python3中樸素貝葉斯

你在這里 (You are here)

If you’re reading this, odds are: (1) you’re interested in bayesian statistics but (2) you have no idea how Markov Chain Monte Carlo (MCMC) sampling methods work, and (3) you realize that all but the simplest, toy problems require MCMC sampling so you’re a bit unsure of how to move forward.

如果您正在閱讀此書(shū)，則可能是：(1)您對(duì)貝葉斯統(tǒng)計(jì)感興趣，但(2)您不知道馬爾可夫鏈蒙特卡洛(MCMC)采樣方法的工作原理，以及(3)您意識(shí)到，除了最簡(jiǎn)單的玩具問(wèn)題需要MCMC采樣，因此您不確定如何前進(jìn)。

Not to worry, we’ll explore this tricky concept using a 1-dimensional normal distribution, using nothing more than python, user-defined functions, and the random module (specifically, the uniform distribution.) We’ll discuss all this nonsense in terms of bars, beers, and a night out with your pals.

不用擔(dān)心，我們將使用一維正態(tài)分布探索這個(gè)棘手的概念，僅使用python，用戶定義函數(shù)和隨機(jī)模塊(特別是統(tǒng)一分布)。我們將在下面討論所有這些廢話。酒吧，啤酒和與朋友共度夜晚的條件

在酒吧想象自己 (Picture yourself at a bar)

It’s Friday night, you and your gang head out for burgers, beer and to catch a televised baseball game. Let’s say you arbitrarily pick a first bar (we’ll call it Larry’s Sports Bar), you sit down, grab a menu and consider your options. If the food and drinks are affordable, that’s a good reason to stay; but if there’s standing room only, you’ve got a reason to leave. If the bar’s filled with big screen TVs all dialed into the right game, another reason to stay. If the food or drinks aren’t appealing, you’ve got another reason to leave. Etc etc, you get the point.

這是星期五晚上，您和幫派前往漢堡，啤酒和觀看電視轉(zhuǎn)播的棒球比賽。 假設(shè)您隨意選擇第一個(gè)酒吧(我們稱(chēng)其為L(zhǎng)arry's Sports Bar )，您坐下來(lái)，拿起一個(gè)菜單并考慮您的選擇。 如果食物和飲料負(fù)擔(dān)得起，那就是留下的充分理由。 但是如果只有客廳，那您就有理由離開(kāi)。 如果酒吧的大屏幕電視都播放了正確的游戲，這是留下的另一個(gè)原因。 如果食物或飲料不受歡迎，您還有其他理由離開(kāi)。 等等，您明白了。

Now, let’s say one (or several) of your friends have gripes with the current establishment — the food is cold, the beers are overpriced, whatever the reason. So he proposes the gang leave Larry’s Sports Bar in favor of Tony’s Pizzeria and Beer Garden because the food there is better, fresher, etc. So the gang deliberates, asking the questions (A) how favorable is Larry’s? (B) How favorable is Tony’s? And (C) how favorable is Tony’s relative to Larry’s?

現(xiàn)在，比方說(shuō)，您的一個(gè)(或幾個(gè))朋友對(duì)當(dāng)前的餐館感到不快-不管是什么原因，食物都很冷，啤酒價(jià)格過(guò)高。 因此，他建議該團(tuán)伙離開(kāi)拉里的體育館 ，轉(zhuǎn)而使用托尼的比薩店和啤酒花園，因?yàn)槟抢锏氖澄锔?#xff0c;更新鮮，等等。于是，該團(tuán)伙在考慮以下問(wèn)題：(A)拉里一家有多有利？ (B)托尼的優(yōu)惠程度如何？ (C) 托尼相對(duì)于拉里有多有利？

This relative comparison is really the most important detail. If Tony’s is only marginally better than Larry’s (or far worse) there’s a good chance that it’s not worth the effort to relocate. But if Tony’s is unambiguously the better of the two, there’s only a slim chance that you might stay. This is the real juice that makes Metropolis-Hastings “work.”

相對(duì)比較確實(shí)是最重要的細(xì)節(jié)。 如果Tony的房屋僅比Larry的房屋略好(或更差)，則很有可能不值得重新安置。 但是，如果托尼(Tony's)無(wú)疑是兩者中的佼佼者，那么您留下的可能性很小。 這是使Metropolis-Hastings“工作”的真正汁液 。

算法 (The algorithm)

Metropolis-Hastings algorithm does:

Metropolis-Hastings算法可以：

Start with a random sample

從隨機(jī)樣本開(kāi)始

Determine the probability density associated with the sample

確定與樣本相關(guān)的概率密度

Propose a new, arbitrary sample (and determine its probability density)

提出一個(gè)新的任意樣本(并確定其概率密度)

Compare densities (via division), quantifying the desire to move

比較密度(通過(guò)除法)，量化移動(dòng)的欲望

Generate a random number, compare with desire to move, and decide: move or stay

生成一個(gè)隨機(jī)數(shù)，與移動(dòng)的欲望進(jìn)行比較，并決定：移動(dòng)還是停留

Repeat

重復(fù)

The real key is (as we’ve discussed) quantifying how desirable the move is as an action/inaction criteria, then (new stuff alert!) observe a random event, compare to said threshold, and make a decision.

真正的關(guān)鍵是(正如我們已經(jīng)討論過(guò)的那樣)量化移動(dòng)作為行動(dòng)/不作為標(biāo)準(zhǔn)的期望程度，然后(新事物警報(bào)！) 觀察隨機(jī)事件 ，與所述閾值進(jìn)行比較，然后做出決定。

隨機(jī)事件 (The random event)

For our purposes, our threshold is the ratio of the proposed sample’s probability density to the current sample’s probability density. If this threshold is near (or above) 1, it means that the previous location was highly undesirable (a number close to 0, very improbable) and the proposed location is highly desirable (as close to 1 as is possible, near distribution’s expectation). Now, we need to generate a number in the range [0,1] from the uniform distribution. If the number produced is less than or equal to the threshold, we move. Otherwise, we stay. That’s it!

出于我們的目的，我們的閾值是建議樣本的概率密度與當(dāng)前樣本的概率密度之比。 如果此閾值接近(或高于)1，則意味著先前的位置非常不理想(數(shù)字接近0，非常不可能)，建議的位置也非常理想(盡可能接近1，接近分布的期望值) 。 現(xiàn)在，我們需要根據(jù)均勻分布生成一個(gè)在[0,1]范圍內(nèi)的數(shù)字。 如果產(chǎn)生的數(shù)字小于或等于閾值，則移動(dòng)。 否則，我們留下。 而已！

那硬東西呢？ (What about the hard stuff?)

This is starting to sound a little too good to be true, right? We haven’t discussed Markov Chains or Monte Carlo simulations yet but fret not. Both are huge topics in their own right and we only need the most basic familiarity with each to make use of MCMC magic.

這聽(tīng)起來(lái)似乎太好了，難以置信，對(duì)吧？ 我們尚未討論馬爾可夫鏈或蒙特卡洛模擬，但不用擔(dān)心。 兩者本身都是很重要的話題，我們只需要對(duì)它們有最基本的了解就可以使用MCMC魔術(shù)。

A Markov Chain is is a chain of discrete events where the probability of the next event is conditioned only upon the current event. (ex: I just finished studying, do I go to sleep or go to the bar? These are my only choices of a finite set.) In this system of discrete choices, there exists a transition matrix, which quantifies the probability of transitioning from any given state to any given state. A Monte Carlo method is really just a fancy name for a simulation/experiment that relies on usage of random numbers.

馬爾可夫鏈?zhǔn)请x散事件的鏈，其中下一個(gè)事件的概率僅取決于當(dāng)前事件。 (例如：我剛剛完成學(xué)習(xí)，我要去睡覺(jué)還是去酒吧？ 這是我對(duì)有限集合的唯一選擇。 )在這個(gè)離散選擇的系統(tǒng)中，存在一個(gè)轉(zhuǎn)換矩陣，該矩陣量化從任何給定狀態(tài)到任何給定狀態(tài)。 蒙特卡洛方法實(shí)際上只是依賴(lài)于隨機(jī)數(shù)使用的模擬/實(shí)驗(yàn)的奇特名稱(chēng)。

As discussed, we’ll be sampling from the normal distribution — a continuous, not discrete, distribution. So how can there be a transition matrix? Surprise!— there’s no transition matrix at all. (It’s actually called a Markov kernel, and it’s just the comparison of probability densities, at least for our purposes.)

正如討論的那樣，我們將從正態(tài)分布中進(jìn)行采樣-連續(xù)而不是離散的分布。 那么如何有一個(gè)過(guò)渡矩陣呢？ 驚喜！ —根本沒(méi)有過(guò)渡矩陣。 (實(shí)際上，它被稱(chēng)為Markov核 ，它只是概率密度的比較，至少出于我們的目的。)

代碼 (The code)

below, we define three functions: (1) Normal, which evaluates the probability density of any observation given the parameters mu and sigma. (2) Random_coin, which references a fellow TDS writer’s post (lined below). And (3) Gaussian_mcmc, which samples executes the algorithm as described above.

下面，我們定義三個(gè)函數(shù)：(1)正態(tài)，在給定參數(shù)mu和sigma的情況下評(píng)估任何觀測(cè)值的概率密度。 (2)Random_coin，它引用了TDS同行作家的帖子(如下所示)。 以及(3)采樣的高斯_mcmc執(zhí)行上述算法。

As promised, we’re not calling any Gaussian or normal function from numpy, scipy, etc. In the third function, we initialize a current sample as an instance of the uniform distribution (where the lower and upper boundaries are +/- 5 standard deviations from the mean.) Likewise, movement is defined in the same way. Lastly, we move (or stay) based on the random event’s observed value in relation to acceptance, which is the probability density comparison discussed at length elsewhere.

如所承諾的，我們不會(huì)從numpy，scipy等中調(diào)用任何高斯函數(shù)或正態(tài)函數(shù)。在第三個(gè)函數(shù)中，我們將當(dāng)前樣本初始化為均勻分布的實(shí)例(上下邊界為+/- 5標(biāo)準(zhǔn)同樣，以相同的方式定義運(yùn)動(dòng)。 最后，我們根據(jù)與接受相關(guān)的隨機(jī)事件的觀察值來(lái)移動(dòng)(或停留)，這是在其他地方詳細(xì)討論的概率密度比較。

Github Gist吉斯圖·吉斯特(Github Gist) Histogram vs actual distribution直方圖與實(shí)際分布

Lastly, one must always give credit where credit is due: Rahul Agarwal’s post defining a Beta distribution MH sampler was instrumental to my development of the above Gaussian distribution MH sampler. I encourage you to read his post as well for a more detailed exploration of the foundational concepts, namely Markov Chains and Monte Carlo simulations.

最后，必須始終在應(yīng)歸功的地方給予信譽(yù)： Rahul Agarwal的職位定義了Beta分布MH采樣器，這對(duì)我開(kāi)發(fā)上述高斯分布MH采樣器至關(guān)重要。 我鼓勵(lì)您也閱讀他的文章，以更詳細(xì)地探索基本概念，即馬爾可夫鏈和蒙特卡洛模擬。

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感謝您的閱讀-如果您認(rèn)為我的內(nèi)容還可以，請(qǐng)訂閱！ :)

翻譯自: https://towardsdatascience.com/bayesian-statistics-metropolis-hastings-from-scratch-in-python-c3b10cc4382d

python3中樸素貝葉斯

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