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3.5. Theorem of hypotheses (Bayes formula)

Lecture



A consequence of the multiplication theorem and the full probability formula is the so-called theorem of hypotheses, or the Bayes formula.

Let's set the following task.

There is a complete group of incompatible hypotheses.   3.5.  Theorem of hypotheses (Bayes formula) . The probabilities of these hypotheses to experience are known and equal, respectively.   3.5.  Theorem of hypotheses (Bayes formula) . An experiment was made, as a result of which an event was observed   3.5.  Theorem of hypotheses (Bayes formula) . The question is, how should the probabilities of hypotheses be changed due to the occurrence of this event?

Here, essentially, it is about finding the conditional probability   3.5.  Theorem of hypotheses (Bayes formula) for each hypothesis.

From the multiplication theorem we have:

  3.5.  Theorem of hypotheses (Bayes formula)   3.5.  Theorem of hypotheses (Bayes formula) ,

or by discarding the left side,

  3.5.  Theorem of hypotheses (Bayes formula)   3.5.  Theorem of hypotheses (Bayes formula) ,

from where

  3.5.  Theorem of hypotheses (Bayes formula)   3.5.  Theorem of hypotheses (Bayes formula) .

Expressing   3.5.  Theorem of hypotheses (Bayes formula) using the total probability formula (3.4.1), we have:

  3.5.  Theorem of hypotheses (Bayes formula)   3.5.  Theorem of hypotheses (Bayes formula) . (3.5.1)

Formula (3.5.1) is also called the Bayes formula or hypothesis theorem.

Example 1. The device can be assembled from high-quality parts and from parts of normal quality; in general, about 40% of devices are assembled from high-quality parts. If the device is assembled from high-quality parts, its reliability (probability of failure-free operation) during   3.5.  Theorem of hypotheses (Bayes formula) equal to 0.95; if from parts of normal quality, its reliability is 0.7. The instrument was tested over time.   3.5.  Theorem of hypotheses (Bayes formula) and worked flawlessly. Find the probability that it is assembled from high-quality parts.

Decision. Two hypotheses are possible:

  3.5.  Theorem of hypotheses (Bayes formula) - the device is assembled from high quality parts,

  3.5.  Theorem of hypotheses (Bayes formula) - the device is assembled from parts of normal quality.

The probability of these hypotheses to experience:

  3.5.  Theorem of hypotheses (Bayes formula) .

As a result of the experience, an event was observed.   3.5.  Theorem of hypotheses (Bayes formula) - the device worked smoothly time   3.5.  Theorem of hypotheses (Bayes formula) .

Conditional probabilities of this event under hypotheses   3.5.  Theorem of hypotheses (Bayes formula) and   3.5.  Theorem of hypotheses (Bayes formula) are equal:

  3.5.  Theorem of hypotheses (Bayes formula)

By the formula (3.5.1) we find the probability of the hypothesis   3.5.  Theorem of hypotheses (Bayes formula) after the experience:

  3.5.  Theorem of hypotheses (Bayes formula) .

Example 2. Two shooters independently fire at one target, each making one shot. The probability of hitting the target for the first arrow is 0.8, for the second 0.4. After shooting a single hole was found in the target. Find the probability that this hole belongs to the first arrow.

Decision. Before the experiment, the following hypotheses are possible:

  3.5.  Theorem of hypotheses (Bayes formula) - neither the first nor the second shooter will fall,

  3.5.  Theorem of hypotheses (Bayes formula) - both arrows will hit,

  3.5.  Theorem of hypotheses (Bayes formula) - the first shooter will fall, and the second will not,

  3.5.  Theorem of hypotheses (Bayes formula) - the first shooter will not fall, and the second will.

The probability of these hypotheses:

  3.5.  Theorem of hypotheses (Bayes formula)

Conditional probabilities of the observed event   3.5.  Theorem of hypotheses (Bayes formula) with these hypotheses are equal:

  3.5.  Theorem of hypotheses (Bayes formula)

After experiencing the hypothesis   3.5.  Theorem of hypotheses (Bayes formula) and   3.5.  Theorem of hypotheses (Bayes formula) become impossible and the probabilities of hypotheses   3.5.  Theorem of hypotheses (Bayes formula) and   3.5.  Theorem of hypotheses (Bayes formula) will be equal to:

  3.5.  Theorem of hypotheses (Bayes formula)

Consequently, the probability that the hole belongs to the first arrow is equal to   3.5.  Theorem of hypotheses (Bayes formula) .

Example 3. Some object is monitored using two observation stations. The object can be in two different states.   3.5.  Theorem of hypotheses (Bayes formula) and   3.5.  Theorem of hypotheses (Bayes formula) , casually going from one to another. By long-term practice it has been established that approximately 30% of the time an object is in   3.5.  Theorem of hypotheses (Bayes formula) and 70% are able   3.5.  Theorem of hypotheses (Bayes formula) . Observation station No. 1 transmits erroneous information in approximately 2% of all cases, and observation station No. 2 in 8%. At some point, Observation Station No. 1 reported: the object is in a state   3.5.  Theorem of hypotheses (Bayes formula) , and the observation station number 2: the object is in a state   3.5.  Theorem of hypotheses (Bayes formula) .

The question is: which of the messages to believe?

Decision. Naturally, to believe the message that is more likely to be true. Let's apply the Bayes formula. To do this, we make hypotheses about the state of the object:

  3.5.  Theorem of hypotheses (Bayes formula) - the object is in a state   3.5.  Theorem of hypotheses (Bayes formula) ,

  3.5.  Theorem of hypotheses (Bayes formula) - the object is in a state   3.5.  Theorem of hypotheses (Bayes formula) .

Observed event   3.5.  Theorem of hypotheses (Bayes formula) consists of the following: station number 1 reported that the object is in a state   3.5.  Theorem of hypotheses (Bayes formula) , and station number 2 - that he is in a state   3.5.  Theorem of hypotheses (Bayes formula) . Probabilities of hypotheses prior to experience

  3.5.  Theorem of hypotheses (Bayes formula)

Find the conditional probabilities of the observed event.   3.5.  Theorem of hypotheses (Bayes formula) with these hypotheses. Under hypothesis   3.5.  Theorem of hypotheses (Bayes formula) for an event to happen   3.5.  Theorem of hypotheses (Bayes formula) , you need the first station to transmit the correct message, and the second one is erroneous:

  3.5.  Theorem of hypotheses (Bayes formula) .

Similarly

  3.5.  Theorem of hypotheses (Bayes formula) .

Applying the Bayes formula, we find the probability that the true state of the object is   3.5.  Theorem of hypotheses (Bayes formula) :

  3.5.  Theorem of hypotheses (Bayes formula) ,

those. of the two messages, the message of the first station is more likely.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis