Bayesian Inference

Bayesian inference is a decision making process that enables you to update your hypothesis as new information becomes available. The Bayesian identity is as follows.

P(H/E) = P(E/H)/P(E) x P(H)

P(H) is the probability of your hypothesis being true prior to receipt of the new evidence
P(E) is the probability of the evidence occurring in general
P(E/H) is the probability of the evidence assuming your hypothesis is true
P(H/E) is the probability that your hypothesis is true given the new evidence

P(E/H)/P(E) is the key term above as it tells you to what extent the new evidence impacts the likelihood that your hypothesis is true. Both P(E/H) and P(E) are between 0 and 1. If P(E/H) > P(E) then the new evidence strengthens your hypothesis and vice versa.

A trivial investment related example might be where H is your view that a specific stock is worth significantly more than the market price and E is an earnings upgrade. The chance of an earnings upgrade given the stock is undervalued (P(E/H)) is greater than the chance of an earnings upgrade in general (P(E)) and so E causes P(H) to rise. Similarly, if E is an earnings downgrade then the converse is true and P(H/E) is less than P(H). So the formula shifts the probabilities in the right direction depending on good or bad news.

The identity tells us the best thing we could hope for if we are invested in a stock is that we get news that is likely if the stock is undervalued, but unlikely if it is not. The worst type of news we could get is common in general, but rare in the case of undervalued stocks. Simply getting positive or negative news is not useful without context. An example of what is often perceived as a positive announcement might be an MOU with a large customer or partner or an update on product development. These sorts of announcements are common, but I would guess that they are not usually released by undervalued companies. Consequently, what seem like positive announcements may actually reduce the probability of a stock being undervalued.

Whether or not you should trade based on news also depends on P(H). Clearly, P(H) might be low for a highly rated stock and so an earnings upgrade might be insufficient to shift P(H/E) to a buy threshold (somewhere over 0.5 depending on alternative opportunities and confidence in your assessment) especially after the inevitable price move. All Bayesian theory practically means from an investing viewpoint is that we should revise our valuation upon receipt of new information and to interpret the relevance of such information in the context of stocks in general as well as the specific stock in question.

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