## Questions and Answers

In the context of Bayesian decision theory, what does the term 'state of nature' refer to?

The possible classes or categories an observation can belong to

What assumption does Bayesian decision theory make about the decision problem?

Both A and B

What is the role of probability in Bayesian decision theory?

To quantify the trade-offs between various classification decisions

What is the purpose of a decision rule in Bayesian decision theory?

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In the fish classification example, what do the symbols $\omega_1$ and $\omega_2$ represent?

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What is the purpose of considering prior probabilities in Bayesian decision theory?

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What does the probability of error depend on when making a decision between two hypotheses (m1 and m2) based solely on their prior probabilities?

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If the prior probabilities P(m1) and P(m2) are equal, what is the probability of making the correct decision?

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What is the purpose of incorporating additional information, such as a lightness measurement x, into the decision-making process?

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What is the 'class-conditional probability density function' p(x|m) referring to?

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What is meant by the 'state of nature' in the context of this problem?

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What is the purpose of using a probabilistic approach in this classification problem?

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What does the term 'prior probability' refer to?

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According to the given scenario, what is the decision rule suggested if one must decide the type of fish without being allowed to see it?

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Why might the decision rule of always choosing the option with the higher prior probability seem strange when judging many fish?

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What is the assumption made in the given scenario regarding the cost or consequence of incorrect classification?

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What is the 'state of nature' referred to in the context of this scenario?

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What factors might influence the prior probabilities of catching sea bass or salmon?

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## Study Notes

### Bayesian Decision Theory

- A fundamental statistical approach to pattern classification, quantifying tradeoffs between classification decisions using probability and costs.
- Assumes the decision problem is posed in probabilistic terms, and all relevant probability values are known.

### Introduction to Pattern Classification

- In pattern classification, a decision must be made between two possible states: sea bass or salmon.
- Let m denote the state of nature, with m = m1 for sea bass and m = m2 for salmon.

### Prior Probabilities

- P(m1) and P(m2) are the prior probabilities of sea bass and salmon, respectively.
- These probabilities reflect our prior knowledge of how likely we are to get a sea bass or salmon before the fish appears.
- The prior probabilities sum to one, i.e., P(m1) + P(m2) = 1.

### Decision Rule

- When deciding with little information, a logical decision rule is to decide m1 if P(m1) > P(m2), and otherwise decide m2.
- This rule makes sense for a single decision, but may not be suitable for repeated decisions.

### Impact of Prior Probabilities

- If P(m1) is much greater than P(m2), the decision in favor of m1 will be right most of the time.
- If P(m1) = P(m2), there is only a fifty-fifty chance of being right.
- The probability of error is the smaller of P(m1) and P(m2).

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