WEBVTT 1 00:00:00.000 --> 00:00:08.477 [MUSIC] [淺談貝氏統計] 2 00:00:08.477 --> 00:00:12.890 In this lecture we'll talk about Bayesian statistics. 這一講來談談貝氏統計 3 00:00:12.890 --> 00:00:16.190 In a previous lecture we talked about likelihoods and expressing relative 前一講介紹似然性，以及如何表達證據支持兩種假設的相對程度 4 00:00:16.190 --> 00:00:19.410 evidence for one hypothesis compared to an alternative hypothesis. 前一講介紹似然性，以及如何表達證據支持兩種假設的相對程度 5 00:00:19.410 --> 00:00:24.810 And we've used evidence in this sense just based on the data that we have at hand. 這讓我們能用手上的資料做為證據 6 00:00:24.810 --> 00:00:27.960 But we'll see that very often you have some prior belief that you might want to 貝氏統計能結合證據與根據假設的事前信念(prior)，計算推論的可靠性 7 00:00:27.960 --> 00:00:31.460 incorporate with the evidence and Bayesian statistics allows you to do this. 貝氏統計能結合證據與根據假設的事前信念(prior)，計算推論的可靠性 8 00:00:32.560 --> 00:00:34.490 Now let's say that you flip a coin three times. 假想一枚硬幣丟了三次 9 00:00:35.680 --> 00:00:38.860 The coin comes up heads every single time. 每一次都是正面朝上的結果 10 00:00:38.860 --> 00:00:41.420 Now do you think that this is a fair coin or not? 根據這個結果能判斷這是枚公正硬幣嗎？ 11 00:00:42.670 --> 00:00:46.440 The data that you have at hand shows that this coin came up heads three times. 因為眼前的資料是三次正面朝上 12 00:00:46.440 --> 00:00:52.380 So if you would be a newborn baby then you don't have any prior beliefs and 對一位全無任何事前信念的新生兒來說 13 00:00:52.380 --> 00:00:54.520 all you have is the data that you just observed. 只有眼前看到的資料 14 00:00:54.520 --> 00:00:59.810 So for a newborn it comes up heads every single time. 新生兒會認為這枚硬幣總會擲出正面朝上 15 00:00:59.810 --> 00:01:02.030 This is just what this coin does. 正如現在看到的事實 16 00:01:02.030 --> 00:01:05.100 But you have prior beliefs that most coins are fair. 對於會看這部影片的大人來說，都有多數硬幣是公正的信念 17 00:01:05.100 --> 00:01:07.230 You might think 你也許認為就算現在是這種結果 18 00:01:07.230 --> 00:01:11.030 Three heads in a row that's a perfectly likely observation. 也只是剛好看到三枚正面朝上而已 19 00:01:11.030 --> 00:01:15.250 I'm not changing my prior beliefs that this is a fair coin yet. 你不會因此改變這是一枚公正硬幣的信念 20 00:01:15.250 --> 00:01:16.490 You would want to see more evidence. 你會想要再做更多試驗 21 00:01:18.670 --> 00:01:24.010 Now data is possible in this Bayesian sense. 貝氏統計能結合個人信念與事實證據 22 00:01:24.010 --> 00:01:27.260 It's not possible when you just calculate P-values. 這是p值做不到的功能 23 00:01:27.260 --> 00:01:31.136 Remember that the P-value expresses the probability of the data or 請記住p值表達以虛無假設為真的前提，不論手上資料是否極端，確實存在的程度 24 00:01:31.136 --> 00:01:34.678 more extreme data assuming that the null hypothesis is true. 請記住p值表達以虛無假設為真的前提，不論手上資料是否極端，確實存在的程度 25 00:01:34.678 --> 00:01:38.544 And some people say actually not what you want to know. 有人認為p值不能代表研究者想要得到的最終解答 26 00:01:38.544 --> 00:01:42.727 What you want to know is the probability that the null hypothesis is true 研究者要追求的是根據手上資料，虛無假設確實存在的程度 27 00:01:42.727 --> 00:01:45.092 given some data that you have collected. 研究者要追求的是根據手上資料，虛無假設確實存在的程度 28 00:01:45.092 --> 00:01:47.077 And this is a posterior probability. 這就是事後機率(posterior probability) 29 00:01:47.077 --> 00:01:50.699 Given that you've collected some data maybe some prior beliefs that you have. 根據手上的資料，加上已存在研究者腦中的信念 30 00:01:50.699 --> 00:01:53.532 What is now the probability that the null hypothesis or 能不能獲得虛無假設或對立假設為真的機率? 31 00:01:53.532 --> 00:01:56.240 the alternative hypothesis is true? 能不能獲得虛無假設或對立假設為真的機率? 32 00:01:56.240 --> 00:01:59.090 So Bayesian statistics allows you to express this probability. 貝氏統計正是用來計算事後機率 33 00:02:01.090 --> 00:02:02.630 It works quite straightforward. 貝氏統計的原則很簡單 34 00:02:02.630 --> 00:02:05.890 You have some prior belief you have some data. 把腦中的事前信念加上手上的資料 35 00:02:05.890 --> 00:02:08.110 You combine these into a posterior belief. 就能獲得目前證據的事後機率 36 00:02:10.530 --> 00:02:14.935 We can calculate posterior odds that the alternative hypothesis is true given 根據手上資料計算的事後機率比(posterior odds) 可以比較對立假設為真與虛無假設為真的程度 37 00:02:14.935 --> 00:02:15.535 the data 根據手上資料計算的事後機率比(posterior odds) 可以比較對立假設為真與虛無假設為真的程度 38 00:02:15.535 --> 00:02:19.884 compared to the probability that the null hypothesis is true given the data. 根據手上資料計算的事後機率比(posterior odds) 可以比較對立假設為真與虛無假設為真的程度 39 00:02:19.884 --> 00:02:25.499 This is a very useful way to split apart these different aspects of the formula. 對照公式的每個部分與貝氏原則可以看到 40 00:02:25.499 --> 00:02:31.090 So the posterior equals the likelihood ratio times the prior. 事後機率就是似然性比與事前信念的乘積 41 00:02:31.090 --> 00:02:33.224 So this likelihood ratio 公式裡的似然性比 42 00:02:33.224 --> 00:02:35.138 that we discussed in a previous lecture 就是前一講介紹的重點 43 00:02:35.138 --> 00:02:38.818 is an essential part of the probability calculations in Bayesian statistics. 可以看到似然性比是計算事後機率的重要成份 44 00:02:38.818 --> 00:02:42.672 But it's also combined with this prior so how can we do this? 不過要結合事前機率，那麼要怎麼知道事前機率呢？ 45 00:02:42.672 --> 00:02:47.050 We have to come up with a prior distribution. 在此介紹一種事前機率分配 46 00:02:47.050 --> 00:02:51.670 In the case of binomial probabilities a beta distribution is used. 所謂的beta分配是用在資料有二項性質的狀況 47 00:02:51.670 --> 00:02:55.450 The beta prior is determined by two parameters which are referred to as beta事前分配由alpha與beta兩種參數決定 48 00:02:55.450 --> 00:02:56.200 alpha and beta. beta事前分配由alpha與beta兩種參數決定 49 00:02:57.380 --> 00:03:01.580 Now this can get a little confusing because we already talked about alphas and 在這裡你也許會和之前提過代表型一與型二錯誤率的alpha與beta搞混 50 00:03:01.580 --> 00:03:04.079 betas as type one errors and type two errors. 在這裡你也許會和之前提過代表型一與型二錯誤率的alpha與beta搞混 51 00:03:05.700 --> 00:03:09.840 Regrettably are not very creative when it comes to 這是因為統計學家使用希臘字母做統計符號的創意不夠所致 52 00:03:09.840 --> 00:03:13.280 thinking up greek letters that they want to use for their statistics. 這是因為統計學家使用希臘字母做統計符號的創意不夠所致 53 00:03:13.280 --> 00:03:15.530 So there are some double use in the literature 有時一種符號在同一篇論文裡有很多意思 54 00:03:15.530 --> 00:03:17.320 this is one of these examples. beta分配就是一個例子 55 00:03:17.320 --> 00:03:19.820 The beta prior is determined by alpha and beta. 這裡的alpha與beta只是決定beta分配的性質 56 00:03:19.820 --> 00:03:23.010 These are just two numbers that have nothing to do with error rates whatsoever. 與任何形式的錯誤率全無關係 57 00:03:24.900 --> 00:03:28.670 Let's take a look a different versions of prior distributions. 來看看幾種不同的beta事前分配 58 00:03:28.670 --> 00:03:32.761 If we set the alpha and the beta in the beta probability to 1 這道beta分配的alpha與beta都是1 59 00:03:32.761 --> 00:03:35.659 we see that we got the perfectly flat line. 呈現出一條完美的水平線 60 00:03:35.659 --> 00:03:40.259 This means that every value of theta which is plotted on 也就是說得到範圍內任一統計值(theta)的機率都是相等的 61 00:03:40.259 --> 00:03:43.659 the horizontal axis is equally likely. 也就是說得到範圍內任一統計值(theta)的機率都是相等的 62 00:03:43.659 --> 00:03:46.100 So we don't have any expectations and 這種分配符合不預期特定結果的信念 63 00:03:46.100 --> 00:03:49.160 we say anything that might happen is possible. 可以說任何結果都有可能發生 64 00:03:50.580 --> 00:03:52.570 Now in the case of a coin flip 在投擲硬幣的例子裡 65 00:03:52.570 --> 00:03:55.920 you might not be convinced that this is what's most likely. 就是代表不會有經常觀察到某種結果的信念 66 00:03:55.920 --> 00:04:00.110 In the case of flipping a coin you can have a lot of different beliefs. 當然對於硬幣是否公正的信念可以因人而異 67 00:04:01.280 --> 00:04:06.000 One of them is the belief that this is a coin that always comes up heads. 其中一種信念是想信這枚硬幣永遠擲出正面朝上 68 00:04:06.000 --> 00:04:09.990 If you have a very strong belief that this is the case 若你相信如此，這道beta分配完美體現你的想法 69 00:04:09.990 --> 00:04:12.665 You think that it will be heads almost always 雖然少數情況會出現不同的結果(theta接近0)，多數情況都會預期正面朝上(theta接近1) 70 00:04:12.665 --> 00:04:17.341 although you are willing to give at least some probability of different values. 雖然少數情況會出現不同的結果(theta接近0)，多數情況都會預期正面朝上(theta接近1) 71 00:04:17.341 --> 00:04:21.183 You might also think that this is a perfectly fair coin 你也可能根據多數硬幣是公正的，而相信眼前這一枚也是 72 00:04:21.183 --> 00:04:23.320 most coins are fair after all. 你也可能根據多數硬幣是公正的，而相信眼前這一枚也是 73 00:04:23.320 --> 00:04:27.670 So you think that this coin should come up heads 50% of the time. 你認為每次擲出正面的機率是50% 74 00:04:27.670 --> 00:04:29.300 You have a very strong belief in this. 把這樣的信念轉換成beta分配 75 00:04:30.320 --> 00:04:33.210 And it's illustrated in this graph where there's a very high peak 分配就是theta等於0.5之處有一個高峰 76 00:04:33.210 --> 00:04:34.810 at 0.5 on the theta. 分配就是theta等於0.5之處有一個高峰 77 00:04:35.940 --> 00:04:39.450 But you're also willing to allow for some variation. 當然你也可以認為存在其它可能的結果 78 00:04:39.450 --> 00:04:42.580 It might be that the coin is not perfectly fair. 因為一些原因，眼前這枚硬幣並不公正 79 00:04:42.580 --> 00:04:45.080 It might have been used and it might be an old coin and 也許被反覆使用，也許太舊了 80 00:04:45.080 --> 00:04:48.249 might come up heads a little bit more often or a little less often. 使得擲出正面的次數多一些 81 00:04:51.710 --> 00:04:55.030 Now let's combine our prior belief whatever it is 來看看怎樣把這些代表事前信念的beta分配 82 00:04:55.030 --> 00:04:57.230 with the data that we've observed. 與觀察獲得的資料結合在一起吧 83 00:04:57.230 --> 00:05:01.810 So we have beta prior distribution and we have the likelihood function 現在有beta分配代表事前機率 手上資料可轉換成似然性函數 84 00:05:01.810 --> 00:05:06.030 and we can quite easily combine these into a posterior distribution. 結合兩者就能立刻獲得事後機率分配 85 00:05:08.240 --> 00:05:12.840 The posterior is also a beta distribution with an alpha and a beta value. 事後機率分配也是一道有alpha與beta值的beta分配 86 00:05:12.840 --> 00:05:16.650 And the alpha and the beta are determined simply by 要計算事後分配的alpha和beta，只要把事前的參數加上似然性函數估計的參數，再減去1就行了 87 00:05:16.650 --> 00:05:21.640 adding the alpha of the prior to the alpha of the likelihood function minus 1. 要計算事後分配的alpha和beta，只要把事前的參數加上似然性函數估計的參數，再減去1就行了 88 00:05:21.640 --> 00:05:23.470 The same is true for the beta. 要計算事後分配的alpha和beta，只要把事前的參數加上似然性函數估計的參數，再減去1就行了 89 00:05:23.470 --> 00:05:26.190 The beta of the prior distribution added to the beta 要計算事後分配的alpha和beta，只要把事前的參數加上似然性函數估計的參數，再減去1就行了 90 00:05:26.190 --> 00:05:27.964 of the likelihood distribution minus 1. 要計算事後分配的alpha和beta，只要把事前的參數加上似然性函數估計的參數，再減去1就行了 91 00:05:29.590 --> 00:05:30.710 Let's see how this works. 來看實際的例子 92 00:05:32.110 --> 00:05:36.760 In this case we have a prior distribution that's uninformative 這道事前分配是無資訊的(uninfomative)，是最容易理解的情況 93 00:05:37.920 --> 00:05:40.972 So this is a prior before we collect some data. 代表未收集資料前認定任何結果發生的機率都相等 94 00:05:40.972 --> 00:05:44.600 And now we collect some data and we have a likelihood function. 收集資料後有了這道似然性函數(藍色虛線) 95 00:05:44.600 --> 00:05:47.510 In this case we flip the coin ten times. 這道函數表示投了硬幣十次 96 00:05:47.510 --> 00:05:50.810 Six out of the ten times we observed heads. 其中六次看到正面朝上 97 00:05:50.810 --> 00:05:53.550 So that's expressed by this likelihood function and 所以看到高峰出現在theta等於6 98 00:05:53.550 --> 00:05:57.160 now we will combine these two functions into a posterior distribution. 接著把兩道分配結合在一起，就會看到事後分配的模樣 99 00:05:59.590 --> 00:06:04.600 You now see a black line that falls exactly on top of the likelihood function. 事後分配的黑線與似然性函數的曲線完美疊合 100 00:06:04.600 --> 00:06:07.730 This is what's meant with an uninformative prior. 這是因為事前分配是無資訊的 101 00:06:07.730 --> 00:06:10.423 Everything was equally likely beforehand and 事前認為任何結果都有可能 102 00:06:10.423 --> 00:06:13.123 it doesn't influence our judgments in any way. 這樣的信念不影響對結果的判斷 103 00:06:13.123 --> 00:06:16.214 We just believe whatever we have observed exactly the same. 所以事後分配與實際資料一致 104 00:06:16.214 --> 00:06:19.800 So the prior does not influence the posterior distribution in any way. 因此這個例子事前信念毫不影響事後分配 105 00:06:21.180 --> 00:06:23.630 Let's take another example where this is the case 接著來看另一個事前信念確實影響事後分配的例子 106 00:06:23.630 --> 00:06:26.680 where the prior does influence the posterior distribution. 接著來看另一個事前信念確實影響事後分配的例子 107 00:06:28.620 --> 00:06:32.570 Here we have some belief that the coin is probably fair. 這道事前分配表示硬幣是公正的 108 00:06:32.570 --> 00:06:36.680 We think that values around 0.5 are somewhat more likely but 設定統計值在0.5多少表達這樣的信念 109 00:06:36.680 --> 00:06:39.220 we don't have a very strong conviction about this. 不過分配的變異表示對此信念不會過度堅持 110 00:06:40.730 --> 00:06:42.600 We again observe the same data. 獲得的資料也是10次之中擲出6次正面 111 00:06:42.600 --> 00:06:47.450 We flip the coin ten times six out of ten the coin lands heads. 獲得的資料也是10次之中擲出6次正面 112 00:06:47.450 --> 00:06:50.170 So this is exactly the same likelihood function that we had in 所以得到與前一個例子同樣的似然性函數 113 00:06:50.170 --> 00:06:53.770 the previous example but we have a slightly different prior. 只是擁有不大相同的事前信念 114 00:06:53.770 --> 00:06:58.030 Combining our prior belief with the observed data yields a posterior 兩者結合之後就能得到一道新的事後機率分配 115 00:06:58.030 --> 00:07:02.040 distribution that no longer falls exactly on top of the likelihood function. 不過這道分配和似然性函數完全不疊合 116 00:07:03.700 --> 00:07:06.250 Indeed we see that the posterior 這道事後機率分配稍微偏向事前信念的beta分配 117 00:07:06.250 --> 00:07:10.720 is slightly shrinking towards the prior belief that we held. 這道事後機率分配稍微偏向事前信念的beta分配 118 00:07:15.080 --> 00:07:18.850 Now the posterior distribution 有了事前機率分配與事後率分配 119 00:07:18.850 --> 00:07:22.170 we can calculate something that's known as a Bayes factor. 就能計算貝氏因子(Bayes Factor) 120 00:07:22.170 --> 00:07:26.590 The Bayes factor is the relative evidence for one model compared to another model. 貝式因子表示手上證據支持兩種模型的相對程度 121 00:07:28.170 --> 00:07:30.239 Now compare this to likelihood ratios. 現在我們用似然性比值計算支持程度 122 00:07:30.239 --> 00:07:33.345 In a likelihood ratio we only have one distribution and 透過這道似然性函數，比較兩個統計值的似然性 123 00:07:33.345 --> 00:07:37.749 we're comparing two different values of theta on the likelihood function. 透過這道似然性函數，比較兩個統計值的似然性 124 00:07:37.749 --> 00:07:41.600 But here we'll use the likelihood from the prior distribution and 現在要用這道函數計算事前分配的似然性 125 00:07:41.600 --> 00:07:45.900 compare it to the likelihood of the posterior distribution. 與事後分配的似然性相互比較 126 00:07:45.900 --> 00:07:46.430 Let's take a look. 接著看要怎麼做 127 00:07:48.660 --> 00:07:51.450 We have data from 20 coin flips. 現在手上有擲20枚硬幣的資料 128 00:07:51.450 --> 00:07:52.960 They came up heads 10 times. 其中10次正面朝上 129 00:07:54.430 --> 00:07:56.670 Let's take a look at two different priors. 對此有兩種事前信念 130 00:07:56.670 --> 00:08:01.581 Either a Beta prior of (1 means it's completely uninformative 一種是不知這枚硬幣公正與否的均勻分配beta(1 131 00:08:01.581 --> 00:08:02.817 the uniform prior. 一種是不知這枚硬幣公正與否的均勻分配beta(1 132 00:08:02.817 --> 00:08:05.517 But we can also compare it to a Beta of (4 另一種是認為這是枚公正硬幣的beta(4 133 00:08:05.517 --> 00:08:09.343 4) where we have some expectation that this is a fair coin. 另一種是認為這是枚公正硬幣的beta(4 134 00:08:09.343 --> 00:08:13.910 Now see the prior distribution in 這張圖中代表事前分配的灰色線正是均勻分佈 135 00:08:13.910 --> 00:08:17.810 the gray line that's uniformly distributed. 這張圖中代表事前分配的灰色線正是均勻分佈 136 00:08:17.810 --> 00:08:20.480 We don't have very strong prior beliefs in this case. 這表示沒有對特定結果存在強烈信念 137 00:08:20.480 --> 00:08:22.510 Everything is equally likely. 任何結果都可以接受 138 00:08:22.510 --> 00:08:26.010 Then we've collected some data now we have a posterior distribution 收集資料後得到的事後分配 139 00:08:26.010 --> 00:08:28.859 which in this case falls exactly on top of the likelihood function. 與由此獲得的似然性函數疊合 140 00:08:29.960 --> 00:08:33.770 You can see that there is a relative difference of the likelihood of 這裡可以看到兩個分配在任意一點的似然性差異 141 00:08:33.770 --> 00:08:36.770 both of these functions at specific points. 這裡可以看到兩個分配在任意一點的似然性差異 142 00:08:36.770 --> 00:08:39.970 Now hypothesis that we're interested in. 現在必須要決定真正有興趣的假設是針對那一個值 143 00:08:39.970 --> 00:08:43.600 Let's say that we want to know whether this is a fair coin or not. 比方說要知道這枚硬幣是否公正 144 00:08:43.600 --> 00:08:48.262 So in this case the hypothesis that theta is 0.5. 要測試這個假設就要看theta為0.5的情況 145 00:08:48.262 --> 00:08:53.006 We can compare the prior distribution with the posterior distribution to see how 比較事前與事後分配的差異可以了解收集的資料改變信念的程度 146 00:08:53.006 --> 00:08:56.156 much our belief has changed by collecting some data. 比較事前與事後分配的差異可以了解收集的資料改變信念的程度 147 00:08:56.156 --> 00:09:01.830 So the prior at 0.5 was much lower than the posterior is at 0.5. 這張圖明顯表示在0.5的事前分配似然性低於事後分配 148 00:09:01.830 --> 00:09:06.149 And the ratio of these two points is 3.7. 得到的似然性比值是3.7 149 00:09:09.470 --> 00:09:13.190 This is a slightly different version where we only have a different prior. 這張圖的事前分配是beta(4 150 00:09:13.190 --> 00:09:17.910 We've observed exactly the same data but the prior now is slightly more informed. 雖然資料也是相同的，但是這個事前分配包含有關這枚硬幣的信念 151 00:09:17.910 --> 00:09:20.400 We already expected that this coin would be fair. 這個情況已事先預期這是枚公正硬幣 152 00:09:20.400 --> 00:09:23.760 We collect the same amount of data and we again see that for 以同樣的資料計算兩個分配的似然性 153 00:09:23.760 --> 00:09:28.160 the theta of 0.5 the hypothesis that this is a fair coin. 並以theta是0.5的似然性比檢測假設 154 00:09:28.160 --> 00:09:32.860 Our belief in the hypothesis that the coin is fair has increased. 事後分配表示對於硬幣是公正的信念增加了 155 00:09:32.860 --> 00:09:35.070 But because we already had quite a strong prior 不過因為事前先有特定的預期 156 00:09:35.070 --> 00:09:38.699 the data only increases our belief with 1.9. 增加的比值只有1.9 157 00:09:41.430 --> 00:09:43.786 So after looking at the data 儘管資料呈現的似然性函數是相同的 158 00:09:43.786 --> 00:09:49.257 the hypothesis that the theta is 0.5 has become either 1.91 or theta為0.5的似然性比是1.91還是3.70 159 00:09:49.257 --> 00:09:54.460 3.70 times more likely depending on the prior that we had. 完全取決於接受那種事前信念 160 00:09:55.620 --> 00:10:01.148 So we can see how base vectors tell you something about the increased evidence or 由此可以了解貝氏因子如何告訴你增加的證據力有多少 161 00:10:01.148 --> 00:10:06.100 the belief that you have about the specific hypothesis compared to your 或者根據事前與事後分配，推估證據比較支持那個假設 162 00:10:06.100 --> 00:10:07.932 prior and the posterior. 或者根據事前與事後分配，推估證據比較支持那個假設 163 00:10:07.932 --> 00:10:12.039 Now we can also look at Bayesian statistics not from a hypothesis testing 除了由假設檢定的角度了解貝氏統計如何根據事前與事後分配測試假設 164 00:10:12.039 --> 00:10:16.233 viewpoint where you test the hypothesis and the prior and the posterior. 除了由假設檢定的角度了解貝氏統計如何根據事前與事後分配測試假設 165 00:10:16.233 --> 00:10:20.990 But we just want to estimate which values do we think is most likely. 也可以估計想測量的真實數值 166 00:10:20.990 --> 00:10:24.410 So this is not Bayesian hypothesis testing 這樣的方法就叫做貝氏估計 167 00:10:25.700 --> 00:10:28.960 We also use the posterior distribution to estimate. 估計同樣需要事後分配 168 00:10:30.370 --> 00:10:33.990 So instead of testing two different models 不同於用事前與事後分配檢驗兩種模型 169 00:10:33.990 --> 00:10:38.090 we will use the posterior just to estimate plausible values. 現在要用事後分配估計最可信的數值 170 00:10:39.170 --> 00:10:43.850 Which values do you believe are most likely based on your prior belief and 要怎樣根據事前信念與觀察到的資料估計數值呢? 171 00:10:43.850 --> 00:10:44.910 the data you have observed? 要怎樣根據事前信念與觀察到的資料估計數值呢? 172 00:10:47.410 --> 00:10:53.510 In this graph we again see a prior distribution and a posterior distribution. 這張圖也有之前看過的事前與事後機率分配 173 00:10:53.510 --> 00:10:57.320 And in this case an uninformative prior 因為事前分配是無資訊的 174 00:10:57.320 --> 00:10:59.770 the posterior fall exactly on top of each other. 似然性函數與事後分配再一次完美疊合 175 00:11:01.220 --> 00:11:05.580 We can calculate the mean of the posterior distribution and we can calculate what 我們可以計算事後分配的平均值 176 00:11:05.580 --> 00:11:10.400 in Bayesian statistics is known as 95% credible interval. 以及貝氏統計稱呼的95%可信區間(credible interval) 177 00:11:10.400 --> 00:11:16.260 A credible interval contains 95% of the values that you find most plausible. 可信區間涵蓋的數值有95%的可能性是真實的 178 00:11:17.880 --> 00:11:20.530 So this is an expression of your belief. 也就是你相信為真的數值範圍 179 00:11:20.530 --> 00:11:24.620 Which values do you believe are most likely given your prior and 範圍中那些數值是你最相信的 全由事前信念與手上資料決定 180 00:11:24.620 --> 00:11:25.410 the observed data? 範圍中那些數值是你最相信的 全由事前信念與手上資料決定 181 00:11:26.800 --> 00:11:30.230 Now have used an uninformative prior 這個例子裡的事前信念是無資訊的 182 00:11:30.230 --> 00:11:34.820 you can see that the likelihood function is basically everything that matters. 所以由手上資料構成的似然性函數 183 00:11:35.860 --> 00:11:38.810 The posterior distribution is completely determined by the data. 決定事後分配的模樣 184 00:11:40.070 --> 00:11:45.860 And this is a situation where the 95% credible interval in Bayesian statistics 只有這種狀況才會看到貝氏統計的可信區間 185 00:11:45.860 --> 00:11:51.900 exactly matches with the 95% confidence interval in frequentist statistics. 與次數主義統計的信賴區間是完全一致的 186 00:11:51.900 --> 00:11:53.040 But that's not always the case. 不過除非所有研究的事前信念都是無資訊的 187 00:11:53.040 --> 00:11:56.890 This is only true when we use an uninformative prior. 這種狀況一輩子很難碰到幾次 188 00:11:56.890 --> 00:11:59.960 Let's see how this changes when we use a more informative prior. 接著來看看有資訊的事前信念會有什麼結果 189 00:12:02.030 --> 00:12:04.650 Now let's take a look at this slightly different situation. 這張圖呈現另一種狀況的分析結果 190 00:12:04.650 --> 00:12:09.530 Rather extreme example where we had a very strong prior belief that 這個狀況的事前信念是相信這枚硬幣有2/3的機會擲出正面 191 00:12:09.530 --> 00:12:15.238 the coin would come up heads about two thirds of the time 這個狀況的事前信念是相信這枚硬幣有2/3的機會擲出正面 192 00:12:15.238 --> 00:12:18.530 You can see the grey distribution peaking at this point 所以灰色的事前分配最高峰就在0.666 193 00:12:18.530 --> 00:12:22.050 this was our prior belief before we collected some data. 表達收集資料前對硬幣的信念 194 00:12:22.050 --> 00:12:24.833 But then we collect data and we actually find a completely different pattern. 但是收集的資料卻呈現與事前信念不同的模樣 195 00:12:24.833 --> 00:12:28.583 We see that the coin came up heads about 40% of the time. 投擲硬幣很多次之後 196 00:12:28.583 --> 00:12:30.748 And we collected quite a lot of data 得到的似然性函數最高點出現在0.4 197 00:12:30.748 --> 00:12:33.910 the blue likelihood function is completely different. 和事前分配完全不一樣 198 00:12:35.080 --> 00:12:38.640 We can also see that the black posterior distribution is now 以黑色曲線標示的事後分配則落在兩者之間 199 00:12:38.640 --> 00:12:40.990 a little bit in between of both of these functions. 以黑色曲線標示的事後分配則落在兩者之間 200 00:12:42.710 --> 00:12:47.495 So the data and the prior are merged into a posterior distribution. 代表事前信念與手上資料融合的結果 201 00:12:47.495 --> 00:12:52.459 And the 95% credible interval now contains values that no 95%可信區間與95%信賴區間並無重疊 202 00:12:52.459 --> 00:12:56.656 longer match within 95% confidence interval. 95%可信區間與95%信賴區間並無重疊 203 00:12:56.656 --> 00:13:01.533 A very strong prior has made it so that our believe in the most likely most 有特定預期的事前信念讓可信區間的範圍明顯縮小 204 00:13:01.533 --> 00:13:05.700 plausible values the values that we feel most confident in. 有特定預期的事前信念讓可信區間的範圍明顯縮小 205 00:13:05.700 --> 00:13:08.749 Are no longer exactly matching up with the data 也因為事前信念的影響 206 00:13:08.749 --> 00:13:12.090 they're slightly influenced by the prior that we have. 讓可信區間的範圍與資料不完全重合 207 00:13:13.890 --> 00:13:16.790 This also shows a strength of Bayesian statistics. 這個例子表現了貝氏統計的能耐 208 00:13:16.790 --> 00:13:20.160 We had a prior belief quantified by putting a function on it 透過事前信念能預期最合理的統計值 209 00:13:20.160 --> 00:13:23.090 and then we observed some data. 再考慮收集的資料 210 00:13:23.090 --> 00:13:25.260 And you can see that our belief can be changed. 接著計算事後信念改變的程度 211 00:13:25.260 --> 00:13:30.422 In this case is still sort of halfway in between. 這個例子的事後分配恰好落在中間 212 00:13:30.422 --> 00:13:35.171 But if we collect enough data we should see that the posterior belief 如果收集足夠的資料，就會看到事後分配相當接近似然性函數 213 00:13:35.171 --> 00:13:39.750 becomes very very close to the likelihood function. 如果收集足夠的資料，就會看到事後分配相當接近似然性函數 214 00:13:39.750 --> 00:13:44.440 So this shows that if we collect enough data 收集更多資料之後，對於硬幣是否公正的信念會逐漸改變 215 00:13:44.440 --> 00:13:47.253 we can become convinced about something else. 逐漸相信真實數值不同於事前信念 216 00:13:47.253 --> 00:13:49.576 This is a very useful way to do science. 這種推論方式非常適合科學 217 00:13:49.576 --> 00:13:53.349 And between science and religion. 也體現科學與宗教信仰的差異 218 00:13:53.349 --> 00:13:57.536 In religion any option to update your belief. 身為教徒者不可改變對教義的信念 219 00:13:57.536 --> 00:14:01.489 But we have a very logical framework 貝氏統計卻提供一套有邏輯的架構 220 00:14:01.489 --> 00:14:06.091 where data can change our prior beliefs and lead to new posterior beliefs. 能根據資料調整事前信念，邁向合理的事後信念 221 00:14:08.610 --> 00:14:15.020 So this 95% credible interval contains the values that you find most plausible. 95%可信區間包括所有你能找到的最可能結果 222 00:14:15.020 --> 00:14:19.611 It's all about expressing and quantifying your belief in specific values. 以具體數值表現與計量最為可信的統計值 223 00:14:19.611 --> 00:14:24.004 Now in this lecture we've seen how can you use Bayesian statistics to 總結而言貝氏統計能量化你事先認定的合理數值 依理論收集資料，再根據資料更新合理數值的信念 224 00:14:24.004 --> 00:14:27.186 quantify your prior beliefs collect data 總結而言貝氏統計能量化你事先認定的合理數值 依理論收集資料，再根據資料更新合理數值的信念 225 00:14:27.186 --> 00:14:31.445 then update your beliefs based on the data that you've observed. 總結而言貝氏統計能量化你事先認定的合理數值 依理論收集資料，再根據資料更新合理數值的信念 226 00:14:31.445 --> 00:14:36.579 [MUSIC] [課程結束]