Decision Theory and Bayesian Inference

An organization uses a spam filtering software to block email messages that may potentially be spam messages. The spam filter can be set to one of two security modes: High-Security-Mode () or Low-Security-Mode ().

Extensive evaluation using a benchmark corpus consisting exclusively of spam messages yields the following performance statistics for the spam filter:

• 96% of the spam messages are blocked in the High-Security-Mode.
• 90% of the spam messages are blocked in the Low-Security-Mode.

Extensive evaluation using a benchmark corpus consisting exclusively of non-spam (legitimate) messages yields the following performance statistics for the spam filter:

• 10% of the non-spam messages are blocked in the High-Security-Mode
• 4% of the non-spam messages are blocked in the Low-Security-Mode

The organization estimates that 80% of the messages that it receives are spam messages.

(a) Let  denote the conditional probability that a message that is not blocked by the spam filter operating in the High-Security-Mode is actually a spam message. Estimate .

(b) Let  denote the conditional probability that a message that is blocked by the spam filter operating in the High-Security-Mode is actually not a spam message. Estimate .

(c) Let  denote the conditional probability that a message that is not blocked by the spam filter operating in the Low-Security-Mode is actually a spam message. Estimate .

(d) Let  denote the conditional probability that a message that is blocked by the spam filter operating in the Low-Security-Mode is actually not a spam message. Estimate .

(e) If the cost of not blocking a spam message is $1 and the cost of blocking a non-spam message is $10, should the organization operate the spam filter in the High-Security-Mode? Why?

(f) Recall that the cost of not blocking a spam message is $1. At least how high should the cost of blocking a non-spam message be for a risk-neutral rational decision maker to prefer operating the spam filter in the Low-Security-Mode?

(g) Let  be the amortized cost per message of operating the spam filter. Write a short memo the CIO of the organization explaining at most how high  could be for the organization to use the spam filter with the specified performance?