Parameter Update - Artificial Intelligence for RoboticsYouTube↗  IA↗(long shot)
Education  28.7K views 5 likes 3 dislikes   29 May 2012
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Suppose we multiply two Gaussians as in Bayes rule--
ベイズの定理のように2つのガウス分布を 掛け合わせたらどうなるでしょう
a prior and a measurement probability.
事前確率と観測確率です
The prior has a mean of mu and a variance of sigma-squared.
事前分布には平均のμと分散のσ²があり
The measurement has a mean of nu and a covariance of r-squared.
観測の分布には平均のνと共分散のr²があります
Then the new mean, mu prime, is the weighted sum of the old means
そして新しい平均のμ'は古い平均値の加重和で
where mu is weighted by r-squared and nu is weighted by sigma-squared
ここでμはr²、νはσ²でそれぞれ加重されます
normalized by the sum of the weight factors.
さらに加重係数の合計で正規化されます
The new variance term--I'm going to write sigma-squared prime here
新しい分散項をσ²'とします
for the new one after the update--is given by this equation over here.
更新後の新しいものです これはこの式で表されます
Let's put this into action.
計算してみましょう
We have a weighted mean over here.
加重平均があります
Clearly, the prior Guassian has a much higher uncertainty.
事前のガウス分布は 明らかにより高い不確実性を持つので
Therefore sigma-squared is larger.
σ²は大きくなります
That means that nu is weighted much, much larger than the mu.
つまりνはμよりも はるかに大きく加重されるのです
So the mean will be closer to the nu than the mu,
よって平均値はμよりもνに近づくので
which means that it'll be somewhere over here.
ここのどこかにあるということです
Interestingly enough, the variance term is unaffected by the actual means.
非常に興味深いことに分散項は 実際の平均からは影響を受けません
It just uses the previous variances and comes up with a new one that's even peakier.
ただ事前分散を使ってより先端のとがっている 新しいものとなって現れます
The result might look like this.
結果はこのようになるでしょう
This is the Kalman situation for the measurement update step
これが観測更新におけるカルマンの状況です
where this is the prior, this is the measurement probability, and this is the posterior.
これが事前分布、観測の確率、事後分布となります
Let's practice these equations with a simple quiz.
簡単な問題でこの式を解く練習をしましょう
Here are our equations again.
これが先ほどの式です
Suppose I use the following Gaussians:
次のガウス分布を使います
[μ = 10, σ^2 = 4, ν = 12, r^2 = 4] These are Gaussians with equal variance but different means that might look as follows.
分散は等しく平均が異なるガウス分布は このようなものです
Compute for me the new mean after the update and the new sigma-squared.
更新後の新しい平均とσ²を計算してください
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