#### Description

The AIC calculation in Binomial GLM seems to be wrong when there are weights. The result is different from that in R.

The current implementation is:

-2.0 * predictions.map { case (y: Double, mu: Double, weight: Double) => weight * dist.Binomial(1, mu).logProbabilityOf(math.round(y).toInt) }.sum()

Suggest changing this to

-2.0 * predictions.map { case (y: Double, mu: Double, weight: Double) => val wt = math.round(weight).toInt if (wt == 0){ 0.0 } else { dist.Binomial(wt, mu).logProbabilityOf(math.round(y * weight).toInt) } }.sum()

The following is an example to illustrate the problem.

val dataset = Seq( LabeledPoint(0.0, Vectors.dense(18, 1.0)), LabeledPoint(0.5, Vectors.dense(12, 0.0)), LabeledPoint(1.0, Vectors.dense(15, 0.0)), LabeledPoint(0.0, Vectors.dense(13, 2.0)), LabeledPoint(0.0, Vectors.dense(15, 1.0)), LabeledPoint(0.5, Vectors.dense(16, 1.0)) ).toDF().withColumn("weight", col("label") + 1.0) val glr = new GeneralizedLinearRegression() .setFamily("binomial") .setWeightCol("weight") .setRegParam(0) val model = glr.fit(dataset) model.summary.aic

This calculation shows the AIC is 14.189026847171382. To verify whether this is correct, I run the same analysis in R but got AIC = 11.66092, -2 * LogLik = 5.660918.

```
da <- scan(, what=list(y = 0, x1 = 0, x2 = 0, w = 0), sep = ",")
0,18,1,1
0.5,12,0,1.5
1,15,0,2
0,13,2,1
0,15,1,1
0.5,16,1,1.5
da <- as.data.frame(da)
f <- glm(y ~ x1 + x2 , data = da, family = binomial(), weight = w)
AIC(f)
-2 * logLik(f)
```

Now, I check whether the proposed change is correct. The following calculates -2 * LogLik manually and get 5.6609177228379055, the same as that in R.

val predictions = model.transform(dataset) -2.0 * predictions.select("label", "prediction", "weight").rdd.map {case Row(y: Double, mu: Double, weight: Double) => val wt = math.round(weight).toInt if (wt == 0){ 0.0 } else { dist.Binomial(wt, mu).logProbabilityOf(math.round(y * weight).toInt) } }.sum()