Bayes Formula
Ruthe Arguillo
There are names for the different terms in the Bayes' Rule formula. The term $\p(BE)$ is often called the"posterior": it is your updated belief of $B$ after you take into account evidence $E$. The term $\p(B)$ is often called the "prior": it was your belief before seeing any evidence. The term $\p(EB)$ is called the update and $\p(E)$ isoften called the normalization constant.
There are several techniques for handling the case where the denominator is not known. One technique is to use the law of total probability to expand out the term, resulting in another formula, called Bayes' Theorem with Law of Total Probability:$$\p(BE) = \frac\p(E \p(E $$
The numbers in this example are from the Mammogram test for breast cancer. The seriousness of cancer underscores the potential for bayesian probability to be applied to important contexts. The natural occurrence of breast cancer is 8%. The mammogram test returns a positive result 95% of the time for patients who have breast cancer. The test resturns a positive result 7% of the time for people who do not have breast cancer. In this demo you can enter different input numbers and it will recalculate.
The Bayes theorem is based on finding P(A B) when P(B A) is given. Here, we will aim at understanding the use of the Bayes rule in determining the probability of events, its statement, formula, and derivation with the help of examples.
Bayes formula exists for events and random variables. Bayes theorem formulas are derived from the definition of conditional probability. It can be derived for events A and B, as well as continuous random variables X and Y. Let us first see the formula for events.
Bayes theorem is a statistical formula to determine the conditional probability of an event. It describes the probability of an event based on prior knowledge of events that have already happened. Bayes rule is named after the Reverend Thomas Bayes and Bayesian probability formula for random events is \(P(AB) = \dfracP(BP(B)\), where
To determine the probability of an event A given that the related event B has already occurred, that is, P(AB) using the Bayes Theorem, we calculate the probability of the event B, that is, P(B); the probability of event B given that event A has occurred, that is, P(BA); and the probability of the event A individually, that is, P(A). Then, we substitute these values into the Bayes formula \(P(AB) = \dfracA)P(A)P(B)\) to determine the probability.
Bayes' theorem is a formula that describes how to update the probabilities of hypotheses when given evidence. It follows simply from the axioms of conditional probability, but can be used to powerfully reason about a wide range of problems involving belief updates.
While this is an equation that applies to any probability distribution over events \(A\) and \(B\), it has a particularly nice interpretation in the case where \(A\) represents a hypothesis \(H\) and \(B\) represents some observed evidence \(E\). In this case, the formula can be written as
A Bayes-type formula is derived for the nonlinear filter where the observation contains both general Gaussian noise as well as Cox noise whose jump intensity depends on the signal. This formula extends the well-known Kallianpur-Striebel formula in the classical non-linear filter setting. We also discuss Zakai-type equations for both the unnormalized conditional distribution as well as unnormalized conditional density in case the signal is a Markovian jump diffusion.
where is a Brownian motion independent of . Under certain conditions on the drift (see [1, 2]), Kallianpur and Striebel derived a Bayes-type formula for the conditional distribution expressed in terms of the so-called unnormalized conditional distribution. In the special case when the dynamics of the signal follows an Itô diffusion
The objective of the paper is in a first step to extend the Kallianpur-Striebel Bayes type formula to the generalized filter setting from above. When there are no jumps present in the observation dynamics (1.4), the corresponding formula has been developed in [4]. We will extend their way of reasoning to the situation including Cox noise.
The remaining part of the paper is organized as follows. in Section 2 we briefly recall some theory of reproducing kernel Hilbert spaces. In Section 3 we obtain the Kallianpur-Striebel formula, before we discuss the Zakai-type equations in Section 4.
Given a Borel measurable function , our nonlinear filtering problem then comes down to determine the least square estimate of , given the observations up to time . In other words, the problem consists in evaluating the optimal filter In this section we want to derive a Bayes formula for the optimal filter (3.8) by an extension of the reference measure method presented in [4] for the purely Gaussian case. For this purpose, define for each with
Proof. Fix . First note that since almost surely, we have by Theorem 2.2 that almost surely. Further, by the independence of the Gaussian process from and from the random measure it follows thatSince for the random variable is Gaussian with zero mean and variance , it follows again by the independence of from and the martingale property of that , and is a probability measure.
Now take and real numbers , , and consider the joint characteristic function Here, for computational convenience, the part of the characteristic function that concerns is formulated in terms of increments of (where we set ). Now, as in in [4, Theorem 3.1], we get by the independence of from thatwhich is the characteristic function of a Gaussian process with mean zero and covariance function .
Further, by the conditional independent increments of we get like in the proof of in [12, Theorem 6.6] that for . So that for one increment one obtains The generalization to the sum of increments is straightforward, and one obtains the characteristic function of the finite dimensional distribution of a Lévy process (of finite variation):All together we end up with which completes the proof.
Using the Bayes formula from above we now want to proceed further in deriving a Zakai-type equations for the unnormalized filter. This equation is basic in order to obtain the filter recursively. To this end we have to impose certain restrictions on both the signal process and the Gaussian part of the observation process.
The title text suggests that an additional term should be added for the probability that the Modified Bayes Theorem is correct. But that's this equation, so it would make the formula self-referential, unless we call the result the Modified Modified Bayes Theorem. It could also result in an infinite regress -- needing another term for the probability that the version with the probability added is correct, and another term for that version, and so on. If the modifications have a limit, then a Modifiedω Bayes Theorem would be the result, but then another term for whether it's correct is needed, leading to the Modifiedω+1 Bayes Theorem, and so on through every ordinal number.
The basic mathematical formula takes this form: P(BE) = P(B) X P(EB) / P(E), with P standing for probability, B for belief and E for evidence. P(B) is the probability that B is true, and P(E) is the probability that E is true. P(BE) means the probability of B if E is true, and P(EB) is the probability of E if B is true.
The overall $PD_TTC$ for the entire portfolio is 5.74%. Lets say we estimate that in the coming year our $PD_PIT$ will be 8%. We now want to calibrate the probability for each rating to reflect the increase in the overall default rate of the portfolio. I was told that this can be done using the following varsion of the Bayes formula:
This article connects to our class discussion mainly through the utilization of Bayes Theorem to describe the probability of an uncertain event, based on probabilities of conditions related to said event. Though not explicitly explained in the above article, the calculation Price did is as follows and is based on the formula extensively discussed in class:
$$P(A\mid B) = \fracP(A\cap B)P(B) \textprovided that P(B) > 0, \tag1$$is the definition of the conditional probability of $A$ given that$B$ occurred. Similarly,$$P(B\mid A) = \fracP(B\cap A)P(A) = \fracP(A\cap B)P(A) \textprovided that P(A) > 0, \tag2$$is the definition of the conditional probability of $B$ given that$A$ occurred. Now, it is true that it is a trivial matter tosubstitute the value of $P(A\cap B)$ from $(2)$ into $(1)$ toarrive at$$P(A\mid B) = \fracP(B\mid A)P(A)P(B) \textprovided that P(A), P(B) > 0, \tag3$$which is Bayes' formula but notice that Bayes's formula actually connects two different conditional probabilities $P(A\mid B)$and $P(B\mid A)$, and is essentially a formula for "turning theconditioning around". The Reverend Thomas Bayes referred to thisin terms of "inverse probability" and even today, there isvigorous debate as to whether statistical inference should bebased on $P(B\mid A)$ or the inverse probability (calledthe a posteriori or posterior probability).
It is undoubtedly as galling to you as it was to me when I firstdiscovered that Bayes' formula was just a trivial substitution of$(2)$ into $(1)$. Perhaps if you have been born 250 years ago,you (Note: the OP masqueraded under username AlphaBetaGamma when I wrote this answer but has since changed his username) could have made the substitution and thenpeople today would be talking about the AlphaBetaGamma formula and theAlphaBetaGammian heresy and the Naive AlphaBetaGamma method$^*$ insteadof invoking Bayes' name everywhere. Solet me console you on your loss of fame by pointing out a differentversion of Bayes' formula. The Law of Total Probability saysthat$$P(B) = P(B\mid A)P(A) + P(B\mid A^c)P(A^c) \tag4$$and using this, we can write $(3)$ as
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