(Warning: this post is a bit technical.)
Suppose you are a Bayesian reasoning agent. While going about your daily activities, you observe an event of type . Because you're a good Bayesian, you have some internal parameter which represents your belief that will occur.
Now, you're familiar with the Ways of Bayes, and therefore you know that your beliefs must be updated with every new datapoint you perceive. Your observation of is a datapoint, and thus you'll want to modify . But how much should this datapoint influence ? Well, that will depend on how sure you are of in the first place. If you calculated based on a careful experiment involving hundreds of thousands of observations, then you're probably pretty confident in its value, and this single observation of shouldn't have much impact. But if your estimate of is just a wild guess based on something your unreliable friend told you, then this datapoint is important and should be weighted much more heavily in your reestimation of .
Of course, when you reestimate , you'll also have to reestimate how confident you are in its value. Or, to put it a different way, you'll want to compute a new probability distribution over possible values of . This new distribution will be , and it can be computed using Bayes' rule:
Here, since is a parameter used to specify the distribution from which is drawn, it can be assumed that computing is straightforward. is your old distribution over , which you already have; it says how accurate you think different settings of the parameters are, and allows you to compute your confidence in any given value of . So the numerator should be straightforward to compute; it's the denominator which might give you trouble, since for an arbitrary distribution, computing the integral is likely to be intractable.
But you're probably not really looking for a distribution over different parameter settings; you're looking for a single best setting of the parameters that you can use for making predictions. If this is your goal, then once you've computed the distribution , you can pick the value of that maximizes it. This will be your new parameter, and because you have the formula , you'll know exactly how confident you are in this parameter.
In practice, picking the value of which maximizes is usually pretty difficult, thanks to the presence of local optima, as well as the general difficulty of optimization problems. For simple enough distributions, you can use the EM algorithm, which is guarranteed to converge to a local optimum. But for more complicated distributions, even this method is intractable, and approximate algorithms must be used. Because of this concern, it's important to keep the distributions and simple. Choosing the distribution is a matter of model selection; more complicated models can capture deeper patterns in data, but will take more time and space to compute with.
It is assumed that the type of model is chosen before deciding on the form of the distribution . So how do you choose a good distribution for ? Notice that every time you see a new datapoint, you'll have to do the computation in the equation above. Thus, in the course of observing data, you'll be multiplying lots of different probability distributions together. If these distributions are chosen poorly, could get quite messy very quickly.
If you're a smart Bayesian agent, then, you'll pick to be a conjugate prior to the distribution . The distribution is conjugate to if multiplying these two distributions together and normalizing results in another distribution of the same form as .
Let's consider a concrete example: flipping a biased coin. Suppose you use the bernoulli distribution to model your coin. Then it has a parameter which represents the probability of gettings heads. Assume that the value 1 corresponds to heads, and the value 0 corresponds to tails. Then the distribution of the outcome of the coin flip looks like this:
It turns out that the conjugate prior for the bernoulli distribution is something called the beta distribution. It has two parameters, and , which we call hyperparameters because they are parameters for a distribution over our parameters. (Eek!)
The beta distribution looks like this:
Since represents the probability of getting heads, it can take on any value between 0 and 1, and thus this function is normalized properly.
Suppose you observe a single coin flip and want to update your beliefs regarding . Since the denominator of the beta function in the equation above is just a normalizing constant, you can ignore it for the moment while computing , as long as you promise to normalize after completing the computation:
Normalizing this equation will, of course, give another beta distribution, confirming that this is indeed a conjugate prior for the bernoulli distribution. Super cool, right?
If you are familiar with the binomial distribution, you should see that the numerator of the beta distribution in the equation for looks remarkably similar to the non-factorial part of the binomial distribution. This suggests a form for the normalization constant:
The beta and binomial distributions are almost identical. The biggest difference between them is that the beta distribution is a function of , with and as prespecified parameters, while the binomial distribution is a function of , with and as prespecified parameters. It should be clear that the beta distribution is also conjugate to the binomial distribution, making it just that much awesomer.
Another difference between the two distributions is that the beta distribution uses gammas where the binomial distribution uses factorials. Recall that the gamma function is just a generalization of the factorial to the reals; thus, the beta distribution allows and to be any positive real number, while the binomial distribution is only defined for integers. As a final note on the beta distribution, the -1 in the exponents is not philosophically significant; I think it is mostly there so that the gamma functions will not contain +1s. For more information about the mathematics behind the gamma function and the beta distribution, I recommend checking out this pdf: http://www.mhtl.uwaterloo.ca/courses/me755/web_chap1.pdf. It gives an actual derivation which shows that the first equation for is equivalent to the second equation for , which is nice if you don't find the argument by analogy to the binomial distribution convincing.
So, what is the philosophical significance of the conjugate prior? Is it just a pretty piece of mathematics that makes the computation work out the way we'd like it to? No; there is deep philosophical significance to the form of the beta distribution.
Recall the intuition from above: if you've seen a lot of data already, then one more datapoint shouldn't change your understanding of the world too drastically. If, on the other hand, you've seen relatively little data, then a single datapoint could influence your beliefs significantly. This intuition is captured by the form of the conjugate prior. and can be viewed as keeping track of how many heads and tails you've seen, respectively. So if you've already done some experiments with this coin, you can store that data in a beta distribution and use that as your conjugate prior. The beta distribution captures the difference between claiming that the coin has 30% chance of coming up heads after seeing 3 heads and 7 tails, and claiming that the coin has a 30% chance of coming up heads after seeing 3000 heads and 7000 tails.
Suppose you haven't observed any coin flips yet, but you have some intuition about what the distribution should be. Then you can choose values for and that represent your prior understanding of the coin. Higher values of indicate more confidence in your intuition; thus, choosing the appropriate hyperparameters is a method of quantifying your prior understanding so that it can be used in computation. and will act like "imaginary data"; when you update your distribution over after observing a coin flip , it will be like you already saw heads and tails before that coin flip.
If you want to express that you have no prior knowledge about the system, you can do so by setting and to 1. This will turn the beta distribution into a uniform distribution. You can also use the beta distribution to do add-N smoothing, by setting and to both be N+1. Setting the hyperparameters to a value lower than 1 causes them to act like "negative data", which helps avoid overfitting to noise in the actual data.
In conclusion, the beta distribution, which is a conjugate prior to the bernoulli and binomial distributions, is super awesome. It makes it possible to do Bayesian reasoning in a computationally efficient manner, as well as having the philosophically satisfying interpretation of representing real or imaginary prior data. Other conjugate priors, such as the dirichlet prior for the multinomial distribution, are similarly cool.