19.1: Probabilistic Reasoning by Rational Agents
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From a Bayesian perspective, statistical inference is all about belief revision . I start out with a set of candidate hypotheses h about the world. I don’t know which of these hypotheses is true, but do I have some beliefs about which hypotheses are plausible and which are not. When I observe the data d, I have to revise those beliefs. If the data are consistent with a hypothesis, my belief in that hypothesis is strengthened. If the data inconsistent with the hypothesis, my belief in that hypothesis is weakened. That’s it! At the end of this section I’ll give a precise description of how Bayesian reasoning works, but first I want to work through a simple example in order to introduce the key ideas. Consider the following reasoning problem:
I’m carrying an umbrella. Do you think it will rain?
In this problem, I have presented you with a single piece of data (d= I’m carrying the umbrella), and I’m asking you to tell me your beliefs about whether it’s raining. You have two possible hypotheses , h: either it rains today or it does not. How should you solve this problem?
Priors: what you believed before
The first thing you need to do ignore what I told you about the umbrella, and write down your pre-existing beliefs about rain. This is important: if you want to be honest about how your beliefs have been revised in the light of new evidence, then you must say something about what you believed before those data appeared! So, what might you believe about whether it will rain today? You probably know that I live in Australia, and that much of Australia is hot and dry. And in fact you’re right: the city of Adelaide where I live has a Mediterranean climate, very similar to southern California, southern Europe or northern Africa. I’m writing this in January, and so you can assume it’s the middle of summer. In fact, you might have decided to take a quick look on Wikipedia 254 and discovered that Adelaide gets an average of 4.4 days of rain across the 31 days of January. Without knowing anything else, you might conclude that the probability of January rain in Adelaide is about 15%, and the probability of a dry day is 85%. If this is really what you believe about Adelaide rainfall (and now that I’ve told it to you, I’m betting that this really is what you believe) then what I have written here is your prior distribution , written P(h):
| Hypothesis | Degree of Belief |
|---|---|
| Rainy day | 0.15 |
| Dry day | 0.85 |
Likelihoods: theories about the data
To solve the reasoning problem, you need a theory about my behaviour. When does Dan carry an umbrella? You might guess that I’m not a complete idiot, 255 and I try to carry umbrellas only on rainy days. On the other hand, you also know that I have young kids, and you wouldn’t be all that surprised to know that I’m pretty forgetful about this sort of thing. Let’s suppose that on rainy days I remember my umbrella about 30% of the time (I really am awful at this). But let’s say that on dry days I’m only about 5% likely to be carrying an umbrella. So you might write out a little table like this:
| Hypothesis | Umbrella | No umbrella |
|---|---|---|
| Rainy day | 0.30 | 0.70 |
| Dry day | 0.05 | 0.95 |
It’s important to remember that each cell in this table describes your beliefs about what data d will be observed, given the truth of a particular hypothesis h. This “conditional probability” is written P(d|h), which you can read as “the probability of d given h”. In Bayesian statistics, this is referred to as likelihood of data d given hypothesis h. 256
joint probability of data and hypothesis
At this point, all the elements are in place. Having written down the priors and the likelihood, you have all the information you need to do Bayesian reasoning. The question now becomes, how do we use this information? As it turns out, there’s a very simple equation that we can use here, but it’s important that you understand why we use it, so I’m going to try to build it up from more basic ideas.
Let’s start out with one of the rules of probability theory. I listed it way back in Table 9.1, but I didn’t make a big deal out of it at the time and you probably ignored it. The rule in question is the one that talks about the probability that two things are true. In our example, you might want to calculate the probability that today is rainy (i.e., hypothesis h is true) and I’m carrying an umbrella (i.e., data d is observed). The joint probability of the hypothesis and the data is written P(d,h), and you can calculate it by multiplying the prior P(h) by the likelihood P(d|h). Mathematically, we say that:
P(d,h)=P(d|h)P(h)
So, what is the probability that today is a rainy day and I remember to carry an umbrella? As we discussed earlier, the prior tells us that the probability of a rainy day is 15%, and the likelihood tells us that the probability of me remembering my umbrella on a rainy day is 30%. So the probability that both of these things are true is calculated by multiplying the two:
\(\begin{aligned} \text { (rainy, umbrella) } &=P(\text { umbrella } | \text {rainy}) \times P(\text { rainy }) \\ &=0.30 \times 0.15 \\ &=0.045 \end{aligned}\)
In other words, before being told anything about what actually happened, you think that there is a 4.5% probability that today will be a rainy day and that I will remember an umbrella. However, there are of course four possible things that could happen, right? So let’s repeat the exercise for all four. If we do that, we end up with the following table:
| Umbrella | No-umbrella | |
|---|---|---|
| Rainy | 0.045 | 0.105 |
| Dry | 0.0425 | 0.8075 |
This table captures all the information about which of the four possibilities are likely. To really get the full picture, though, it helps to add the row totals and column totals. That gives us this table:
| Umbrella | No-umbrella | Total | |
|---|---|---|---|
| Rainy | 0.0450 | 0.1050 | 0.15 |
| Dry | 0.0425 | 0.8075 | 0.85 |
| Total | 0.0875 | 0.9125 | 1 |
This is a very useful table, so it’s worth taking a moment to think about what all these numbers are telling us. First, notice that the row sums aren’t telling us anything new at all. For example, the first row tells us that if we ignore all this umbrella business, the chance that today will be a rainy day is 15%. That’s not surprising, of course: that’s our prior. The important thing isn’t the number itself: rather, the important thing is that it gives us some confidence that our calculations are sensible! Now take a look at the column sums, and notice that they tell us something that we haven’t explicitly stated yet. In the same way that the row sums tell us the probability of rain, the column sums tell us the probability of me carrying an umbrella. Specifically, the first column tells us that on average (i.e., ignoring whether it’s a rainy day or not), the probability of me carrying an umbrella is 8.75%. Finally, notice that when we sum across all four logically-possible events, everything adds up to 1. In other words, what we have written down is a proper probability distribution defined over all possible combinations of data and hypothesis.
Now, because this table is so useful, I want to make sure you understand what all the elements correspond to, and how they written:
| Umbrella | No-umbrella | ||
|---|---|---|---|
| Rainy | P(Umbrella, Rainy) | P(No-umbrella, Rainy) | P(Rainy) |
| Dry | P(Umbrella, Dry) | P(No-umbrella, Dry) | P(Dry) |
| P(Umbrella) | P(No-umbrella) |
Finally, let’s use “proper” statistical notation. In the rainy day problem, the data corresponds to the observation that I do or do not have an umbrella. So we’ll let d 1 refer to the possibility that you observe me carrying an umbrella, and d 2 refers to you observing me not carrying one. Similarly, h 1 is your hypothesis that today is rainy, and h 2 is the hypothesis that it is not. Using this notation, the table looks like this:
Updating beliefs using Bayes’ rule
The table we laid out in the last section is a very powerful tool for solving the rainy day problem, because it considers all four logical possibilities and states exactly how confident you are in each of them before being given any data. It’s now time to consider what happens to our beliefs when we are actually given the data. In the rainy day problem, you are told that I really am carrying an umbrella. This is something of a surprising event: according to our table, the probability of me carrying an umbrella is only 8.75%. But that makes sense, right? A guy carrying an umbrella on a summer day in a hot dry city is pretty unusual, and so you really weren’t expecting that. Nevertheless, the problem tells you that it is true. No matter how unlikely you thought it was, you must now adjust your beliefs to accommodate the fact that you now know that I have an umbrella. 257 To reflect this new knowledge, our revised table must have the following numbers:
| Umbrella | No-umbrella | |
|---|---|---|
| Rainy | 0 | |
| Dry | 0 | |
| Total | 1 | 0 |
In other words, the facts have eliminated any possibility of “no umbrella”, so we have to put zeros into any cell in the table that implies that I’m not carrying an umbrella. Also, you know for a fact that I am carrying an umbrella, so the column sum on the left must be 1 to correctly describe the fact that P(umbrella)=1.
What two numbers should we put in the empty cells? Again, let’s not worry about the maths, and instead think about our intuitions. When we wrote out our table the first time, it turned out that those two cells had almost identical numbers, right? We worked out that the joint probability of “rain and umbrella” was 4.5%, and the joint probability of “dry and umbrella” was 4.25%. In other words, before I told you that I am in fact carrying an umbrella, you’d have said that these two events were almost identical in probability, yes? But notice that both of these possibilities are consistent with the fact that I actually am carrying an umbrella. From the perspective of these two possibilities, very little has changed. I hope you’d agree that it’s still true that these two possibilities are equally plausible. So what we expect to see in our final table is some numbers that preserve the fact that “rain and umbrella” is slightly more plausible than “dry and umbrella”, while still ensuring that numbers in the table add up. Something like this, perhaps?
| Umbrella | No-umbrella | |
|---|---|---|
| Rainy | 0.514 | 0 |
| Dry | 0.486 | 0 |
| Total | 1 | 0 |
What this table is telling you is that, after being told that I’m carrying an umbrella, you believe that there’s a 51.4% chance that today will be a rainy day, and a 48.6% chance that it won’t. That’s the answer to our problem! The posterior probability of rain P(h|d) given that I am carrying an umbrella is 51.4%
How did I calculate these numbers? You can probably guess. To work out that there was a 0.514 probability of “rain”, all I did was take the 0.045 probability of “rain and umbrella” and divide it by the 0.0875 chance of “umbrella”. This produces a table that satisfies our need to have everything sum to 1, and our need not to interfere with the relative plausibility of the two events that are actually consistent with the data. To say the same thing using fancy statistical jargon, what I’ve done here is divide the joint probability of the hypothesis and the data P(d,h) by the marginal probability of the data P(d), and this is what gives us the posterior probability of the hypothesis given that we know the data have been observed. To write this as an equation: 258
\(P(h | d)=\dfrac{P(d, h)}{P(d)}\)
However, remember what I said at the start of the last section, namely that the joint probability P(d,h) is calculated by multiplying the prior P(h) by the likelihood P(d|h). In real life, the things we actually know how to write down are the priors and the likelihood, so let’s substitute those back into the equation. This gives us the following formula for the posterior probability:
\(\ P(h | d)=\dfrac{P(d | h)P(h)}{P(d)}\)
And this formula, folks, is known as Bayes’ rule . It describes how a learner starts out with prior beliefs about the plausibility of different hypotheses, and tells you how those beliefs should be revised in the face of data. In the Bayesian paradigm, all statistical inference flows from this one simple rule.