Probstat/notes/parameter estimation - ประวัติรุ่นแก้ไข

Jittat: /* Bayes estimators */

2014-12-06T18:23:41Z

Bayes estimators

2014-12-06T18:22:37Z

Bayes estimators

2014-12-06T18:14:38Z

Examples

2014-12-06T18:11:41Z

Examples

2014-12-06T18:11:20Z

Examples

2014-12-06T18:05:28Z

Examples

2014-12-06T18:04:52Z

Examples

2014-12-06T18:04:22Z

Examples

2014-12-06T18:03:27Z

Examples

2014-12-06T18:00:11Z

Examples

@@ แถว 168: / แถว 168: @@
 <center>
 <math>
-E[p|x_1,x_2,\ldots,x_n].
+E[p|X_1=x_1,X_2=x_2,\ldots,X_n=x_n].
 </math>
 </center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 18:22, 6 ธันวาคม 2557
แถว 154:		แถว 154:

	== Bayes estimators ==		== Bayes estimators ==
		+	Let the observed data be <math>x_1,x_2,\ldots,x_n</math>. We fist compute the conditional probability
		+
		+	<center>
		+	<math>
		+	f(p\|x_1,x_2,\ldots,x_n)
		+	= \frac{f(x_1,x_2,\ldots,x_n,p)}{f(x_1,x_2,\ldots,x_n)}
		+	= \frac{f(x_1,x_2,\ldots,x_n\|p)f(p)}{f(x_1,x_2,\ldots,x_n)}.
		+	</math>
		+	</center>
		+
		+	With that distribution function, the Bayes estimator is
		+
		+	<center>
		+	<math>
		+	E[p\|x_1,x_2,\ldots,x_n].
		+	</math>
		+	</center>

@@ แถว 132: / แถว 132: @@
 </math>
 </center>
 '''EX3''': maximum likelihood estimator for the Poisson mean
 == Bayes estimators ==

@@ แถว 129: / แถว 129: @@
 <center>
 <math>\frac{d}{dp}\ln f(x_1,x_2,\ldots,x_n|p)
-= \frac{\sum_i x_i}{p} - \frac{\n - \sum_i x_i}{1-p}.
+= \frac{\sum_i x_i}{p} - \frac{n - \sum_i x_i}{1-p}.
 </math>
 </center>

@@ แถว 113: / แถว 113: @@
 = p^{x_1}(1-p)^{1-x_1}\cdot p^{x_2}(1-p)^{1-x_2} \cdots p^{x_n}(1-p)^{1-x_n}
 = p^{\left(\sum_i x_i\right)}(1-p)^{\left(\sum_i (1-x_i)\right)}.
 </math>
 </center>

@@ แถว 112: / แถว 112: @@
 <math>f(x_1,x_2,\ldots,x_n|p)
 = p^{x_1}(1-p)^{1-x_1}\cdot p^{x_2}(1-p)^{1-x_2} \cdots p^{x_n}(1-p)^{1-x_n}
-= p^{\sum_i x_i}(1-p)^{\sum_i (1-x_i)}.
+= p^{(\sum_i x_i)}(1-p)^{(\sum_i (1-x_i))}.
 </math>
 </center>

@@ แถว 95: / แถว 95: @@
 Since the probability is maximum at <math>\theta=3</math>, the maximum likelihood estimator for the number of blue balls is 3.
-'''EX2''': maximum likelihood estimator for a Bernoulli parameter
+'''EX2''': Maximum likelihood estimator for a Bernoulli parameter
 '''EX3''': maximum likelihood estimator for the Poisson mean
 == Bayes estimators ==