In this series of articles , I intend to discuss Bayesian Decision Theory and its most important basic ideas. The articles are mostly based on the classic book "Pattern Classification" by Duda,Hart and Stork. If you want the ideas in all its glory, go get the book !
As I was reading the book, I realized that in its heart , this field is a set of mostly common sense ideas validated by rigorous mathematics. So I will try to discuss the basic ideas in plain english without much mathematical rigor. Since I am still learning how to best explain these complex ideas, any comments on how to improve will be welcome !
You may ask what happened to PCA . Well, I still intend to write more on it but have not found enough time to sit and write it all. I have written a draft version of it but felt it was too technical and not very intuitive.So I am hoping to rewrite it again.
You would need to know basics of probability to understand the following. In particular the ideas of Prior , Posterior and the idea of Likelihood . Of course if you know all these , you will know the big idea of Bayes Theorem . I will try to explain them lightly but if you have any doubts check out Wikipedia or some old books.
We will take the example used in Duda et al. There are two types of fish : Sea Bass and Salmon. We catch a lot of fish of these two types but which are mixed together and our aim is to separate them by automation. We have a conveyor belt in which the fishes come one by one and we need to decide if the current fish is sea bass or salmon. Of course, we want to make it as accurate as possible but also don’t want to spend lot of money on this project. This is in its heart a classification project. We will be given a few examples of both sea bass and salmon and based on it we need to infer the general characteristics using which we can distinguish them.
Basic Probability Ideas
Now let us be slightly more formal. We say that there are two "classes" of fish – sea bass and salmon. According to our system, there is no other type of fish. If we treat it as a state machine , then our system has two states. The book uses a notation of to represent them. We will use the names seabass and salmon as it is more intuitive.
The first basic idea is that of prior probability . This is represented as which basically give the probability that the next fish in the conveyor is seabass or salmon. Of course, both of them have to sum to one. From the Bayesian perspective, this probability is usually obtained from prior (domain) knowledge. We will not talk about Frequentist interpretation as we will focus on Bayesian decision theory.
Let us assume that we use length of the fish to differentiate the fishes. So whenever the fish comes to the conveyor belt, we calculate its length (how, we don’t really care here ) . So we have transformed the fish into a simple representation using a single number, its length. So the length is a feature that we use to classify and the step of converting the fish into length is called feature extraction .
In a real life scenario, we will have multiple features and the input will converted to a vector. For eg we may use length , lightness of skin , fin length etc as feature. In this case , the fish will be transformed into a triplet. Converting the input to a feature vector makes further processing easy and more robust. We will usually use the letter x to represent the feature. So you can consider is the probability of evidence. Eg lets say we got a fish (we dont know what it is yet) of length 5 inches. Now gives the probability that some fish (either seabass or salmon) has the length 5 inches.
The next idea is that of likelihood . It is also called class conditional probability. It is represented as either . The interpretation is simple. This answers the question that if the fish is seabass what is the probability that it will have length inches (ditto salmon). Alternatively , what is the probability that a 5 inch seabass exists and so on. Or even how "likely" is a 5 inch seabass ?
The posterior probability is the other side of the story. This is represented by . Intuitively, given that we have a fish of length inches , what is the probability that it is a seabass (or salmon). The interesting thing is that knowing prior probability and likelihood we can calculate posterior probability using the famous "Bayes Theorem". We can represent it in words as ,
This gives another rationale for the word "likelihood". Among all other things being equal , the item with higher likelihood is more "likely" to final result. For eg if the likelihood of a 10 inch seabass is more than that of salmon then when we observe an unknown fish of length 10 inches , it is most likely a seabass.
PS : There is an excellent (but long) tutorial on Bayes Theorem at "An Intuitive Explanation of Bayes’ Theorem" . True to its title, it does try to explain the bizarre (atleast initially) result of Bayes Theorem using multiple examples. I highly recommend reading it.
Bayesian Decision Theory
Let us enter into the decision theory at last. In a very high level definition, you can consider decision theory as a field which studies about "decisions" (to classify as seabass or not to be) – more exactly, it considers these decisions in terms of cost or loss functions. (More on that later). In essence , you can think of decision theory as providing a decision rule which tells us what action to taken when we make a particular observation. Decision theory can be thought of as all about evaluating decision rules. (Of course, I am grossly simplifying things, but I think I have conveyed the essence).
Informal Discussion of Decision Theory for Two Class System with Single Feature
Let us take a look at the simplest application of decision theory to our problem. We have a two class system (seabass,salmon) and we are using a single feature (length) to make a decision. Be aware that length is not an ideal feature because many a time you will be having both seabass and salmon of same length (say 5 inches). So when we come across a fish with length 5 inches, we are stuck. We don’t know what decision to take as we know both seabass and salmon can be 5 inches. Decision theory to the rescue !
Instead of providing the theoretical ideas, I will discuss various scenarios and what is the best decision theoretic action to do. In all the scenarios let us assume that we want to be as accurate as possible.
Case I : We don’t know anything and we are not allowed to see the fish
This is the worst case to be in. We have no idea about seabass and salmon (a vegetarian , perhaps ?🙂 ). You are also not allowed to see the fish. But you are asked is the next fish in conveyor a seabass or salmon ? All is not lost – The best thing to do is to randomize. So the decision rule is with probability 50% say the next fish is seabass and with probability 50% say it is salmon.
Convince yourself that this is the best thing to do – Not only when the seabass and salmon are in 50:50 , even when they are in 90:10 ratio.
Case II : You know the prior probability but still you don’t see the fish
We are in a slightly better position here. We don’t get to see the fish yet , but we know the prior probability that the next fish is a seabass or salmon. ie We are give Remember, we want to be as accurate as possible and we want to as reliable about accuracy rate as possible.
A common mistake to do is to randomize again. ie with say that the next fish is seabass and salmon otherwise. For eg let us say, . Let me attempt an informal reasoning – In the (sample) worst case, you will get first 40 as seabass, next 30 as salmon and next 30 as seabass. But you say first 30 as salmon and next 70 as seabass. In this hypothetical example you are only at the most 40% accurate even though you can theoretically do better.
What does decision theory say here ? If then ALWAYS say seabass. Else ALWAYS say salmon. In this case the accuracy rate is . Conversely, the error rate is the minimum of both the prior probabilities. Convince yourself that this is the best you can do . It sure is counterintuitive to always say seabass when you know you will get salmon too. But we can easily prove that this is the best you can do "reliably".
Mathematically, decision rule is decide seabass if else decide salmon .
Error is .
Case III : You know the likelihood function and the length but not the prior probability
This case is really hypothetical. ie we can see the fish and hence find its length. Let say x inches. We have but we don’t know the prior probability. The decision rule here is : For each fish , find the appropriate likelihood values. If the likelihood of seabass is higher than that of salmon , say the fish is seabass and salmon otherwise.
Note that we are making a decision based on our "observation" in contrast to previous cases. Unless, you are really unlucky and the prior probabilities are really skewed you can do well with this decision rule.
Mathematically, decision rule is decide seabass if else decide salmon .
Case IV : You know the length, prior probability and the likelihood function
This is the scenario we are mostly in. We know the length of the fish (say 5 inches). We know the prior probability (say 60% salmon and 40% seabass). We also know the likelihood of them. (say P(5 inches|seabass) is 60% and P(5 inches|salmon) is 10% )
Now we can apply our favorite Bayes rule to get posterior probability. If the length of the fish is 5 inches then what is the probability that it a seabass ? A salmon ? Once you can calculate the posterior , the decision rule becomes simpler. If the posterior probability of seabass is higher than say the fish is seabass else say it is salmon.
Mathematically, decision rule is decide seabass if else decide salmon . This rule is very important and is called as Bayes Decision Rule.
For this decision rule, the error is
We can expand Bayes decision rule using Bayes theorem.
Decide seabass if else decide salmon.
There are two special cases.
1. If likelihood are equal then our decision depends on prior probabilities.
2. If prior probabilities are equal then our decisions depend on likelihoods.
We have only scratched the surface of decision theory. In particular we did not focus much on bounding the error today. Also we did not discuss the cases where there are multiple classes or features. Hopefully, I will discuss them in a future post.
Pattern Classification by Duda,Hart and Stork. Chapter 2.