Being robbed is, of course, only one possible explanation for what you observed, and there are many more explanations. So what is our P D? With every other part of the formula— even though we just guessed at a value for this exercise—we can collect real data to provide a more concrete probability. For our prior, P robbed , we might simply look at historical crime data and pin down a probability that a given house on your street would be robbed any given day.
Likewise, we could, theoretically, investigate past robberies and come up with a more accurate likelihood for observing the evidence you did given a robbery. But how could we ever really even guess at P broken window,open front door,missing laptop?
Instead of researching the probability of the data you observed, we could try to calculate the probabilities of all other possible events that could explain your observations. Since they must sum to 1, we could work backward and find P D. In Chapters 6 and 7, where we calculated the probability that a customer service rep was male and the probability of choosing different colored LEGO studs, respectively, we had plenty of information about P D.
This allowed us to come up with an exact probability of our belief in our hypothesis given what we observed. We can do this by looking at the ratio of our unnormalized posterior distributions.
Because the P D would be a constant, we can safely remove it without changing our analysis. Our new hypothesis consists of three events: 1. A neighborhood kid hit a baseball through the front window. You left your door unlocked. Now we need to solve for the likelihood and prior of this data. The Prior for Our Alternative Hypothesis Our prior represents the possibility of all three events happening. This means we need to first work out the probability of each of these events and then use the product rule to determine the prior.
While bringing a laptop to work and leaving it there might be common, completely forgetting you took it in the first place is less common. Now we need a posterior for each of our hypotheses to compare. A ratio tells us how many times more likely one hypothesis is than the other. Now we have a ratio of the unnormalized posteriors. Because the posterior tells us how strong our belief is, this ratio of posteriors tells us how many times better H1 explains our data than H2 without knowing P D.
In other words, our analysis shows that our original hypothesis H1 explains our data much, much better than our alternate hypothesis H2. This also aligns well with our intuition—given the scene you observed, a robbery certainly sounds like a more likely assessment. How unlikely would you have to believe being robbed is—our prior for H1—in order for the ratio of H1 to H2 to be even?
In practice, however, they often demonstrate how to apply vital background information to fully reason about an uncertain situation. Using probability distributions instead of single values is useful for two major reasons. First, in reality there is often a wide range of possible beliefs we might have and consider. Second, representing ranges of probabilities allows us to state our confidence in a set of hypotheses.
We explored both of these examples when examining the mysterious black box in Chapter 5. What C-3PO is missing in his calculations is that Han is a badass! To represent that range, we need to look at a distribution of beliefs regarding the probability of success, rather than a single value representing the probability.
To C-3PO, the only possible outcomes are successfully navigating the asteroid field or not. Next, we need to determine our prior. Statistics is a tool that aids and organizes our reasoning and beliefs about the world.
Bayesian Priors and Working with Probability Distributions 85 We have a prior belief that Han will make it through the asteroid field, because Han has survived every improbable situation so far. What makes Han Solo legendary is that no matter how unlikely survival seems, he always succeeds!
The prior probability is often very controversial for data analysts outside of Bayesian analysis. But this scene is an object chapter in why dismissing our prior beliefs is even more absurd. Right now, we have many reasons for believing Han will survive, but no numbers to back up that belief. If we believed Han absolutely could not die, the movie would become predictable and boring. Figure shows the distribution for our prior probability that Han will make it.
Distribution of our prior belief of Han Solo surviving Density 0 0. First, our beliefs are very approximate, so we need to concede a variable rate of survival. Second, a beta distribution will make future calculations much easier. Now, with our likelihood and prior in hand, we can calculate our posterior probability in the next section.
By combining beliefs, we create our posterior distribution. The formula for the posterior is actually very simple and intuitive. Because this is so simple, working with the beta distribution is very convenient for Bayesian statistics.
Figure plots our final posterior belief. By combining the C-3PO belief with our Han-is-a-badass belief, we find that we have a far more reasonable position. Wrapping Up In this chapter, you learned how important background information is to analyzing the data in front of you. You also saw that you can use probability distributions, rather than a single probability, to express a range of possible beliefs. A friend finds a coin on the ground, flips it, and gets six heads in a row and then one tails.
Give the beta distribution that describes this. Use integration to determine the probability that the true rate of flipping heads is between 0. Come up with a prior probability that the coin is fair.
Use a beta distribution such that there is at least a 95 percent chance that the true rate of flipping heads is between 0. Now see how many more heads with no more tails it would take to convince you that there is a reasonable chance that the coin is not fair. Bayesian Priors and Working with Probability Distributions 89 Part III Pa r a me t e r E s t im at ion 10 I n t rod u c t i o n t o A v e r a g i n g a nd Pa r a me t e r E s t im at ion This chapter introduces you to parameter estimation, an essential part of statistical inference where we use our data to guess the value of an unknown variable.
For example, we might want to estimate the probability of a visitor on a web page making a purchase, the number of jelly beans in a jar at a carnival, or the location and momentum of a particle. In all of these cases, we have an unknown value we want to estimate, and we can use information we have observed to make a guess. We refer to these unknown values as parameters, and the process of making the best guess about these parameters as parameter estimation.
Nearly everyone understands that taking an average of a set of observations is the best way to estimate a true value, but few people really stop to ask why this works—if it really does at all. We need to prove that we can trust averaging, because in later chapters, we build it into more complex forms of parameter estimation.
You decide to use a ruler to measure the depth at seven roughly random locations in your yard. You come up with the following measurements in inches : 6. Given that, how can we use these measurements to make a good guess as to the actual snowfall? This simple problem is a great example case for parameter estimation. Instead, we have a collection of data that we can combine using probability, to determine the contribution of each observation to our estimate, in order to help us make the best possible guess.
Averaging Measurements to Minimize Error You first instinct is probably to average these measurements. In grade school, we learn to average elements by adding them up and dividing the sum by the total number of elements. After all, each of our measurements is different, and all of them are likely different from the true value of the snow that fell. For many centuries, even great mathematicians feared that averaging data compounds all of these erroneous measurements, making for a very inaccurate estimate.
One error commonly made in statistics is to blindly apply procedures without understanding them, which frequently leads to applying the wrong solution to a problem. Probability is our tool for reasoning about uncertainty, and parameter estimation is perhaps the most common process for dealing with uncertainty.
A simplified view of uniform snowfall Snow depth inches 6 4 2 0 0 2 4 6 Place of measurement Figure Visualizing a perfectly uniform, discrete snowfall Introduction to Averaging and Parameter Estimation 95 This is the perfect scenario. Obviously, averaging works in this case, because no matter how we sample from this data, our answer will always be 6 inches.
Compare that to Figure , which illustrates the data when we include the windblown snow against the left side of your house. This leads us to our first key insight into why averaging works: errors in measurement tend to cancel each other out. Suppose the wind has blown 21 inches of snow to one of the six squares and left only 3 inches at each of the remaining squares, as shown in Figure Now we have a very different distribution of snowfall.
For starters, unlike the preceding example, none of the values we can sample from have the true level of snowfall. Also, our errors are no longer nicely distributed—we have a bunch of lower-than-anticipated measurements and one extremely high measurement. Table shows the possible measurements, the difference from the true value, and the probability of each measurement.
However, we can use probability to show that even in this extreme distribution, our errors still cancel each other out. The probability of each error observed is how strongly we believe in that error. When we want to combine our observations, we can consider the probability of the observation as a value representing the strength of its vote toward the final estimate.
In this case, the error of —3 inches is five times more likely than the error of 15 inches, so —3 gets weighted more heavily. So, if we were taking a vote, —3 would get five votes, whereas 15 would only get one vote.
We combine all of the votes by multiplying each value by its probability and adding them together, giving us a weighted sum. In the extreme case where all the values are the same, we would just have 1 multiplied by the value observed and the result would just be that value.
When we weight our observations by our belief in that observation, the errors tend to cancel each other out. Estimating the True Value with Weighted Probabilities We are now fairly confident that errors from our true measurements cancel out.
We are left with just the t in the end. It is simply the sum of each value weighted by its probability. No matter how the errors are distributed, the probability of errors at one extreme is canceled out by probabilities at the other extreme.
Means for Measurement vs. But the mean is often used as a way to summarize a set of data. Even though mean is a very simple and well-known parameter estimate, it can be easily abused and lead to strange results. If you were building an amusement park and wanted to know what height restrictions to put on a roller coaster so that at least half of all visitors could ride it, then you have a real value you are trying to measure.
However, in that case, the mean suddenly becomes less helpful. A better measurement to estimate is the probability that someone entering your park will be taller than x, where x is the minimum height to ride a roller coaster. Wrapping Up In this chapter, you learned that you can trust your intuition about averaging out your measurements in order to make a best estimate of an unknown value. This is true because errors tend to cancel out.
We can formalize this notion of averaging into the idea of the expectation or mean. When we calculate the mean, we are weighting all of our observations by the probability of observing them. Exercises Try answering the following questions to see how well you understand averaging to estimate an unknown measurement. In the Fahrenheit temperature scale, Say you are taking care of a child that feels warm and seems sick, but you take repeated readings from the thermometer and they all read between You try the thermometer yourself and get several readings between What could be wrong with the thermometer?
Given that you feel healthy and have traditionally had a very consistently normal temperature, how could you alter the measurements , In the previous chapter, you learned that the mean is the best way to guess the value of an unknown measurement, and that the more spread out our observations, the more uncertain we are about our estimate of the mean. Dropping Coins in a Well Say you and a friend are wandering around the woods and stumble across a strange-looking old well.
You peer inside and see that it seems to have no bottom. To test it, you pull a coin from your pocket and drop it in, and sure enough, after a few seconds you hear a splash. From this, you conclude that the well is deep, but not bottomless. With the supernatural discounted, you and your friend are now equally curious as to how deep the well actually is. To gather more data, you grab five more coins from your pocket and drop them in, getting the following measurements in seconds: 3.
Next, your friend wants to try his hand at getting some measurements. Rather than picking five similarly sized coins, he grabs a wider assortment of objects, from small pebbles to twigs. Dropping them in the well, your friend gets the following measurements: 3. For each group, each observation is denoted with a subscript; for example, a 2 is the second observation from group a.
The mean for both a and b is 3. Table displays each observation and its distance from the mean. A first guess at how to quantify the difference between the two spreads might be to just sum up their differences from the mean.
The reason the differences cancel out is that some are negative and some are positive. So, if we convert all the differences to positives, we can eliminate this problem without invalidating the values. This gives us the positive version of our negative numbers without actually changing them. This is a more useful approach for our particular situation, but it applies only when the two sample groups are the same size. Even with these additional observations, the data in group a seems less spread out than the data in group b, but the absolute sum of group a is now To correct for this, we can normalize our values by dividing by the total number of observations.
This means that for group a the average observation is 0. We call the result of this formula the mean absolute deviation MAD. The MAD is a very useful and intuitive measure of how spread out your observations are.
Given that group a has a MAD of 0. This method has at least two benefits over using MAD. The first benefit is a bit academic: squaring values is much easier to work with mathematically than taking their absolute value. Notice that the equation for variance is exactly the same as MAD except that the absolute value function in MAD has been replaced with squaring. Because it has nicer mathematical properties, variance is used much more frequently in the study of probability than MAD.
MAD gave us an intuitive definition: this is the average distance from the mean. Variance, on the other hand, says: this is the average squared difference. Recall that when we used MAD, group b was about 10 times more spread out than group a, but in the case of variance, group b is now times more spread out! Finding the Standard Deviation While in theory variance has many properties that make it useful, in practice it can be hard to interpret the results.
If the MAD of group b is 0. To fix this, we can take the square root of the variance in order to scale it back into a number that works with our intuition a bit better.
Looking at all of the different parts, given that our goal is to numerically represent how spread out our data is, we can see that: 1. Finally, we take the square root of everything so that the numbers are closer to what they would be if we used the more intuitive absolute distance.
Notice that, just like with MAD, the difference in the spread between b and a is a factor of So we now have three different ways of measuring the spread of our data. We can see the results in Table By far the most commonly used value is the standard deviation, because we can use it, together with the mean, to define a normal distribution, which in turn allows us to define explicit probabilities to possible true values of our measurements.
The most intuitive measurement of the spread of values is the mean absolute deviation MAD , which is the average distance of each observation from the mean. The mathematically preferred method is the variance, which is the squared difference of our observations. But when we calculate the variance, we lose the intuitive feel for what our calculation means. Our third option is to use the standard deviation, which is the square root of the variance. The standard deviation is mathematically useful and also gives us results that are reasonably intuitive.
Exercises Try answering the following questions to see how well you understand these different methods of measuring the spread of data. One of the benefits of variance is that squaring the differences makes the penalties exponential.
Give some examples of when this would be a useful property. Calculate the mean, variance, and standard deviation for the following values: 1, 2, 3, 4, 5, 6, 7, 8, 9, On its own, each concept is useful, but together, they are even more powerful: we can use them as parameters for the most famous probability distribution of all, the normal distribution. This allows us to perform more sophisticated reasoning with uncertain values.
We established in the preceding chapter that the mean is a solid method of estimating an unknown value based on existing data, and that the standard deviation can be used to measure the spread of that data. By measuring the spread of our observations, we can determine how confidently we believe in our mean. It makes sense that the more spread out our observations, the less sure we are in our mean.
The normal distribution allows us to precisely quantify how certain we are in various beliefs when taking our observations into account. Measuring Fuses for Dastardly Deeds Imagine a mustachioed cartoon villain wants to set off a bomb to blow a hole in a bank vault.
He knows that if he gets feet away from the bomb, he can escape to safety. It takes him 18 seconds to make it that far. Although the villain has only one bomb, he has six fuses of equal size, so he decides to test out five of the six fuses, saving the last one for the bomb.
The fuses are all the same size and should take the same amount of time to burn through. He sets off each fuse and measures how long it takes to burn through to make sure he has the 18 seconds he needs to get away. Of course, being in a rush leads to some inconsistent measurements. Here are the times he recorded in seconds for each fuse to burn through: 19, 22, 20, 19, So far so good: none of the fuses takes less than 18 seconds to burn. But now we want to determine a concrete probability for how likely it is that, given the data we have observed, a fuse will go off in less than 18 seconds.
Since our villain values his life even more than the money, he wants to be In Chapter 11, you learned that you can quantify how spread out your observations are by calculating the standard deviation. It seems rational that this might also help us figure out how likely the alternatives to our mean might be. For example, suppose you drop a glass on the floor and it shatters.
When observations are scattered visually, we intuitively feel that there might be other observations at the extreme limits of what we can see. We The Normal Distribution are also less confident in exactly where the center is. We can quantify this intuition with the most studied and well-known probability distribution: the normal distribution.
The Normal Distribution The normal distribution is a continuous probability distribution like the beta distribution in Chapter 5 that best describes the strength of possible beliefs in the value of an uncertain measurement, given a known mean and standard deviation.
The normal distribution with a mean of 0 and a standard deviation of 1 0. The width of a normal distribution is determined by its standard deviation. So, if our observations are more scattered, we believe in a wider range of possible values and have less confidence in the central mean.
When the only thing we know about a problem is the mean and standard deviation of the data we have observed, the normal distribution is the most honest representation of our state of beliefs. Normal distribution representing our fuse measurements 0.
To solve this problem, we need to use the probability density function PDF , a concept you first learned about in Chapter 5. Area representing fuse lengths less than or equal to 18 seconds 0. Notice that even though none of the observed values was less than 18, because of the spread of the observations, the normal distribution in Figure shows that a value of 18 or less is still possible.
By integrating over all values less than 18, we can calculate the probability that the fuse will not last as long as our villain needs it to.
Integrating this function by hand is not an easy task. Thankfully, we have R to do the integration for us. Before we do this, though, we need to determine what number to start integrating from. Luckily, as you can see in Figures and , the probability density function becomes an incredibly small value very quickly. We can see that the line in the PDF is nearly flat at 10, meaning there is virtually no probability in this region, so we can just integrate from 10 to Table gives probabilities for these other areas.
As an example, when measuring snowfall in Chapter 10 we had the following measurements: 6. For these measurements, the mean is 6. This means that we can be 95 percent sure that the true value of the snowfall was somewhere between 3. No need to manually calculate an integral or boot up a computer to use R! The Normal Distribution Even when we do want to use R to integrate, this trick can be useful for determining a minimum or maximum value to integrate from or to.
What can we use for our upper bound? We can integrate from 21 to Being three standard deviations from the mean will account for The remaining 0. So if we integrate from 21 to We saw the progression of one, two, and three standard deviations from the mean in Table , which were values at 68, 95, and You can easily intuit from this that an eight-sigma event must be extremely unlikely.
To show the growing rarity of an event as it increases by n sigma, say you are looking at events you might observe on a given day. Some are very common, such as waking up to the sunrise. Others are less common, such as waking up and it being your birthday.
Table shows how many days it would take to expect the event to happen per one sigma increase. Based on that, you might take some issue with the notion that the normal distribution is truly the best method to model parameter estimation given that we know only the mean and standard deviation of any given data set.
Distribution: Normal Beta 2. We can see that for both distributions the center of mass appears in roughly the same place, but the bounds for the normal distribution extend way beyond the limits of our graph.
This demonstrates a key point: only when you know nothing about the data other than its mean and variance is it safe to assume a normal distribution. However, in most cases this is not practically important because measurements out that far The Normal Distribution are essentially impossible in probabilistic terms. But for our example of measuring the probability of an event happening, this missing information is important for modeling our problem.
So, while the normal distribution is a very powerful tool, it is no substitute for having more information about a problem. Wrapping Up The normal distribution is an extension of using the mean for estimating a value from observations. The normal distribution combines the mean and the standard deviation to model how spread out our observations are from the mean.
This is important because it allows us to reason about the error in our measurements in a probabilistic way. Not only can we use the mean to make our best guess, but we can also make probabilistic statements about ranges of possible values for our estimate.
Exercises Try answering the following questions to see how well you understand the normal distribution. What is the probability of observing a value five sigma greater than the mean or more? A fever is any temperature greater than Given the following measurements, what is the probability that the patient has a fever? Suppose in Chapter 11 we tried to measure the depth of a well by timing coin drops and got the following values: 2.
What is the probability that the well is over meters deep? What is the probability there is no well i. There are two good explanations for this probability being higher than it should. Which is more likely to you? This chapter will cover more on the probability density function PDF ; introduce the cumulative distribution function CDF , which helps us more easily determine the probability of ranges of values; and introduce quantiles, which divide our probability distributions into parts with equal probabilities.
For example, a percentile is a quantile, meaning it divides the probability distribution into equal pieces. Estimating the Conversion Rate for an Email Signup List Say you run a blog and want to know the probability that a visitor to your blog will subscribe to your email list.
In marketing terms, getting a user to perform a desired event is referred to as the conversion event, or simply a conversion, and the probability that a user will subscribe is the conversion rate.
As discussed in Chapter 5, we would use the beta distribution to estimate p, the probability of subscribing, when we know k, the number of people subscribed, and n, the total number of visitors. When the beta distribution was introduced, you learned only the basics of what it looked like and how it behaved. We want to not only make a single estimate for our conversion rate, but also come up with a range of possible values within which we can be very confident the real conversion rate lies.
The PDF is a function that takes a value and returns the probability of that value. PDF Beta , Density 0 0. It seems unlikely that the conversion rate is exactly 0. We know the total area under the curve of the PDF must add up to 1, since this PDF represents the probability of all possible estimates. We can estimate ranges of values for our true conversion rate by looking at the area under the curve for the ranges we care about.
This is exactly like how we used integration with the normal distribution in the prior chapter. Tools of Parameter Estimation: The PDF, CDF, and Quantile Function Given that we have uncertainty in our measurement, and we have a mean, it could be useful to investigate how much more likely it is that the true conversion rate is 0. To do this, we can calculate the probability of the actual rate being lower than 0. So, if we take the integral from 0 to 0.
We can ask questions about the other extreme as well, such as: how likely is it that we actually got an unusually bad sample and our true conversion rate is much higher, such as a value greater than, say, 0.
So, in this example, the probability that our conversion rate is 0. For most well-known probability distributions, R supports an equivalent dfunction function for calculating the PDF. If we wanted to determine the probability of getting three or fewer heads in five coin tosses, for example, we would use the CDF for the binomial distribution like this: pbinom 3,5,0. With the visualizations, we simply drew lines from the y-axis and used those to find a point on the x-axis.
As an example, imagine we have a function that squares values. However, reversing the function is exactly what we did in the previous section to estimate the median: we looked at the y-axis for 0.
The inverse of the CDF is an incredibly common and useful tool called the quantile function. To compute an exact value for our median and confidence interval, we need to use the quantile function for the beta distribution. Just like the CDF, the quantile function is often very tricky to derive and use mathematically, so instead we rely on software to do the hard work for us. Probability of subscription Quantile function Beta , 0. The value on the y-axis is the value for that quantile. This function is very useful for quickly answering questions about what values are bounds of our probability distribution.
For example, if we want to know the value that We can then use the quantile function to quickly calculate exact values for confidence intervals for our estimates. To find the 95 percent confidence interval, we can find the values greater than the 2. We can easily calculate these for our data with qbeta : Our lower bound is qbeta 0. We can, of course, increase or decrease these thresholds depending on how certain we want to be.
Now that we have all of the tools of parameter estimation, we can easily pin down an exact range for the conversion rate. The great news is that we can also use this to predict ranges of values for future events. Suppose an article on your blog goes viral and gets , visitors.
Based on our calculations, we know that we should expect between and new email subscribers. These tools form the basis of how we can estimate parameters and calculate our confidence in those estimations. That means we can not only make a good guess as to what an unknown value might be, but also determine confidence intervals that very strongly represent the possible values for a parameter.
Returning to the task of measuring snowfall from Chapter 10, say you have the following measurements in inches of snowfall: 7. A child is going door to door selling candy bars. So far she has visited 30 houses and sold 10 candy bars. She will visit 40 more houses today. What is the 95 percent confidence interval for how many candy bars she will sell the rest of the day?
Most companies that provide email list management services tell you, in real time, how many people have opened an email and clicked the link. Our data so far tells us that of the first five people that open an email, two of them click the link. Figure shows our beta distribution for this data. Beta 2,3 likelihood for possible conversion rates Density 1. We used these numbers because two people clicked and three did not click. Unlike in the previous chapter, where we had a pretty narrow spike in possible values, here we have a huge range of possible values for the true conversion rate because we have very little information to work with.
Figure shows the CDF for this data, to help us more easily reason about these probabilities. The 95 percent confidence interval i. At this point our data tells us that the true conversion rate could be anything between 0.
Almost everything else is fair game. Taking that 80 percent rate at face value seems naive when I consider my own behavior. In Chapter 9, you learned how we could use past information to modify our belief that Han Solo can successfully navigate an asteroid field.
Our data tells us one thing, but our background information tells us another. As you know by now, in Bayesian terms the data we have observed is our likelihood, and the external context information—in this case from our personal experience and our email service—is our prior probability. Our challenge now is to figure out how to model our prior. Luckily, unlike the case with Han Solo, we actually have some data here to help us. The conversion rate of 2.
However, this still leaves us with a range of possible options: Beta 1,41 , Beta 2,80 , Beta 5, , Beta 24, , and so on. So which should we use? The problem now is that even the most liberal option we have, Beta 1,41 , seems a little too pessimistic, as it puts a lot of our probability density in very low values.
Notice that for the likelihood with no prior, we have some belief that our conversion rate could be as high as 80 percent. As mentioned, this is highly suspicious; any experienced email marketer would tell you than an 80 percent conversion rate is unheard of. Adding a prior to our likelihood adjusts our beliefs so that they become much more reasonable.
But I still think our updated beliefs are a bit pessimistic. The way any rational person does: with more data! We wait a few hours to gather more results and now find that out of people who opened your email, 25 have clicked the link! Our prior is still keeping our ego in check, giving us a more conservative estimate for the true conversion rate.
However, as we add evidence to our likelihood, it starts to have a bigger impact on what our posterior beliefs look like. In other words, the additional observed data is doing what it should: slowly swaying our beliefs to align with what it suggests.
In the morning we find that subscribers have opened their email, and 86 of those have clicked through. Figure shows our updated beliefs.
When we had almost no evidence, our likelihood proposed some rates we know are absurd e. In light of little evidence, our prior beliefs squashed any data we had.
But as we continue to gather data that disagrees with our prior, our posterior beliefs shift toward what our own collected data tells us and away from our original prior. Another important takeaway is that we started with a pretty weak prior. Even then, after just a day of collecting a relatively small set of information, we were able to find a posterior that seems much, much more reasonable.
This prior probability distribution was based on real data, so we could be fairly confident that it would help us get our estimate closer to reality. So what do we do then? Prior as a Means of Quantifying Experience Because we knew the idea of an 80 percent click-through rate for emails was laughable, we used data from our email provider to come up with a better estimate for our prior.
A marketer might know from personal experience that you should expect about a 20 percent conversion rate, for example. Given this information from an experienced professional, you might choose a relatively weak prior like Beta 2,8 to suggest that the expected conversion rate should be around 20 percent.
This distribution is just a guess, but the important thing is that we can quantify this assumption. For nearly every business, experts can often provide powerful prior information based simply on previous experience and observation, even if they have no training in probability specifically.
Parameter Estimation with Prior Probabilities By quantifying this experience, we can get more accurate estimates and see how they can change from expert to expert. For example, if a marketer is certain that the true conversion rate should be 20 percent, we might model this belief as Beta , As we gather data, we can compare models and create multiple confidence intervals that quantitatively model any expert beliefs.
Additionally, as we gain more and more information, the difference due to these prior beliefs will decrease. This corresponds to using a very weak prior that holds that each outcome is equally likely: Beta 1,1.
The technical term for a fair prior is a noninformative prior. Beta 1,1 is illustrated in Figure Bayesian Statistics the Fun Way will change that.
This book will give you a complete understanding of Bayesian statistics through simple explanations and un-boring examples. Find out the probability of UFOs landing in your garden, how likely Han Solo is to survive a flight through an asteroid belt, how to win an argument about conspiracy theories, and whether a burglary really was a burglary, to name a few examples.
By using these off-the-beaten-track examples, the author actually makes learning statistics fun. Next time you find yourself with a sheaf of survey results and no idea what to do with them, turn to Bayesian Statistics the Fun Way to get the most value from your data. This book will give you a complete understanding of Bayesian statistics through simple explanations and un-boring examples. Topics covered include measuring your own uncertainty in a belief, applying Bayes' theorem, and calculating distributions" A study of those statistical ideas that use a probability distribution over parameter space.
This engaging book explains the ideas that underpin the construction and analysis of Bayesian models, with particular focus on computational methods and schemes. Think Bayes is an introduction to Bayesian statistics using computational methods. The premise of this book, and the other books in the Think X series, is that if you know how to program, you can use that skill to learn other topics. Comprehension and computation in Bayesian problem solving. Frontiers in Psychology, 27 July, 6: Bayesian Statistics : The Fun Way.
We will not cover Bayes' rule, An easy introduction to Bayes' theorem and Bayesian statistics is the following: Kurt, W. Bayesian statistics the fun way.
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