Priors

In the context of Bayes's Theorem, Priorspriors refer generically to the beliefs an agent holds regarding a fact, hypothesis or consequence, before being presented with evidence. Upon being presented with new evidence, the agent can multiply their prior with a likelihood distribution to calculate a new (posterior) probability for their belief.

Examples

In the context of Bayes's Theorem, Priors refer generically to the beliefs an agent holds regarding a fact, hypothesis or consequence, before being presented with evidence. More technically, in order for thisUpon being presented with new evidence, the agent can multiply their prior with a likelihood distribution to calculate a posteriornew (posterior) probability using Bayes's Theorem, this referred prior probability and likelihood distribution are needed.for their belief.

Examples

Suppose you had a barrel containing some number of red and white balls. You start with the belief that each ball was independently assigned red color (vs. white color) at some fixed probability. Furthermore, you start out ignorant of this fixed probability (the parameter could be anywhere between 0 and 1). Each red ball you see then makes it more likely that the next ball will be red (followingred, following a Laplacian Rule of SucessionSuccession). For example, seeing 6 red balls out of 10 suggests that the initial probability used for assigning the balls a red color was .6, and that there's also a probability of .6 for the next ball being red.

Thus our prior can affectaffects how we interpret the evidence. The first prior is an inductive prior - things that happened before are predicted to happen again with greater probability. The second prior is anti-inductive - the more red balls we see, the fewer we expect to see in the future.

In both cases, you started out believing something about the barrel - presumably because someone else told you, or because you saw it with your own eyes. But then their words, or even your own eyesight, was evidence, and you must have had prior beliefs about probabilities and likelihoods in order to interpret the evidence. So it seems that an ideal Bayesian would need some sort of inductive prior at the very moment they were born. Where an ideal Bayesian would get this prior, has occasionally been a matter of considerable controversy in the philosophy of probability.

As a real life example, consider two leaders from different political parties. Each one has his own beliefs - priors - about social organization and the roles of people and government in society. These differences in priors can be attributed to a wide range of factors, ranging from genetic variabilitytheir educational backgrounds to education influencehereditary differences in their personalities and condition the politics and laws they want to implement.personality. However, neither can show that his beliefs are better than those of the other, unless he can show that his priors were generated by sources which track reality better¹.

Because carrying out any reasoning at all seems to require a prior of some kind, ideal Bayesians would need some sort of priors from the moment that they were born. The question of where an ideal Bayesian would get this prior from has occasionally been a matter of considerable controversy in the philosophy of probability.

It's important to notice that piors represent a commitment to a certain belief. That is,In informal discussion, people often talk about "updating" their priors. This is technically incorrect, as seen in this Less Wrong discussion, you can't shift your prior. What happens is that, after being presented with the evidence, you update yourone does not change their prior probability, thus actually becomingbut rather uses it to calculate a posterior probability.

It should be noticed, however, that it can make sense to informally talk about updating priors when dealing with a sequence of inferences. In such cases, However, as this posterior probability happens to become athen becomes the prior probability for the next inference, so it can make it easier to refer to it in that way.talking about "updating one's priors" is often a convenient shorthand.

It should be noticed, however, that it can make sense to informally talk about updating priors when dealing with a sequence of inferences. In such cases, posterior probability actually becomeshappens to become a prior for the next inference, so it can make it easier to refer to it in that way.

It's important to notice that piors represent a commitment to a certain belief. That is, as seen in this Cyan puts itLess Wrong discussion, you can't shift your prior. What happens is that, after being presented with the evidence, you update your prior probability, thus actually becoming a posterior probability.

Prior probabilityUpdating prior probabilities

This specific term usually refersIt's important to notice that piors represent a commitment to a prior already based on considerable evidence - for example when we estimatecertain belief. That is, as Cyan puts it, you can't shift your prior. What happens is that, after being presented with the number of red balls after doing 100 similar experiments or hearing about how the box was created.

As a complementary example, suppose there are a hundred boxes, one of which contains a diamond - and this is allevidence, you know about the boxes. Thenupdate your prior probabilityprobability, thus actually becoming a posterior probability.

It should be noticed, however, that a box contains a diamond is 1%, or prior odds of 1:99.

Later you may run a diamond-detector over a box, which is 88% likelyit can make sense to beepinformally talk about updating priors when a box contains a diamond, and 8% likely to beep (false positive) when a box doesn't contain a diamond. If the detector beeps, then represents the introduction of evidence,dealing with a likelihood ratiosequence of 11:1 in favor of a diamond, which sends the prior odds of 1:99 to posterior odds of 11:99 = 1:9. But if someone asks you "What was your prior probability?" you would still say "My prior probability was 1%, but I saw evidence which raised theinferences. In such cases, posterior probability to 10%."

Your prior probability in this case was actually becomes a prior belief based on a certain amount of information - i.e., someone told you that one out of a hundred boxes contained a diamond. Indeed, someone told you howfor the detector worked - what sort of evidence a beep represented. In conclusion, the term prior probability is more likelynext inference, so it can make it easier to refer to a single summary judgment of some variable's prior probability, versus the above Bayesian's general notion of priors.it in that way.

Examples

As a concrete example, supposeSuppose you had a barrel containing some number of red and white balls. You start with the belief that each ball was independently assigned red color (vs. white color) at some fixed probability. Furthermore, you start out ignorant of this fixed probability (the parameter could be anywhere between 0 and 1). Each red ball you see then makes it more likely that the next ball will be red (following a Laplacian Rule of Sucession).

References

AAs a real life example may come in hand to better understand the consequences of the priors on the reasoning about any subject. Considerexample, consider two leaders from different political parties - eachparties. Each one has his own beliefs about social organization and the roles of people and government in the society. ThisThese differences can be attributed to a wide range of factors, from genetic variability to education influence in their personalities and condition the politics and laws they want to implement. However, both leaders - and, most importantly, the voters - should noteneither can show that neither has reason to belief his reasoning isbeliefs are better than those of the other, unless he can demonstrateshow that his priors lead to awere generated by sources which track reality better political model and improvements in the society.¹.

In the context of Bayes's Theorem, Priors refersrefer generically to the beliefs an agent holds regarding a fact, hypothesis or consequence, before being presented with evidence. More technically, in order for this agent to calculate a posterior probability using Bayes's Theorem, this referred prior probability and likelihood distribution are needed.