bayesian decision theory. Due to computational constraints, this is impossible to do perfectly, but naturally evolved brains [http://en.wikipedia.org/wiki/Bayesian_brain do seem to mirror] these probabilistic methods when they adapt to an uncertain environment. Although they can't follow it perfectly and often err, brains do seem to often approximate the correct theory by constructing Bayesian models of their environment and then using these models to make decisions. Such models and distributions are constantly being updated and reconfigured according to feedback from the environment.
What Less Wrong refers to as [[Rationality]] is an effort to make conscious thoughts and decisions a better approximation of Bayesian decision theory, in order to better understand the world and achieve one's goals. This process can involve [http://lesswrong.com/lw/8uj/compressing_reality_to_math/ applying math to reality] in a simplified and extremely useful way:
 
  
You receive a phone call from a friend inviting you to do something tomorrow - playing a board game, as you usually do. The question arises: where to play? At your apartment or ouside in the park?
 
 
You check the weather forecast and conclude there's a 50% chance of rain. Since playing the game implies being stuck in the place you chose, you add another decision to your model - to just talk instead of playing. That way you can move according to the weather.
 
 
If you attribute different preferences to both the activity and the location while considering the influence of the probability of raining, you can build a model to help you decide and clarify the problem. That way you can define both variables in a more informed and balanced mode, thus being able to make better decisions.
 
  
 
==Further Reading & References==
 
==Further Reading & References==

Revision as of 16:35, 27 October 2012

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Bayesian decision theory refers to a decision theory which is informed by Bayesian probability. It is a statistical system that tries to quantify the tradeoff between various decisions, making use of probabilities and costs. An agent operating under such a decision theory uses the concepts of Bayesian statistics to estimate the expected value of its actions, and update its expectations based on new information. These agents can and are usually referred to as estimators.

From the perspective of Bayesian decision theory, any kind of probability distribution - such as the distribution for tomorrow's weather - represents a prior distribution. That is, it represents how we believe today the weather is going to be tomorrow. This contrasts with frequentist inference, the classical probability interpretation, where conclusions about an experiment are drawn from a set of repetitions of such experience, each producing statistically independent results. For a frequentist, a probability function would be a simple distribution function with no special meaning.

Suppose we know that we're going to meet a friend tomorrow, and that there's a .5 chance for it raining. If we are choosing between various options of how to spend the next day, with the pleasantness of some of the options (such as going to the park) being affected by the possibility of rain, we can assign values to the different options with or with rain. We can then pick the option whose expected value is the highest, given the probability of rain.

One definition of Rationality rationality, used both on Less Wrong and in economics and psychology, is behavior which obeys the rules of Bayesian decision theory. Due to computational constraints, this is impossible to do perfectly, but naturally evolved brains do seem to mirror these probabilistic methods when they adapt to an uncertain environment. Although they can't follow it perfectly and often err, brains do seem to often approximate the correct theory by constructing Bayesian models of their environment and then using these models to make decisions. Such models and distributions are constantly being updated and reconfigured according to feedback from the environment.


Further Reading & References

  • Berger, James O. (1985). Statistical decision theory and Bayesian Analysis (2nd ed.). New York: Springer-Verlag. ISBN 0-387-96098-8. MR 0804611
  • Bernardo, José M.; Smith, Adrian F. M. (1994). Bayesian Theory. Wiley. ISBN 0-471-92416-4. MR 1274699

See also