樸素貝葉斯和貝葉斯估計(jì)

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Maybe you’re an investor trying to decide whether a stock is worth investing in. Maybe you’ve only recently heard of Bayesian inference and want to get a sense of how it can be applied in the real world. Maybe you’re a seasoned analyst who stumbled upon this article and found the title interesting. Regardless of where you come from, I thank you for giving this piece a read. I’m going to talk about the normal-normal model, one of the foundational models in Bayesian statistics, and how it can be used to estimate the growth rate of a company’s revenue. That estimate can then be used to decide whether or not the company is a worthwhile investment.

也許您是試圖確定股票是否值得投資的投資者。也許您只是最近才聽說過貝葉斯推理，并想了解如何將其應(yīng)用到現(xiàn)實(shí)世界中。 也許您是一位經(jīng)驗(yàn)豐富的分析師，偶然發(fā)現(xiàn)了這篇文章，并發(fā)現(xiàn)標(biāo)題很有趣。 無論您來自何處，我都感謝您閱讀本文。 我將討論貝葉斯統(tǒng)計(jì)中的基本模型之一，正常-正常模型，以及如何將其用于估計(jì)公司收入的增長率。 然后，可以使用該估算值來確定公司是否值得投資。

The first objective of this piece is to demonstrate how the normal-normal model can be used to incorporate a subjective overlay into data analysis. The second is to provide some intuition behind the normal-normal model and Bayesian inference in general without getting too bogged down in the mechanics. I’ll say it here and again at the end of the article, but this piece does not constitute investment advice. It is meant to be educational.

本文的第一個(gè)目的是演示如何使用正常-正常模型將主觀疊加納入數(shù)據(jù)分析。 第二個(gè)目的是在法線-法線模型和貝葉斯推理之后提供一些直覺，而又不會(huì)過于迷惑力學(xué)。 我將在本文的結(jié)尾處一再說，但這并不構(gòu)成投資建議。 這是為了教育。

With that disclaimer out of the way, let’s get to it!

有了這個(gè)免責(zé)聲明，讓我們開始吧！

手頭的任務(wù) (The Task at Hand)

Financial modeling generally refers to projecting fundamental values for a company in order to arrive at a fair price estimate for the company’s stock. Some of the most common metrics used to arrive at valuations are revenue, earnings, and cash flow. The company we’re going to look at is MongoDB, a software services company. It began trading publicly back in 2017, and its revenue growth has been tremendous.

財(cái)務(wù)建模通常是指預(yù)測公司的基本價(jià)值，以便得出公司股票的合理價(jià)格估計(jì)。 用于得出估值的一些最常見的指標(biāo)是收入，收益和現(xiàn)金流量。 我們要看的公司是軟件服務(wù)公司MongoDB。 它于2017年開始公開交易，其收入增長巨大。

Given how young the company is and how it’s in a growth-oriented phase of its existence, it’s reasonable to focus on revenue in order to value the company. Data in the company’s 10-K filings, the annual financial reports, shows revenue numbers on a quarterly basis starting in fiscal 2016. Annual numbers are present from the year 2014. To give us more data than the six annual numbers (which translate into five growth numbers), I’ve computed rolling one-year revenue growth on a quarterly basis. That data is shown below.

考慮到公司的年輕程度以及它處于生存發(fā)展階段的方式，合理地關(guān)注收入以對(duì)公司進(jìn)行估值是合理的。 該公司10-K檔案(年度財(cái)務(wù)報(bào)告)中的數(shù)據(jù)顯示了從2016財(cái)年開始的季度收入數(shù)字。從2014年開始提供年度數(shù)字。為我們提供的數(shù)據(jù)要比六個(gè)年度數(shù)字(這意味著五個(gè)增長數(shù)字)，我已經(jīng)計(jì)算出了一個(gè)季度滾動(dòng)的一年收入增長。 該數(shù)據(jù)如下所示。

Closer to the end of this piece, I’ll compare the results of our analysis using year-end data versus quarterly data. (Although I haven’t run a formal analysis, I assume there’s a degree of serial correlation in the quarterly data. This won’t matter in terms of explaining the concepts of the normal-normal model, but it is certainly something to be mindful of in practice.)

在本文的最后，我將比較使用年末數(shù)據(jù)和季度數(shù)據(jù)進(jìn)行分析的結(jié)果。 (盡管我沒有進(jìn)行正式的分析，但我假設(shè)季度數(shù)據(jù)中存在一定程度的序列相關(guān)性。這在解釋法線-法線模型的概念方面并不重要，但一定要注意在實(shí)踐中。)

A common way to project revenue for a company is to use the average historical revenue growth rate over a certain amount of time. For companies with many years of data, this isn’t necessarily a bad practice, especially if the growth rates follow a normal distribution. Given how little sample data we have and the histogram of the data which I’ll plot below, we may feel that using the sample mean in this case is unwise.

預(yù)測公司收入的一種常用方法是使用一定時(shí)間內(nèi)的平均歷史收入增長率。 對(duì)于擁有多年數(shù)據(jù)的公司而言，這不一定是壞習(xí)慣，尤其是當(dāng)增長率遵循正態(tài)分布時(shí)。 考慮到我們只有很少的樣本數(shù)據(jù)以及我將在下面繪制的數(shù)據(jù)直方圖，我們可能會(huì)覺得在這種情況下使用樣本均值是不明智的。

Bayesian inference is particularly useful in situations where our sample size is small and we hold a subjective belief that our sample data does not appropriately represent what a larger sample would look like.

在樣本量較小并且我們主觀認(rèn)為樣本數(shù)據(jù)不能適當(dāng)代表較大樣本的情況下，貝葉斯推斷特別有用。

To conduct Bayesian inference, we’ll need a prior distribution and a sampling model. Before defining those distributions in our context, I’ll go over some of the basics of Bayesian inference and how the prior distribution and sampling model come into play. Feel free to skip this section if you’re familiar with Bayes’ theorem and how it applies to distributions.

要進(jìn)行貝葉斯推斷，我們需要先驗(yàn)分布和采樣模型。 在我們的上下文中定義這些分布之前，我將介紹貝葉斯推斷的一些基礎(chǔ)知識(shí)以及先驗(yàn)分布和采樣模型如何發(fā)揮作用。 如果您熟悉貝葉斯定理及其在分布中的應(yīng)用，請(qǐng)隨時(shí)跳過本節(jié)。

貝葉斯定理和分布 (Bayes’ Theorem and Distributions)

In its simplest form, Bayes’ theorem is defined as

以最簡單的形式，貝葉斯定理定義為

which is equivalent to

相當(dāng)于

This is all well and good if we have neatly defined probabilities to use, but distributions complicate the process a little.

如果我們有明確定義的使用概率，那么這一切都很好，但是分布會(huì)使過程復(fù)雜化了一點(diǎn)。

First, let’s substitute A with θ and B with Y. In this case, Y refers to the points in our sample data, and θ refers to the true average growth rate in revenue for MongoDB. Re-writing the second form of the formula with our substitutions, we have

首先，讓我們用θ替換A并用Y替換B。 在這種情況下， Y表示示例數(shù)據(jù)中的點(diǎn)， θ表示MongoDB的收入的真實(shí)平均增長率。 用我們的替換來重寫公式的第二種形式，我們有

In words, the distribution we’re trying to model is the distribution of average revenue growth rate GIVEN our sample growth rates. We will use our sample data and a little bit of judgement to define this distribution P(Y|θ). We will also need a prior distribution P(θ) for our average growth rate and the marginal distribution of our data P(Y). The onus is on us to define our sampling distribution as well as define a prior distribution for θ. Once we have a sampling distribution P(Y|θ), the correct way to obtain P(Y) would be to solve for the integral below:

換句話說，我們要建模的分布是根據(jù)我們的樣本增長率得出的平均收入增長率的分布。 我們將使用樣本數(shù)據(jù)和一些判斷來定義此分布P ( Y | θ )。 對(duì)于平均增長率和數(shù)據(jù)的邊際分布P ( Y )，我們還將需要先驗(yàn)分布P ( θ )。 我們有責(zé)任定義采樣分布以及θ的先驗(yàn)分布。 一旦有了采樣分布P ( Y | θ )，獲得P ( Y )的正確方法就是求解以下積分：

In practice, this may be difficult to do, but we can use a shortcut. Since Y is only conditional on θ in this instance, P(Y) is an unconditional probability distribution and encompasses all possibilities of Y. This means that the area under the distribution will be equal to 1 (the sum of all probabilities for an event equals 1), and the integral will be equal to 1 multiplied by a normalizing constant. Rather than solve for this normalizing constant, we can instead say

實(shí)際上，這可能很難做到，但是我們可以使用快捷方式。 由于在這種情況下Y僅以θ為條件，因此P ( Y )是無條件的概率分布，并且包含Y的所有可能性。 這意味著分布下的面積將等于1(一個(gè)事件的所有概率之和等于1)，并且積分將等于1乘以歸一化常數(shù)。 除了解決這個(gè)標(biāo)準(zhǔn)化常數(shù)外，我們可以說

P(θ|Y)∝P(Y|θ)P(θ)

P ( θ | Y )∝ P ( Y | θ ) P ( θ )

where ∝ stands for “is proportional to.” In other words, we don’t need to worry about P(Y). With one task eliminated, we only have to define our sampling and prior distributions.

∝代表“正比于”。 換句話說，我們不必?fù)?dān)心P ( Y )。 消除一項(xiàng)任務(wù)后，我們只需定義采樣和先驗(yàn)分布即可。

(Note: technically, Y is conditional on sample variance. In this case, we are going to assume that the variance is known and constant. Because our variance is assumed to be known and a constant, we can omit it from the notation.)

(注意：從技術(shù)上講， Y以樣本方差為條件。在這種情況下，我們將假設(shè)方差是已知的并且是常數(shù)。因?yàn)槲覀兊姆讲畋患俣橐阎⑶沂浅?shù)，所以可以從符號(hào)中忽略它。)

定義我們的抽樣模型和先驗(yàn)分布 (Defining Our Sampling Model and Prior Distribution)

We’re going to use a normal model for our sampling distribution. Having looked at the histogram for our data, one may think that there are distributions available to us that better represent the data. I like the normal distribution in this case because it is continuous and has support along all real numbers (revenue growth could theoretically be negative or positive).

我們將使用正常模型進(jìn)行抽樣分配。 在查看了我們數(shù)據(jù)的直方圖之后，我們可能會(huì)認(rèn)為有一些可用的分布更好地表示了數(shù)據(jù)。 在這種情況下，我喜歡正態(tài)分布，因?yàn)樗沁B續(xù)的并且在所有實(shí)數(shù)上都有支持(理論上收入增長可以是負(fù)數(shù)或正數(shù))。

To define this sampling model, we compute the mean and variance for this data set and use these as the parameters for our sampling model. The form this will take is

為了定義該采樣模型，我們計(jì)算該數(shù)據(jù)集的均值和方差，并將其用作我們的采樣模型的參數(shù)。 采取的形式是

where the first term represents the unknown true average growth rate for MongoDB’s revenue and the second term represents the variance of the growth rates; we will treat this variance as known. We could just as easily assume that we know our mean but not our variance or that we know neither; all three classes of situations are well-documented and have substantial literature regarding how to work them. The normal-normal model applies to the situation with known variance and unknown mean, hence why we are making our current assumptions.

其中第一項(xiàng)代表MongoDB收入的未知真實(shí)平均增長率，第二項(xiàng)代表增長率的方差； 我們將這種差異視為已知。 我們可以很容易地假設(shè)我們知道我們的平均值，但是我們不知道方差，或者我們都不知道。 這三類情況都有充分的文獻(xiàn)記錄，并有大量有關(guān)如何工作的文獻(xiàn)。 正常-正常模型適用于方差已知且均值未知的情況，因此我們?yōu)槭裁匆M(jìn)行當(dāng)前的假設(shè)。

Next, we need to define a prior distribution for θ. For the same reasons that we’re using a normal distribution for the sampling model (continuous, support along positive and negative values), we’re going to use a normal distribution as our prior. We need to define a mean and a variance for the variable θ. We’ll define this distribution as

接下來，我們需要定義θ的先驗(yàn)分布。 出于同樣的原因，我們?cè)诔闃幽Ｐ椭惺褂谜龖B(tài)分布(連續(xù)的，沿正值和負(fù)值的支持)，因此我們將使用正態(tài)分布作為先驗(yàn)。 我們需要為變量θ定義均值和方差。 我們將這種分布定義為

where the first term is the prior mean and the second term is the prior variance. There is significant literature dedicated to selecting priors; the main focus of this piece is how to apply the normal-normal model, so I didn’t put extensive effort in defining my prior distribution.

其中第一項(xiàng)是先驗(yàn)均值，第二項(xiàng)是先驗(yàn)方差。 有大量文獻(xiàn)致力于選擇先驗(yàn)。 本文的主要重點(diǎn)是如何應(yīng)用正態(tài)-正態(tài)模型，因此我沒有花太多精力來定義我的先前分布。

To select a value for the prior mean, I looked at the average revenue growth rate of sales of the S&P 500 index over the last 19 years (multpl.com) and then multiplied it by the β of MongoDB. In the world of equities, β refers to the covariance of an individual stock’s returns with the return of broader basket of stocks (often called an index) divided by the variance of the index returns. MongoDB has a β of about 1.26 according to Seeking Alpha, a research site with news, data, and analyses of many stocks. Whenever we see a β > 1, we can assume that the stock we are looking at is more volatile than the index it is being compared to; for this reason, I multiply the revenue growth of the index by β. Other approaches could involve looking at slightly older companies in the software service industries or similar age companies across industries. No method is perfect, and all are viable.

為了為先前的均值選擇一個(gè)值，我查看了標(biāo)普500指數(shù)過去19年的平均銷售收入增長率(multpl.com) ，然后將其乘以MongoDB的β。 在股票世界中，β指的是單個(gè)股票收益與更廣泛的一籃子股票(通常稱為指數(shù))的收益除以指數(shù)收益的方差的協(xié)方差。 根據(jù)提供新聞，數(shù)據(jù)和許多股票分析的研究網(wǎng)站Seeking Alpha的數(shù)據(jù)，MongoDB的β約為1.26。 每當(dāng)我們看到β> 1時(shí)，我們就可以假設(shè)我們所看的股票的波動(dòng)性大于它所比較的??指數(shù)； 因此，我將指數(shù)的收入增長乘以β。 其他方法可能涉及查看軟件服務(wù)行業(yè)中稍老的公司或跨行業(yè)的類似年齡的公司。 沒有一種方法是完美的，并且所有方法都是可行的。

The next parameter we have to assign is the prior variance. Just to be clear, this is not what we presume is the variance in growth rates, but the presumed variance of the AVERAGE growth rate; this prior variance is meant to reflect our certainty in the accuracy of the prior mean. If we had full confidence that this was the correct mean to use, we could set our variance effectively equal to 0 (for computation purposes, we can’t actually use 0, but we can use a very small number such as .00001). On the other hand, if we have very little confidence in our estimate, we can use a large variance to indicate this level of certainty. In this case, where our prior mean is about 4.5%, I don’t have much of an opinion of how confident I am with this estimate. To define my distribution, I’ll use a standard deviation of 10%. With this, I’m effectively stating that I’m 95% confident that the true value for theta lies between -15.5% and 24.5% (4.5+/-2 standard deviations). This estimate may seem highly conservative given how MongoDB’s average growth rate has been about 61%, but this is exactly why Bayesian inference is powerful. MongoDB has spent the majority of its time trading in a bull market that was particularly favorable for software names. The prior distribution reflects data from multiple market cycles and consequently multiple phases of growth and contraction. Between the possibility of economic contraction, the chance MongoDB doesn’t execute its strategy effectively, and revenue growth slowing simply due to scale, I’m holding the subjective belief that MongoDB’s true average growth rate is less than what the sample data suggests. The prior distribution I’ve selected represents that belief. Now, we can study the output of our analysis.

我們必須分配的下一個(gè)參數(shù)是先驗(yàn)方差。 需要明確的是，這不是我們所假定的增長率的方差，而是假定的平均增長率的方差。 此先驗(yàn)方差旨在反映我們對(duì)先驗(yàn)均值準(zhǔn)確性的確定性。 如果我們完全有信心使用這是正確的平均值，則可以將方差有效地設(shè)置為0(出于計(jì)算目的，我們實(shí)際上不能使用0，但是可以使用非常小的數(shù)字，例如.00001)。 另一方面，如果我們對(duì)估計(jì)的信心很小，則可以使用較大的方差來表示此確定性級(jí)別。 在這種情況下，我們之前的均值約為4.5％，我對(duì)這個(gè)估計(jì)有多自信沒有多少看法。 為了定義我的分布，我將使用10％的標(biāo)準(zhǔn)偏差。 借此，我有效地表明，我有95％的信心認(rèn)為theta的真實(shí)值在-15.5％和24.5％之間(4.5 +/- 2標(biāo)準(zhǔn)偏差)。 考慮到MongoDB的平均增長率如何達(dá)到61％左右，這個(gè)估計(jì)值似乎非常保守，但這正是貝葉斯推斷強(qiáng)大的原因。 MongoDB大部分時(shí)間都在牛市中交易，這對(duì)軟件名稱特別有利。 先前的分布反映了來自多個(gè)市場周期的數(shù)據(jù)，因此反映了增長和收縮的多個(gè)階段。 在經(jīng)濟(jì)收縮的可能性，MongoDB無法有效執(zhí)行其戰(zhàn)略的機(jī)會(huì)以及僅僅是由于規(guī)模而導(dǎo)致的收入增長放緩之間，我主觀地認(rèn)為MongoDB的真實(shí)平均增長率低于樣本數(shù)據(jù)所表明的水平。 我選擇的先前分配代表了這一信念。 現(xiàn)在，我們可以研究分析的結(jié)果。

To recap, here are the forms for our two models:

回顧一下，這是我們兩個(gè)模型的表格：

Great, let’s move on to our analysis!

太好了，讓我們繼續(xù)進(jìn)行分析！

后驗(yàn)分析和直覺 (Posterior Analysis and Intuition)

I’ll focus more on the intuition offered by these forms rather than walk through a derivation by hand. Anyone truly interested in using the normal-normal model should study the derivation of the above parameters. Wikipedia has some good documentation, and most introductory textbooks to Bayesian statistics cover the derivations in detail.

我將更多地關(guān)注這些形式提供的直覺，而不是手工進(jìn)行推導(dǎo)。 任何對(duì)使用法線-法線模型感興趣的人都應(yīng)該研究上述參數(shù)的推導(dǎo)。 Wikipedia有一些很好的文檔，并且有關(guān)貝葉斯統(tǒng)計(jì)的大多數(shù)入門教科書都詳細(xì)介紹了派生方法。

When we have a normal distribution for our sampling model as well as a normal for our prior distribution on the sample mean, the resulting posterior distribution is a product of two normal models. The power of the normal-normal model is that the product of these distributions is also a normal distribution, albeit with updated parameters. In Bayesian jargon, a normal prior distribution is a conjugate prior distribution, meaning that it and its resulting posterior distribution have the same form. The fact that our posterior distribution is a normal distribution may not seem like that big of a deal, but depending on the data we’re trying to model and the parameters we’re trying to estimate, there are many instances where our posterior does not take such a familiar form. Because this posterior distribution is well-defined, we can sample from it directly and consequently compute summary statistics on it easily.

當(dāng)我們的采樣模型具有正態(tài)分布，并且樣本均值具有先驗(yàn)分布的正態(tài)分布時(shí)，所得后驗(yàn)分布是兩個(gè)正態(tài)模型的乘積。 正態(tài)-正態(tài)模型的功效在于，盡管具有更新的參數(shù)，但這些分布的乘積也是正態(tài)分布。 在貝葉斯行話中，正態(tài)先驗(yàn)分布是共軛先驗(yàn)分布，這意味著它和它的后驗(yàn)分布具有相同的形式。 后驗(yàn)分布是正態(tài)分布這一事實(shí)似乎沒什么大不了的，但是根據(jù)我們要建模的數(shù)據(jù)和我們要估算的參數(shù)，在很多情況下我們的后驗(yàn)分布不是采取這樣熟悉的形式 由于此后驗(yàn)分布是定義明確的，因此我們可以直接從中進(jìn)行采樣，從而輕松地計(jì)算出其后的摘要統(tǒng)計(jì)量。

The notations and re-parametrizations below are from Chapter 5 in Peter Hoff’s textbook, “A First Course in Bayesian Statistics,” the book I used in my first undergraduate Bayesian statistics course and the book I’ve been studying in recent times.

下面的表示法和重新參數(shù)化來自彼得·霍夫(Peter Hoff)教科書“貝葉斯統(tǒng)計(jì)學(xué)的第一門課程”的第5章，這是我在我的第一門貝葉斯統(tǒng)計(jì)學(xué)課程中使用的書，也是我最近所研究的書。

Our posterior distribution takes the form

我們的后驗(yàn)分布形式為

where the first term refers to the posterior mean and second term refers to the posterior variance. The formulas to calculate these updated parameters are

其中第一項(xiàng)指的是后均值，第二項(xiàng)指的是后方方差。 計(jì)算這些更新參數(shù)的公式是

and

和

These formulas may look somewhat intimidating, but hopefully you see some similarities between them. A common practice and a particularly helpful one for gaining intuition about these formulas is to look at the formulas in terms of precision rather than variance. Precision is the inverse of variance.

這些公式可能看起來有些嚇人，但希望您能看到它們之間的相似之處。 獲得這些公式的直覺的一種常見實(shí)踐和一種特別有用的方法是，從精度而不是方差的角度來看這些公式。 精度是方差的倒數(shù)。

In this case, we have three relevant precisions to observe:

在這種情況下，我們需要觀察三個(gè)相關(guān)的精度：

If we invert the posterior variance formula to calculate posterior precision, we see that the posterior precision in terms of standard deviations is

如果我們反轉(zhuǎn)后驗(yàn)方差公式以計(jì)算后驗(yàn)精度，則可以看到以標(biāo)準(zhǔn)差表示的后驗(yàn)精度為

This can be written in terms of precisions as

這可以用精度來表示為

In this form we can clearly see that the posterior precision is the sum of the prior precision and the sample precision multiplied by the sample size. We can also re-write the posterior mean in terms of precisions:

在這種形式下，我們可以清楚地看到后驗(yàn)精度是先驗(yàn)精度與樣本精度的和乘以樣本大小。 我們還可以根據(jù)精度重寫后驗(yàn)均值：

Here, we can clearly see that the posterior mean is a weighted average of the prior mean and sample mean.

在這里，我們可以清楚地看到后驗(yàn)均值是先驗(yàn)均值和樣本均值的加權(quán)平均值。

For our data, the posterior parameters are:

對(duì)于我們的數(shù)據(jù)，后驗(yàn)參數(shù)為：

And there we have them — our updated parameters. Our posterior estimate for the average growth rate is about 52.7% — a decent bit lower than our sample average, but not overwhelmingly lower. We’ve taken a subjective belief, represented that belief with a distribution, and used that distribution to augment our analysis. Hooray! This is the power of Bayesian inference. As long as we can define our beliefs, we can incorporate them in a rigorous way in our analysis. Let’s talk a little more about what we have and also what we don’t have.

有了它們-我們更新的參數(shù)。 我們對(duì)平均增長率的后驗(yàn)估計(jì)約為52.7％，雖然比我們的樣本平均值低了很多，但絕不算低。 我們采用了主觀信念，用分布表示了該信念，并使用該分布來擴(kuò)大我們的分析。 萬歲！ 這就是貝葉斯推理的力量。 只要我們能夠定義我們的信念，我們就可以將其嚴(yán)格地納入我們的分析中。 讓我們?cè)僬勔恍╆P(guān)于我們擁有和不擁有的東西。

With our posterior standard deviation, we can compute a credible interval for our estimate. For those new to Bayesian statistics, a credible interval is not the same thing as a confidence interval even though they are computed in a similar manner. Our 95% credible interval for the posterior mean is .527+/?2?.0391.527+/?2?.0391 which leads to points of 44.88% and 60.52%. With this credible interval, we’re making the statement that we’re 95% sure that the true value of the posterior mean falls within the interval. Even at this point, we don’t treat this updated mean as a known entity. Furthermore, we are not saying that 52.7% is our forecast for revenue growth rate over the next rolling one-year period. If we wanted to make a forecast within this framework, we’d use the posterior predictive distribution. Since that is a separate topic, I won’t touch on it here, but the process of deriving that distribution is similar to deriving the posterior distribution.

利用我們的后驗(yàn)標(biāo)準(zhǔn)差，我們可以計(jì)算出可信的區(qū)間。 對(duì)于貝葉斯統(tǒng)計(jì)新手來說，可信區(qū)間與置信區(qū)間并不相同，即使它們是以類似方式計(jì)算的。 我們的后驗(yàn)平均值的95％可信區(qū)間為.527 +/- 2 * .0391.527 + /-/ 2 * .0391，得出的分?jǐn)?shù)分別為44.88％和60.52％。 在此可信區(qū)間內(nèi)，我們聲明95％的后驗(yàn)均值的真實(shí)值落在該區(qū)間內(nèi)。 即使在這一點(diǎn)上，我們也不會(huì)將這種更新的均值視為已知實(shí)體。 此外，我們并不是說52.7％是我們對(duì)下一個(gè)滾動(dòng)的一年期內(nèi)收入增長率的預(yù)測。 如果我們想在此框架內(nèi)進(jìn)行預(yù)測，則可以使用后驗(yàn)預(yù)測分布。 由于這是一個(gè)單獨(dú)的主題，因此在此不再贅述，但是推導(dǎo)該分布的過程類似于推導(dǎo)后驗(yàn)分布。

Two key implications should be noted from this analysis: the first is that as sample size grows larger, the posterior mean and posterior variance are more and more determined by the sample data. I’m not going to state that there’s an explicit cutoff, but at some amount of data, adding a prior doesn’t move the needle much all else equal. Intuitively, this is reasonable. If you have rich enough sampling data, the sampling data likely represents the actual structure in the data, and you may not see the need to utilize a prior distribution.

此分析應(yīng)注意兩個(gè)關(guān)鍵含義：首先是隨著樣本量的增加，后均值和后方差越來越多地由樣本數(shù)據(jù)決定。 我不會(huì)說有一個(gè)明確的界限，但是在一定數(shù)量的數(shù)據(jù)下，添加一個(gè)先驗(yàn)不會(huì)使其他所有條件都變差。 憑直覺，這是合理的。 如果您有足夠豐富的采樣數(shù)據(jù)，則采樣數(shù)據(jù)可能表示數(shù)據(jù)中的實(shí)際結(jié)構(gòu)，并且您可能看不到需要利用先驗(yàn)分布。

To emphasize the first point, we can re-run our analysis using strictly the year-end data which would leave us with a sample size of five data points. Using the same prior distribution, our new sampling mean and variance are about 59.8% and .012 (or 11.1% standard deviation), and our posterior mean and variance are 23% and .0019 (or 4.45% standard deviation). This posterior estimate for the mean is much lower than what we saw in our first iteration; with our sample size cut significantly, the prior plays a much heavier role in the output. The standard deviation didn’t change as much, but we can see that it’s larger even though our sampling standard deviation was smaller the second time around. We have a much lower estimate, and we have slightly less confidence in the estimate (wider credible interval).

為了強(qiáng)調(diào)第一點(diǎn)，我們可以嚴(yán)格使用年終數(shù)據(jù)來重新運(yùn)行分析，這將使我們擁有五個(gè)數(shù)據(jù)點(diǎn)的樣本量。 使用相同的先驗(yàn)分布，我們的新采樣均值和方差分別為59.8％和.012(或11.1％標(biāo)準(zhǔn)偏差)，而后驗(yàn)均值和方差分別為23％和.0019(或4.45％標(biāo)準(zhǔn)偏差)。 該均值的后驗(yàn)估計(jì)值比我們?cè)诘谝淮蔚锌吹降囊偷枚唷?由于我們的樣本量大大減少，因此先驗(yàn)數(shù)據(jù)在輸出中起著舉足輕重的作用。 標(biāo)準(zhǔn)偏差變化不大，但是即使第二次采樣標(biāo)準(zhǔn)偏差較小，我們也可以看到它更大。 我們的估算值要低得多，而我們對(duì)估算值的信心則稍差(可信區(qū)間更大)。

The second implication of our analysis is that the smaller the prior variance, the greater the prior precision and the greater impact it has on both the posterior mean and posterior variance. The more confidence we have in our prior, the more it will affect our posterior estimates. To illustrate this point, I re-ran our original analysis with different values for the prior variance. The values for the prior mean are all .045, and the sampling mean and variance come from our rolling revenue data. The table below shows the results of this experiment.

我們的分析的第二個(gè)含義是，先驗(yàn)方差越小，先驗(yàn)精度越高，它對(duì)后均值和后方方差的影響越大。 我們對(duì)先驗(yàn)的信心越高，對(duì)后驗(yàn)估計(jì)的影響就越大。 為了說明這一點(diǎn)，我使用先前的方差的不同值重新運(yùn)行了我們的原始分析。 先前均值均為0.045，而抽樣均值和方差來自我們的滾動(dòng)收入數(shù)據(jù)。 下表顯示了該實(shí)驗(yàn)的結(jié)果。

I’ll also plot the distributions.

我還將繪制分布。

Notice how much closer to the prior mean our posterior distribution with prior variance set to .05 is. As we increase our prior variance (effectively signifying less confidence in the prior mean), the center of our posterior distribution moves closer to the sample mean. Also, while the magnitude of the changes in the posterior variances may not appear that great in the table, from the distribution plots above, we can see how the distributions get progressively wider; in other words, the credible interval for the true value of average growth widens.

注意，先驗(yàn)方差設(shè)置為.05的后驗(yàn)分布離先驗(yàn)均值有多近。 隨著我們?cè)黾酉闰?yàn)方差(有效地表示對(duì)先驗(yàn)均值的置信度降低)，我們后驗(yàn)分布的中心移近樣本均值。 同樣，盡管后驗(yàn)方差變化的幅度在表格中可能看起來不太大，但從上面的分布圖來看，我們可以看到分布如何逐漸變寬。 換句話說，平均增長真實(shí)值的可信區(qū)間變寬了。

摘要 (Summary)

Just to recap, we were analyzing a young company and wanted to estimate the true growth rate of its revenue. Given the small amount of sample data we had and a subjective belief that the average growth rate will be less than what the sample data suggests, we used Bayesian inference to augment our analysis. We defined a sampling model for our data, defined a prior for the average growth rate that reflected our subjective view, and utilized the normal-normal model to arrive at a posterior estimate and interval for the company’s average growth rate. I hope you found this brief introduction to Bayesian inference as well as the analysis of the results useful. I don’t recommend using the specific numbers in this piece for any valuation of MongoDB, but hopefully you can apply the concepts to your own analysis. I’m attaching a link to the GitHub repository for the code; nothing is particularly complicated, but I’ll share it in the spirit of transparency and reproducibility.

回顧一下，我們正在分析一家年輕的公司，并希望估計(jì)其收入的真實(shí)增長率。 考慮到我們擁有的樣本數(shù)據(jù)量很少，并且主觀認(rèn)為平均增長率將低于樣本數(shù)據(jù)表明的速度，因此我們使用貝葉斯推斷來增強(qiáng)我們的分析。 我們?yōu)閿?shù)據(jù)定義了一個(gè)采樣模型，為反映我們主觀觀點(diǎn)的平均增長率定義了先驗(yàn)，并利用正常-正常模型得出了公司平均增長率的后驗(yàn)估計(jì)和區(qū)間。 我希望您對(duì)貝葉斯推理的簡要介紹以及對(duì)結(jié)果的分析有用。 對(duì)于MongoDB的任何評(píng)估，我不建議使用本文中的特定數(shù)字，但希望您可以將這些概念應(yīng)用于您自己的分析。 我正在將代碼的鏈接附加到GitHub存儲(chǔ)庫； 沒有什么特別復(fù)雜，但是我將本著透明和可復(fù)制的精神來分享。

https://github.com/vinai-oddiraju/TDS_Blog_Post1.git

https://github.com/vinai-oddiraju/TDS_Blog_Post1.git

Lastly, I want to thank the friends and family members who took time to read my drafts and provide feedback throughout the process. As this is my first time writing about a project in this manner, their support is especially appreciated. Thanks, and take care!

最后，我要感謝花時(shí)間閱讀我的草稿并在整個(gè)過程中提供反饋的朋友和家人。 由于這是我第一次以此方式撰寫項(xiàng)目，因此特別感謝他們的支持。 謝謝，保重！

免責(zé)聲明 (Disclaimer)

The thoughts and views expressed in this report are mine alone and do not necessarily reflect the views of my firm. This report is intended to be educational in nature and should not be construed as individual investment advice nor as a recommendation to buy, sell, or hold any security or to adopt any investment strategy.

本報(bào)告中表達(dá)的思想和觀點(diǎn)僅屬于我個(gè)人，不一定反映我公司的觀點(diǎn)。 本報(bào)告本質(zhì)上是具有教育意義的報(bào)告，不應(yīng)解釋為個(gè)人投資建議，也不能解釋為購買，出售或持有任何證券或采用任何投資策略的建議。

資料來源 (Sources)

[1] Hoff, Peter D. A First Course in Bayesian Statistical Methods (2007). Print.

[1] Hoff，PeterD。 貝葉斯統(tǒng)計(jì)方法的第一門課程 (2007年)。 打印。

翻譯自: https://towardsdatascience.com/a-bayesian-approach-to-estimating-revenue-growth-55d029efe2dd

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