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(10-8)!} \theta^8\left(1 \end{equation} The last element to fill in is $\theta$. If we do so for Sarah, who postulates $\theta = 0.5$, we get `r round(dbinom(8, 10, 0.5), 4)`. For Paul, who postulates $\theta = 0.8$, we get `r round(dbinom(8, 10, 0.8), 4)`. If we do this for every possible outcome, and create a bar graph of each likelihood, we get the following two figures that illustrate what each model deems likely (the yellow bar indicates each models' likelihood of the example of 8 heads out of 10 flips): -```{r two-models-likelihoods-binomial, fig.cap='The likelihoods of all possible outcomes of 10 coin flips, under Sarahs model and under Pauls model. The yellow bar indicates the likelihood of the observed data (8 heads).', fig.align='center', out.width='90%', echo = FALSE} +```{r two-models-likelihoods-binomial, fig.cap="The likelihoods of all possible outcomes of 10 coin flips, under Sarah's model and under Paul's model. The yellow bar indicates the likelihood of the observed data (8 heads).", fig.align='center', out.width='90%', echo = FALSE} #| label: fig-two-models-likelihoods-binomial par(mfrow = c(1, 2), cex.main = 0.95) cols <- viridis::viridis(6) @@ -135,14 +135,14 @@ Lastly, but perhaps most importantly, we can also consider a model that tries to par(mfrow = c(1, 2), cex.main = 0.95) cols <- viridis::viridis(6) -plot(1, 1, type ="n", xlim = c(0,1), ylim = c(0,4), bty = "n", main = "Alex' Model", +plot(1, 1, type ="n", xlim = c(0,1), ylim = c(0,4), bty = "n", main = "Alex's Model", las = 1, xlab = expression(theta), ylab = "Density") # curve(tBetFun(x, 1, 1), add = TRUE, col = cols[4], lwd = 4) mySeq <- seq(0, 1, length.out = 1e3) polygon(x = c(mySeq, rev(mySeq)), y = c(rep(0, 1e3), tBetFun(mySeq, 1, 1, side = "neutral")), col = cols[4]) barplot(rep(1/11, 11), names.arg = 0:10, xlab = "Number of heads", ylab = "Likelihood", - main = "Likely Outcomes under Alex' Model", col = cols[4], ylim = c(0,0.32)) + main = "Likely Outcomes under Alex's Model", col = cols[4], ylim = c(0,0.32)) barplot(c(rep(0, 8), 1/11), add = TRUE, col = cols[6]) # barplot(c(rep(0, 8), 1/11), add = TRUE, col = cols[6]) @@ -150,13 +150,13 @@ barplot(c(rep(0, 8), 1/11), add = TRUE, col = cols[6]) ## More Model Comparisons {#sec-more-model-comparison} -We can apply the same principles from @sec-model-comparison to compare how well each of the additional models has predicted the observed data of 8 heads out of 10 coin flips. To do so, we can simply take the ratio of each of the yellow bars in the figures that depict how likely each model considers the various possible outcomes of 10 coin flips. For instance, Alex' model has a marginal likelihood of `r round((1/11), 4)` for 8 heads, whereas Betty's model has a marginal likelihood of `r round( (table(sampsU)/1e5)[9], 4)` for 8 heads. If we want to compare the predictions of Betty and Alex, we can look at the ratio of these values to obtain $\text{BF}_{AB} =$ `r round( round(1/11, 4)/ round((table(sampsU)/1e5)[9], 4), 1)`, which is equivalent to $\text{BF}_{BA} =$ `r round( round((table(sampsU)/1e5)[9], 4) / round(1/11, 4) , 1)`. This means that the data are about twice as likely under Betty's model than under Alex' model, which can be considered weak evidence in favor of Betty's model over Alex' model. +We can apply the same principles from @sec-model-comparison to compare how well each of the additional models has predicted the observed data of 8 heads out of 10 coin flips. To do so, we can simply take the ratio of each of the yellow bars in the figures that depict how likely each model considers the various possible outcomes of 10 coin flips. For instance, Alex's model has a marginal likelihood of `r round((1/11), 4)` for 8 heads, whereas Betty's model has a marginal likelihood of `r round( (table(sampsU)/1e5)[9], 4)` for 8 heads. If we want to compare the predictions of Betty and Alex, we can look at the ratio of these values to obtain $\text{BF}_{AB} =$ `r round( round(1/11, 4)/ round((table(sampsU)/1e5)[9], 4), 1)`, which is equivalent to $\text{BF}_{BA} =$ `r round( round((table(sampsU)/1e5)[9], 4) / round(1/11, 4) , 1)`. This means that the data are about twice as likely under Betty's model than under Alex's model, which can be considered weak evidence in favor of Betty's model over Alex's model. If we were to use the betting analogy again, we could say that while both Alex and Betty had bet some money on the outcome of 8 heads, Betty had bet more money on this particular outcome than Alex, and is therefore rewarded more. Because Betty has a more specific belief (namely that the coin is biased towards heads), she had more money at her disposal for betting on the considered values (i.e., values between 0.5 and 1). In contrast, Alex played it very safely: they win some money for any outcome because they spread their betting money across all values. However, because of this, their reward is lower for having correctly predicted the observed data compared to someone who made a more specific bet on the observed data. The phenomenon of more specific models being rewarded more (when predicting well) than their non-specific competitor is known as **parsimony**, and will be discussed in more depth in Chapter 4. -A last model comparison we can make is to compare Alex' model to Sarah's model. In a typical (two-sided) statistical test about a proportion, this is the most often-used comparison: Sarah's model is considered to be the null model, and Alex' model is considered the two-sided alternative model. As we saw, Alex' marginal likelihood is `r round((1/11), 4)`, while Sarah's marginal likelihood is `r round(dbinom(8, 10, 0.5), 4)`, so the Bayes factor comparing these two models, $\text{BF}_{AS}$, equals `r round(round((1/11), 4) /round(dbinom(8, 10, 0.5), 4), 2)`. This means the data are about twice as likely under Alex' model compared to Sarah's model. +A last model comparison we can make is to compare Alex's model to Sarah's model. In a typical (two-sided) statistical test about a proportion, this is the most often-used comparison: Sarah's model is considered to be the null model, and Alex's model is considered the two-sided alternative model. As we saw, Alex's marginal likelihood is `r round((1/11), 4)`, while Sarah's marginal likelihood is `r round(dbinom(8, 10, 0.5), 4)`, so the Bayes factor comparing these two models, $\text{BF}_{AS}$, equals `r round(round((1/11), 4) /round(dbinom(8, 10, 0.5), 4), 2)`. This means the data are about twice as likely under Alex's model compared to Sarah's model. -As a bonus, when we know $\text{BF}_{BA}$ and $\text{BF}_{AS}$, we automatically know $\text{BF}_{BS}$. Since we know how much more likely the data are under Betty's model than under Alex' model (about 2 times), and how much more likely the data are under Alex' model than under Sarah's model (about 2 times), we also now know that Betty's model is about $2 \times 2 = 4$ times more likely than Sarah's model! This property it known as **transitivity**. +As a bonus, when we know $\text{BF}_{BA}$ and $\text{BF}_{AS}$, we automatically know $\text{BF}_{BS}$. Since we know how much more likely the data are under Betty's model than under Alex's model (about 2 times), and how much more likely the data are under Alex's model than under Sarah's model (about 2 times), we also now know that Betty's model is about $2 \times 2 = 4$ times more likely than Sarah's model! This property it known as **transitivity**. ## Concluding Thoughts diff --git a/03-estimation.qmd b/03-estimation.qmd index a8312b2..f1d2ba1 100644 --- a/03-estimation.qmd +++ b/03-estimation.qmd @@ -26,7 +26,7 @@ tBetFun <- function(x, shape1 = 1, shape2 = 1, side = "pos") { ## Models Have Beliefs -Belief is a funny word and we tend to associate the word with things that have seemingly little to do with science or statistics. However, we have beliefs about anything in the world -- they might be quite weak, but we still have *some* starting point for reasoning about some phenomenon. For instance, I know very little about the number of penguins in the world, but I do know there are more than 0, probably more than $1{,}000$, and fewer than $1{,}000{,}000{,}000$ of them.[^03-estimation-1] I could characterize my very uninformed belief by means of a proability distribution, where the probabilty mass depicts how plausible I deem certain values. This is exactly the same as we did in the previous chapter by characterizing the different peoples' beliefs about the fairness of a coin. In this case, even if we would know absolutely nothing about the coin in question, we still have *some* information about the parameter $\theta$. For instance, we know that it will be between 0 and 1 because it is a probability. If we do not have any knowledge beyond this, we could reflect our prior belief about $\theta$ by means of a uniform distribution, just like Alex did. In fact, we will now illustrate belief updating by looking at how Alex updates their beliefs, as a result of observing 8 heads out of 10 flips. +Belief is a funny word and we tend to associate the word with things that have seemingly little to do with science or statistics. However, we have beliefs about anything in the world -- they might be quite weak, but we still have *some* starting point for reasoning about some phenomenon. For instance, I know very little about the number of penguins in the world, but I do know there are more than 0, probably more than $1{,}000$, and fewer than $1{,}000{,}000{,}000$ of them.[^03-estimation-1] I could characterize my very uninformed belief by means of a probability distribution, where the probabilty mass depicts how plausible I deem certain values. This is exactly the same as we did in the previous chapter by characterizing the different peoples' beliefs about the fairness of a coin. In this case, even if we would know absolutely nothing about the coin in question, we still have *some* information about the parameter $\theta$. For instance, we know that it will be between 0 and 1 because it is a probability. If we do not have any knowledge beyond this, we could reflect our prior belief about $\theta$ by means of a uniform distribution, just like Alex did. In fact, we will now illustrate belief updating by looking at how Alex updates their beliefs, as a result of observing 8 heads out of 10 flips. [^03-estimation-1]: How would you characterize your beliefs about the number of penguins in the world? What would your belief distribution look like? @@ -41,7 +41,7 @@ As you might recall from @sec-open-minded-model, Alex was keeping an open mind a par(mfrow = c(1, 1), cex.main = 1) cols <- viridis::viridis(6) -plot(1, 1, type ="n", xlim = c(0,1), ylim = c(0,4), bty = "n", main = "Alex' Model", +plot(1, 1, type ="n", xlim = c(0,1), ylim = c(0,4), bty = "n", main = "Alex's Model", las = 1, xlab = expression(theta), ylab = "Density") mySeq <- seq(0, 1, length.out = 1e3) polygon(x = c(mySeq, rev(mySeq)), y = c(rep(0, 1e3), tBetFun(mySeq, 1, 1, side = "neutral")), col = cols[4]) @@ -73,7 +73,7 @@ It is important to note that the likelihood is **not** a probability distributio ### The Marginal Likelihood {#sec-marginal-likelihood} -Now we can take a look at the second part of the updating factor: $p( \text{data})$. This part is known as the **marginal likelihood**. In contrast to the first part, the marginal likelihood is a single number. Namely, it is the average of all the likelihoods, where the likelihood of each value is weighted by the prior belief placed on that value by the model. This is the same procedure as in the previous chapter (see @sec-more-models). In fact, for all of the models, the marginal likelihood was indicated by the yellow bar: it is the likelihood of the observed data, weighted by each model's specific beliefs. In the case of Alex' model, the prior belief is equal across all values, so the marginal likelihood is "simply" the average likelihood.[^03-estimation-3] In this case, the marginal likelihood is equal to `r round(1/11, 4)` - precisely the height of the yellow bar in @fig-uninformed-model-binomial-prediction. +Now we can take a look at the second part of the updating factor: $p( \text{data})$. This part is known as the **marginal likelihood**. In contrast to the first part, the marginal likelihood is a single number. Namely, it is the average of all the likelihoods, where the likelihood of each value is weighted by the prior belief placed on that value by the model. This is the same procedure as in the previous chapter (see @sec-more-models). In fact, for all of the models, the marginal likelihood was indicated by the yellow bar: it is the likelihood of the observed data, weighted by each model's specific beliefs. In the case of Alex's model, the prior belief is equal across all values, so the marginal likelihood is "simply" the average likelihood.[^03-estimation-3] In this case, the marginal likelihood is equal to `r round(1/11, 4)` - precisely the height of the yellow bar in @fig-uninformed-model-binomial-prediction. [^03-estimation-3]: Technical sidenote: *integrating* over the likelihood will give the marginal likelihood for Alex. @@ -130,7 +130,7 @@ abline(h = margLik, lwd = 4, col = "purple") ## Updated Beliefs -We started this section with the prior beliefs of Alex' model, which we updated by looking at which values predicted the data well, in order to form posterior beliefs. Just as with prior beliefs, our posterior beliefs are in the form of a posterior distribution. The figure below shows both distributions, including the vertical lines from @fig-likelihood-binomial-8-heads-marginal-redgreen to illustrate the boost/penalty in plausibility. +We started this section with the prior beliefs of Alex's model, which we updated by looking at which values predicted the data well, in order to form posterior beliefs. Just as with prior beliefs, our posterior beliefs are in the form of a posterior distribution. The figure below shows both distributions, including the vertical lines from @fig-likelihood-binomial-8-heads-marginal-redgreen to illustrate the boost/penalty in plausibility. ```{r prior-posterior-binomial, fig.cap='Prior and posterior distribution. ', fig.align='center', out.width='90%', echo = FALSE} #| label: fig-prior-posterior-binomial @@ -138,7 +138,7 @@ prior <- dbeta(ss, 1, 1) posterior <- prior * dd / margLik par(cex = 1.4, cex.lab = 1.4, mar = c(5.1, 4.5, 4.1, 2.1)) plot(ss, posterior, type = "l", bty = "n", las = 1, ylab = "Density", xlab = bquote(theta), - main = bquote("Alex' Prior and Posterior Distribution of" ~theta), + main = bquote("Alex's Prior and Posterior Distribution of" ~theta), lwd = 3, col = "darkorange") ll <- unique(bords)[1] uu <- unique(bords)[length(unique(bords))] @@ -148,13 +148,13 @@ curve(dbeta(x, 1, 1), from = 0, to = 1, lwd = 3, col ="blue", add = TRUE) legend("topleft", legend = c("Prior", "Posterior"), col = c("blue", "darkorange"), lty = 1, lwd = 4, bty = "n") ``` -We have now updated Alex' prior knowledge to posterior knowledge! The posterior distribution enables us to make probabilistic statements about values of $\theta$ because it is a probability distribution.[^03-estimation-4] For instance, the median of this posterior distribution is $0.764$, which means that under Alex' model, there is a 50% probability that $\theta$ is greater than $0.764$. Or, we can use the posterior distribution to make an *interval estimation*. In the Bayesian framework, this is known as the **credible interval**. This interval entails taking the middle $x$% of the posterior distribution. For instance, we can take a 95% credible interval, which ranges from $0.482$ to $0.940$ in this case. This means that under Alex' model, there is a 95% probability that the true value of $\theta$ is between $0.482$ and $0.940$. Such a straightfoward interpretation is one of the strengths of the Bayesian framework, compared to the frequentist framework. +We have now updated Alex's prior knowledge to posterior knowledge! The posterior distribution enables us to make probabilistic statements about values of $\theta$ because it is a probability distribution.[^03-estimation-4] For instance, the median of this posterior distribution is $0.764$, which means that under Alex's model, there is a 50% probability that $\theta$ is greater than $0.764$. Or, we can use the posterior distribution to make an *interval estimation*. In the Bayesian framework, this is known as the **credible interval**. This interval entails taking the middle $x$% of the posterior distribution. For instance, we can take a 95% credible interval, which ranges from $0.482$ to $0.940$ in this case. This means that under Alex's model, there is a 95% probability that the true value of $\theta$ is between $0.482$ and $0.940$. Such a straightfoward interpretation is one of the strengths of the Bayesian framework, compared to the frequentist framework. [^03-estimation-4]: Note the contrast to the sampling distribution, which is a distribution of the sample statistic $\hat{p}$, rather than the parameter. ## More Models -Up until this point, we have only looked at Alex' model: it had a specific starting point (i.e., prior belief), which was updated with the observed data to form an ending point. However, in the previous chapter we saw that there are all sorts of different models. Each of those models have different prior beliefs (as reflected by their prior distributions). Even though the Bayesian knowledge updating procedure is exactly the same for all models, having a different starting point means that they also have a different ending point. Specifically, for each model, their prior distribution is different, and their marginal likelihood is different (as indicated by the yellow bars in the Figures in the previous chapter). +Up until this point, we have only looked at Alex's model: it had a specific starting point (i.e., prior belief), which was updated with the observed data to form an ending point. However, in the previous chapter we saw that there are all sorts of different models. Each of those models have different prior beliefs (as reflected by their prior distributions). Even though the Bayesian knowledge updating procedure is exactly the same for all models, having a different starting point means that they also have a different ending point. Specifically, for each model, their prior distribution is different, and their marginal likelihood is different (as indicated by the yellow bars in the Figures in the previous chapter). The models shown so far differ in their "learning ability". Generally, the uninformed model is a great learner: because it has a relatively weak starting point, it lets the data speak for itself. This is reflected by the posterior distribution being highly similar to the likelihood.[^03-estimation-5] In order to illustrate the differences in the models, and how they learn, we can go over how Sarah's model and David's model learned from the data. @@ -188,7 +188,7 @@ Let's define the marginal likelihood once more: Sarah has prior beliefs about $\ ### David's Learning Process -Somewhere in the middle of Alex' ambivalence and Sarah's fanaticism, there is David's prior belief that only values between 0.5 and 1 are possible, and that values closer to 1 are more plausible than values closer to 0.5. As you can see in @fig-david-model-binomial below, this means that values between 0 and 0.5 are assigned 0 prior density - a process known as **truncation**. Truncation makes a model **one-sided**, since only values to one side of 0.5 have been assigned prior density, while the other side is set to 0. From Sarah's learning process we saw that this means that David's *posterior* beliefs will also be 0 for values below 0.5. So what does David's learning process look like? +Somewhere in the middle of Alex's ambivalence and Sarah's fanaticism, there is David's prior belief that only values between 0.5 and 1 are possible, and that values closer to 1 are more plausible than values closer to 0.5. As you can see in @fig-david-model-binomial below, this means that values between 0 and 0.5 are assigned 0 prior density - a process known as **truncation**. Truncation makes a model **one-sided**, since only values to one side of 0.5 have been assigned prior density, while the other side is set to 0. From Sarah's learning process we saw that this means that David's *posterior* beliefs will also be 0 for values below 0.5. So what does David's learning process look like? ```{r david-model-binomial, fig.cap="David's model for a coin toss. David believes that only values between 0.5 and 1 are possible, and that values closer to 1 are more plausible, a priori. When certain ranges of values have their density set to 0, we refer to the distribution as a *truncated distribution*.", fig.align='center', out.width='90%', echo = FALSE} #| label: fig-david-model-binomial @@ -232,7 +232,7 @@ plot(ss, dd, type = "l", bty = "n", las = 1, ylab = bquote("P(Data |" ~ theta * abline(h = davMarLik, lwd = 3, col = "purple") ``` -When we apply the Bayesian knowledge updating, we arrive at David's posterior knowledge. As before, the values that predicted better than average will have a higher posterior density than prior density, while the reverse holds for values of $\theta$ that predicted the data worse than average. Note that David excluded some values of $\theta$ a priori (similar to Sarah), namely those values between 0 and $0.5$. Since the prior density for those values is 0, the posterior density will also be 0, regardless of the observed data. For David's posterior distribution, the median equals $0.801$, and the 95% credible interval ranges from $0.568$ to $0.95$. Although this is fairly similar to Alex' posterior statistics (median $=0.764$, 95% CI = $[0.482, 0.940]$), they are not identical because both started with different prior beliefs and so will have different posterior beliefs. The interpretation of the median and CI will be the same, but now conditional on David's model: under David's model, there is a 95% probability that $\theta$ is between 0.568 and 0.95. +When we apply the Bayesian knowledge updating, we arrive at David's posterior knowledge. As before, the values that predicted better than average will have a higher posterior density than prior density, while the reverse holds for values of $\theta$ that predicted the data worse than average. Note that David excluded some values of $\theta$ a priori (similar to Sarah), namely those values between 0 and $0.5$. Since the prior density for those values is 0, the posterior density will also be 0, regardless of the observed data. For David's posterior distribution, the median equals $0.801$, and the 95% credible interval ranges from $0.568$ to $0.95$. Although this is fairly similar to Alex's posterior statistics (median $=0.764$, 95% CI = $[0.482, 0.940]$), they are not identical because both started with different prior beliefs and so will have different posterior beliefs. The interpretation of the median and CI will be the same, but now conditional on David's model: under David's model, there is a 95% probability that $\theta$ is between 0.568 and 0.95. ```{r prior-posterior-binomial-david, fig.cap='Prior and posterior distribution for David. Values that predicted the data better than average have received a boost in plausibility (i.e., posterior density > prior density), whie the reverse holds for values that predicted the data poorly.', fig.align='center', out.width='90%', echo = FALSE} #| label: fig-prior-posterior-binomial-david @@ -280,7 +280,7 @@ Basically, these are the dynamics: In the context of a prior distribution for a proportion, the $a$ and $b$ can be interpreted as previously observed heads and tails. For instance, having a uniform prior distribution (like Alex) can be seen as having observed 1 heads and 1 tails already. David's prior distribution is in fact a beta distribution with $a = 3$ and $b = 1$, truncated for lower values (meaning values $< 0.5$ receive 0 probability mass). This corresponds to David having already seen 3 heads and 1 tails. In order to obtain Sarah's prior distribution, we would have a beta distribution with $a = b = \infty$, which means that Sarah believes as if she has already seen incredibly many heads and tails. -In addition to the prior distribution, the posterior distribution for $\theta$ is also a beta distribution. Specifically, it is a beta distribution where $a$ is equal to the $a$ of the prior distribution, plus the number of observed succeses/heads. The $b$ is equal to the $b$ of the prior distribution, plus the number of observed failures/tails. For instance, Alex' posterior distribution (@fig-prior-posterior-binomial) is a beta distribution with $a = 1 + 8 = 9$, and $b = 1 + 2 = 3$, while David's posterior distribution (@fig-prior-posterior-binomial-david) is a (trunctated) beta distribution with $a = 3 + 8 = 11$, and $b = 1 + 2 = 3$. Representing the knowledge updating in terms of the beta distribution also illustrates how Sarah's prior and posterior distribution are the same: her posterior $a$ is equal to $\infty + 8 = \infty$, and her posterior $b$ is equal to $\infty + 2 = \infty$. +In addition to the prior distribution, the posterior distribution for $\theta$ is also a beta distribution. Specifically, it is a beta distribution where $a$ is equal to the $a$ of the prior distribution, plus the number of observed succeses/heads. The $b$ is equal to the $b$ of the prior distribution, plus the number of observed failures/tails. For instance, Alex's posterior distribution (@fig-prior-posterior-binomial) is a beta distribution with $a = 1 + 8 = 9$, and $b = 1 + 2 = 3$, while David's posterior distribution (@fig-prior-posterior-binomial-david) is a (trunctated) beta distribution with $a = 3 + 8 = 11$, and $b = 1 + 2 = 3$. Representing the knowledge updating in terms of the beta distribution also illustrates how Sarah's prior and posterior distribution are the same: her posterior $a$ is equal to $\infty + 8 = \infty$, and her posterior $b$ is equal to $\infty + 2 = \infty$. ### Two-sided vs One-sided Estimation @@ -323,11 +323,11 @@ So far we have been talking about models. In order to conduct a hypothesis test, ### The Savage-Dickey Density Ratio -When employing such a null/alternative hypothesis setup, there exists a convenient computational trick to obtain the Bayes factor of one alternative model over the null model. When we take the updating of the alternative model, for instance Alex', we can take the prior density at the value of testing ($0.5$) and the posterior density at the value of testing. The ratio of these two values is equal to the Bayes factor comparing Alex and Sarah's models. The prior density at 0.5 equals 1 (i.e. $P(\theta = 0.5) = 1$) and the posterior density at 0.5 equals approximately 0.5 (i.e., ($P(\theta = 0.5 \mid \text{data}) \approx 0.5$)), so their ratio is approximately $1 / 0.5 = 2$, in favor of the alternative hypothesis (i.e., $\text{BF}_{10} \approx 2$). The Savage-Dickey density ratio also implies the following: when the prior density is higher than the posterior density at the value of testing, we will find evidence in favor of the alternative hypothesis (the magnitude of evidence depends on how much lower the prior density is). This makes sense: the value of testing is the only value that is considered by the null hypothesis, so if that value has suffered a *decline* in plausibility as a result of the data, then that means the null hypothesis as a whole did not do well. @fig-prior-posterior-binomial-sd-ratio) below shows Alex' prior and posterior distribution, with the grey dots indicating the density ratio. +When employing such a null/alternative hypothesis setup, there exists a convenient computational trick to obtain the Bayes factor of one alternative model over the null model. When we take the updating of the alternative model, for instance Alex's, we can take the prior density at the value of testing ($0.5$) and the posterior density at the value of testing. The ratio of these two values is equal to the Bayes factor comparing Alex and Sarah's models. The prior density at 0.5 equals 1 (i.e. $P(\theta = 0.5) = 1$) and the posterior density at 0.5 equals approximately 0.5 (i.e., ($P(\theta = 0.5 \mid \text{data}) \approx 0.5$)), so their ratio is approximately $1 / 0.5 = 2$, in favor of the alternative hypothesis (i.e., $\text{BF}_{10} \approx 2$). The Savage-Dickey density ratio also implies the following: when the prior density is higher than the posterior density at the value of testing, we will find evidence in favor of the alternative hypothesis (the magnitude of evidence depends on how much lower the prior density is). This makes sense: the value of testing is the only value that is considered by the null hypothesis, so if that value has suffered a *decline* in plausibility as a result of the data, then that means the null hypothesis as a whole did not do well. @fig-prior-posterior-binomial-sd-ratio) below shows Alex's prior and posterior distribution, with the grey dots indicating the density ratio. -For a different point of testing, for instance $0.8$, we can take the Savage-Dickey density ratio at that point. With a test value of 0.8, the null model becomes equal to Paul's model (his prior distribution was concentrated on the value 0.8), so we are comparing Alex' model to Paul's model. In @fig-prior-posterior-binomial-sd-ratio, the density values are indicated by the purple dots. Here, the null model (i.e., Paul's model) did very well, since $0.8$ fits the observed data very well, as indicated by the posterior density being much higher than the prior density at that point ($\text{BF}_{01} \approx 3$). +For a different point of testing, for instance $0.8$, we can take the Savage-Dickey density ratio at that point. With a test value of 0.8, the null model becomes equal to Paul's model (his prior distribution was concentrated on the value 0.8), so we are comparing Alex's model to Paul's model. In @fig-prior-posterior-binomial-sd-ratio, the density values are indicated by the purple dots. Here, the null model (i.e., Paul's model) did very well, since $0.8$ fits the observed data very well, as indicated by the posterior density being much higher than the prior density at that point ($\text{BF}_{01} \approx 3$). -```{r prior-posterior-binomial-sd-ratio, fig.cap="Prior and posterior distribution for Alex, with the grey/purple dots indicating the prior/posterior density values for two test values: 0.5 and 0.8. The ratio of the grey values is equal to the Bayes factor comparing Alex' and Sarah's models, while the ratio of the purple values is equal to the Bayes factor comparing Alex' and Paul's models. This ratio is known as the Savage-Dickey density value.", fig.align='center', out.width='90%', echo = FALSE} +```{r prior-posterior-binomial-sd-ratio, fig.cap="Prior and posterior distribution for Alex, with the grey/purple dots indicating the prior/posterior density values for two test values: 0.5 and 0.8. The ratio of the grey values is equal to the Bayes factor comparing Alex's and Sarah's models, while the ratio of the purple values is equal to the Bayes factor comparing Alex's and Paul's models. This ratio is known as the Savage-Dickey density value.", fig.align='center', out.width='90%', echo = FALSE} #| label: fig-prior-posterior-binomial-sd-ratio margLik <- 1/11 ss <- seq(0, 1, length.out = 1e4) @@ -336,7 +336,7 @@ prior <- dbeta(ss, 1, 1) posterior <- prior * dd / margLik par(cex = 1.4, cex.lab = 1.4, mar = c(5.1, 4.5, 4.1, 2.1)) plot(ss, posterior, type = "l", bty = "n", las = 1, ylab = "Density", xlab = bquote(theta), - main = bquote("Alex' Prior and Posterior Distribution of" ~theta), + main = bquote("Alex's Prior and Posterior Distribution of" ~theta), lwd = 3, col = "darkorange") prDens <- 1 postDens <- mean(posterior[round(ss, 3) == 0.5]) diff --git a/docs/02-models.html b/docs/02-models.html index 379a941..46541fe 100644 --- a/docs/02-models.html +++ b/docs/02-models.html @@ -214,7 +214,7 @@

-
Figure 2.2: The likelihoods of all possible outcomes of 10 coin flips, under Sarahs model and under Pauls model. The yellow bar indicates the likelihood of the observed data (8 heads).
+
Figure 2.2: The likelihoods of all possible outcomes of 10 coin flips, under Sarah’s model and under Paul’s model. The yellow bar indicates the likelihood of the observed data (8 heads).
@@ -279,10 +279,10 @@

2.4 More Model Comparisons

-

We can apply the same principles from Section 2.2 to compare how well each of the additional models has predicted the observed data of 8 heads out of 10 coin flips. To do so, we can simply take the ratio of each of the yellow bars in the figures that depict how likely each model considers the various possible outcomes of 10 coin flips. For instance, Alex’ model has a marginal likelihood of 0.0909 for 8 heads, whereas Betty’s model has a marginal likelihood of 0.1758 for 8 heads. If we want to compare the predictions of Betty and Alex, we can look at the ratio of these values to obtain \(\text{BF}_{AB} =\) 0.5, which is equivalent to \(\text{BF}_{BA} =\) 1.9. This means that the data are about twice as likely under Betty’s model than under Alex’ model, which can be considered weak evidence in favor of Betty’s model over Alex’ model.

+

We can apply the same principles from Section 2.2 to compare how well each of the additional models has predicted the observed data of 8 heads out of 10 coin flips. To do so, we can simply take the ratio of each of the yellow bars in the figures that depict how likely each model considers the various possible outcomes of 10 coin flips. For instance, Alex’s model has a marginal likelihood of 0.0909 for 8 heads, whereas Betty’s model has a marginal likelihood of 0.1773 for 8 heads. If we want to compare the predictions of Betty and Alex, we can look at the ratio of these values to obtain \(\text{BF}_{AB} =\) 0.5, which is equivalent to \(\text{BF}_{BA} =\) 2. This means that the data are about twice as likely under Betty’s model than under Alex’s model, which can be considered weak evidence in favor of Betty’s model over Alex’s model.

If we were to use the betting analogy again, we could say that while both Alex and Betty had bet some money on the outcome of 8 heads, Betty had bet more money on this particular outcome than Alex, and is therefore rewarded more. Because Betty has a more specific belief (namely that the coin is biased towards heads), she had more money at her disposal for betting on the considered values (i.e., values between 0.5 and 1). In contrast, Alex played it very safely: they win some money for any outcome because they spread their betting money across all values. However, because of this, their reward is lower for having correctly predicted the observed data compared to someone who made a more specific bet on the observed data. The phenomenon of more specific models being rewarded more (when predicting well) than their non-specific competitor is known as parsimony, and will be discussed in more depth in Chapter 4.

-

A last model comparison we can make is to compare Alex’ model to Sarah’s model. In a typical (two-sided) statistical test about a proportion, this is the most often-used comparison: Sarah’s model is considered to be the null model, and Alex’ model is considered the two-sided alternative model. As we saw, Alex’ marginal likelihood is 0.0909, while Sarah’s marginal likelihood is 0.0439, so the Bayes factor comparing these two models, \(\text{BF}_{AS}\), equals 2.07. This means the data are about twice as likely under Alex’ model compared to Sarah’s model.

-

As a bonus, when we know \(\text{BF}_{BA}\) and \(\text{BF}_{AS}\), we automatically know \(\text{BF}_{BS}\). Since we know how much more likely the data are under Betty’s model than under Alex’ model (about 2 times), and how much more likely the data are under Alex’ model than under Sarah’s model (about 2 times), we also now know that Betty’s model is about \(2 \times 2 = 4\) times more likely than Sarah’s model! This property it known as transitivity.

+

A last model comparison we can make is to compare Alex’s model to Sarah’s model. In a typical (two-sided) statistical test about a proportion, this is the most often-used comparison: Sarah’s model is considered to be the null model, and Alex’s model is considered the two-sided alternative model. As we saw, Alex’s marginal likelihood is 0.0909, while Sarah’s marginal likelihood is 0.0439, so the Bayes factor comparing these two models, \(\text{BF}_{AS}\), equals 2.07. This means the data are about twice as likely under Alex’s model compared to Sarah’s model.

+

As a bonus, when we know \(\text{BF}_{BA}\) and \(\text{BF}_{AS}\), we automatically know \(\text{BF}_{BS}\). Since we know how much more likely the data are under Betty’s model than under Alex’s model (about 2 times), and how much more likely the data are under Alex’s model than under Sarah’s model (about 2 times), we also now know that Betty’s model is about \(2 \times 2 = 4\) times more likely than Sarah’s model! This property it known as transitivity.

2.5 Concluding Thoughts

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zG;$<^Lwq!*G1pZLVDkQ-f4q;Afq_Awq6&lLuduRQ_;vW3Ac}ghMhoC6=>6x-=OT=OnddNS5&1!vaAbaaSoS1C RRbbRHc)I$ztaD0e0sw^uw_^YR diff --git a/docs/03-estimation.html b/docs/03-estimation.html index b4e5d83..fb139bd 100644 --- a/docs/03-estimation.html +++ b/docs/03-estimation.html @@ -215,7 +215,7 @@

\(\theta\), how well it predicted the observed data, relative to all other possible values of \(\theta\). If this is still sounding rather vague, don’t worry - in this section we will demonstrate how this updating factor operates. However, before we discuss the updating factor, we go back to the start, and discuss the concept of prior knowledge/beliefs.

3.1 Models Have Beliefs

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Belief is a funny word and we tend to associate the word with things that have seemingly little to do with science or statistics. However, we have beliefs about anything in the world – they might be quite weak, but we still have some starting point for reasoning about some phenomenon. For instance, I know very little about the number of penguins in the world, but I do know there are more than 0, probably more than \(1{,}000\), and fewer than \(1{,}000{,}000{,}000\) of them.1 I could characterize my very uninformed belief by means of a proability distribution, where the probabilty mass depicts how plausible I deem certain values. This is exactly the same as we did in the previous chapter by characterizing the different peoples’ beliefs about the fairness of a coin. In this case, even if we would know absolutely nothing about the coin in question, we still have some information about the parameter \(\theta\). For instance, we know that it will be between 0 and 1 because it is a probability. If we do not have any knowledge beyond this, we could reflect our prior belief about \(\theta\) by means of a uniform distribution, just like Alex did. In fact, we will now illustrate belief updating by looking at how Alex updates their beliefs, as a result of observing 8 heads out of 10 flips.

+

Belief is a funny word and we tend to associate the word with things that have seemingly little to do with science or statistics. However, we have beliefs about anything in the world – they might be quite weak, but we still have some starting point for reasoning about some phenomenon. For instance, I know very little about the number of penguins in the world, but I do know there are more than 0, probably more than \(1{,}000\), and fewer than \(1{,}000{,}000{,}000\) of them.1 I could characterize my very uninformed belief by means of a probability distribution, where the probabilty mass depicts how plausible I deem certain values. This is exactly the same as we did in the previous chapter by characterizing the different peoples’ beliefs about the fairness of a coin. In this case, even if we would know absolutely nothing about the coin in question, we still have some information about the parameter \(\theta\). For instance, we know that it will be between 0 and 1 because it is a probability. If we do not have any knowledge beyond this, we could reflect our prior belief about \(\theta\) by means of a uniform distribution, just like Alex did. In fact, we will now illustrate belief updating by looking at how Alex updates their beliefs, as a result of observing 8 heads out of 10 flips.

3.2 Updating Beliefs

@@ -252,7 +252,7 @@

3.2.2 The Marginal Likelihood

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Now we can take a look at the second part of the updating factor: \(p( \text{data})\). This part is known as the marginal likelihood. In contrast to the first part, the marginal likelihood is a single number. Namely, it is the average of all the likelihoods, where the likelihood of each value is weighted by the prior belief placed on that value by the model. This is the same procedure as in the previous chapter (see Section 2.3). In fact, for all of the models, the marginal likelihood was indicated by the yellow bar: it is the likelihood of the observed data, weighted by each model’s specific beliefs. In the case of Alex’ model, the prior belief is equal across all values, so the marginal likelihood is “simply” the average likelihood.3 In this case, the marginal likelihood is equal to 0.0909 - precisely the height of the yellow bar in Figure 2.6.

+

Now we can take a look at the second part of the updating factor: \(p( \text{data})\). This part is known as the marginal likelihood. In contrast to the first part, the marginal likelihood is a single number. Namely, it is the average of all the likelihoods, where the likelihood of each value is weighted by the prior belief placed on that value by the model. This is the same procedure as in the previous chapter (see Section 2.3). In fact, for all of the models, the marginal likelihood was indicated by the yellow bar: it is the likelihood of the observed data, weighted by each model’s specific beliefs. In the case of Alex’s model, the prior belief is equal across all values, so the marginal likelihood is “simply” the average likelihood.3 In this case, the marginal likelihood is equal to 0.0909 - precisely the height of the yellow bar in Figure 2.6.

We can use this average likelihood to see which possible values of \(\theta\) predicted the data better than average, and which values predicted the data worse than average: since the likelihood reflects the predictive quality of each value, the marginal likelihood reflects the average quality across all values. In the following figure, you can again see the likelihood function from Figure 3.2, but now with the marginal likelihood added:

@@ -285,7 +285,7 @@

3.3 Updated Beliefs

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We started this section with the prior beliefs of Alex’ model, which we updated by looking at which values predicted the data well, in order to form posterior beliefs. Just as with prior beliefs, our posterior beliefs are in the form of a posterior distribution. The figure below shows both distributions, including the vertical lines from Figure 3.4 to illustrate the boost/penalty in plausibility.

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We started this section with the prior beliefs of Alex’s model, which we updated by looking at which values predicted the data well, in order to form posterior beliefs. Just as with prior beliefs, our posterior beliefs are in the form of a posterior distribution. The figure below shows both distributions, including the vertical lines from Figure 3.4 to illustrate the boost/penalty in plausibility.

@@ -296,11 +296,11 @@

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We have now updated Alex’ prior knowledge to posterior knowledge! The posterior distribution enables us to make probabilistic statements about values of \(\theta\) because it is a probability distribution.4 For instance, the median of this posterior distribution is \(0.764\), which means that under Alex’ model, there is a 50% probability that \(\theta\) is greater than \(0.764\). Or, we can use the posterior distribution to make an interval estimation. In the Bayesian framework, this is known as the credible interval. This interval entails taking the middle \(x\)% of the posterior distribution. For instance, we can take a 95% credible interval, which ranges from \(0.482\) to \(0.940\) in this case. This means that under Alex’ model, there is a 95% probability that the true value of \(\theta\) is between \(0.482\) and \(0.940\). Such a straightfoward interpretation is one of the strengths of the Bayesian framework, compared to the frequentist framework.

+

We have now updated Alex’s prior knowledge to posterior knowledge! The posterior distribution enables us to make probabilistic statements about values of \(\theta\) because it is a probability distribution.4 For instance, the median of this posterior distribution is \(0.764\), which means that under Alex’s model, there is a 50% probability that \(\theta\) is greater than \(0.764\). Or, we can use the posterior distribution to make an interval estimation. In the Bayesian framework, this is known as the credible interval. This interval entails taking the middle \(x\)% of the posterior distribution. For instance, we can take a 95% credible interval, which ranges from \(0.482\) to \(0.940\) in this case. This means that under Alex’s model, there is a 95% probability that the true value of \(\theta\) is between \(0.482\) and \(0.940\). Such a straightfoward interpretation is one of the strengths of the Bayesian framework, compared to the frequentist framework.

3.4 More Models

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Up until this point, we have only looked at Alex’ model: it had a specific starting point (i.e., prior belief), which was updated with the observed data to form an ending point. However, in the previous chapter we saw that there are all sorts of different models. Each of those models have different prior beliefs (as reflected by their prior distributions). Even though the Bayesian knowledge updating procedure is exactly the same for all models, having a different starting point means that they also have a different ending point. Specifically, for each model, their prior distribution is different, and their marginal likelihood is different (as indicated by the yellow bars in the Figures in the previous chapter).

+

Up until this point, we have only looked at Alex’s model: it had a specific starting point (i.e., prior belief), which was updated with the observed data to form an ending point. However, in the previous chapter we saw that there are all sorts of different models. Each of those models have different prior beliefs (as reflected by their prior distributions). Even though the Bayesian knowledge updating procedure is exactly the same for all models, having a different starting point means that they also have a different ending point. Specifically, for each model, their prior distribution is different, and their marginal likelihood is different (as indicated by the yellow bars in the Figures in the previous chapter).

The models shown so far differ in their “learning ability”. Generally, the uninformed model is a great learner: because it has a relatively weak starting point, it lets the data speak for itself. This is reflected by the posterior distribution being highly similar to the likelihood.5 In order to illustrate the differences in the models, and how they learn, we can go over how Sarah’s model and David’s model learned from the data.

3.4.1 Sarah’s Learning Process

@@ -326,7 +326,7 @@

3.4.2 David’s Learning Process

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Somewhere in the middle of Alex’ ambivalence and Sarah’s fanaticism, there is David’s prior belief that only values between 0.5 and 1 are possible, and that values closer to 1 are more plausible than values closer to 0.5. As you can see in Figure 3.7 below, this means that values between 0 and 0.5 are assigned 0 prior density - a process known as truncation. Truncation makes a model one-sided, since only values to one side of 0.5 have been assigned prior density, while the other side is set to 0. From Sarah’s learning process we saw that this means that David’s posterior beliefs will also be 0 for values below 0.5. So what does David’s learning process look like?

+

Somewhere in the middle of Alex’s ambivalence and Sarah’s fanaticism, there is David’s prior belief that only values between 0.5 and 1 are possible, and that values closer to 1 are more plausible than values closer to 0.5. As you can see in Figure 3.7 below, this means that values between 0 and 0.5 are assigned 0 prior density - a process known as truncation. Truncation makes a model one-sided, since only values to one side of 0.5 have been assigned prior density, while the other side is set to 0. From Sarah’s learning process we saw that this means that David’s posterior beliefs will also be 0 for values below 0.5. So what does David’s learning process look like?

@@ -349,7 +349,7 @@

\(\theta\) that predicted the data worse than average. Note that David excluded some values of \(\theta\) a priori (similar to Sarah), namely those values between 0 and \(0.5\). Since the prior density for those values is 0, the posterior density will also be 0, regardless of the observed data. For David’s posterior distribution, the median equals \(0.801\), and the 95% credible interval ranges from \(0.568\) to \(0.95\). Although this is fairly similar to Alex’ posterior statistics (median \(=0.764\), 95% CI = \([0.482, 0.940]\)), they are not identical because both started with different prior beliefs and so will have different posterior beliefs. The interpretation of the median and CI will be the same, but now conditional on David’s model: under David’s model, there is a 95% probability that \(\theta\) is between 0.568 and 0.95.

+

When we apply the Bayesian knowledge updating, we arrive at David’s posterior knowledge. As before, the values that predicted better than average will have a higher posterior density than prior density, while the reverse holds for values of \(\theta\) that predicted the data worse than average. Note that David excluded some values of \(\theta\) a priori (similar to Sarah), namely those values between 0 and \(0.5\). Since the prior density for those values is 0, the posterior density will also be 0, regardless of the observed data. For David’s posterior distribution, the median equals \(0.801\), and the 95% credible interval ranges from \(0.568\) to \(0.95\). Although this is fairly similar to Alex’s posterior statistics (median \(=0.764\), 95% CI = \([0.482, 0.940]\)), they are not identical because both started with different prior beliefs and so will have different posterior beliefs. The interpretation of the median and CI will be the same, but now conditional on David’s model: under David’s model, there is a 95% probability that \(\theta\) is between 0.568 and 0.95.

@@ -384,7 +384,7 @@

3.5.2 Beta Interpretation

In the context of a prior distribution for a proportion, the \(a\) and \(b\) can be interpreted as previously observed heads and tails. For instance, having a uniform prior distribution (like Alex) can be seen as having observed 1 heads and 1 tails already. David’s prior distribution is in fact a beta distribution with \(a = 3\) and \(b = 1\), truncated for lower values (meaning values \(< 0.5\) receive 0 probability mass). This corresponds to David having already seen 3 heads and 1 tails. In order to obtain Sarah’s prior distribution, we would have a beta distribution with \(a = b = \infty\), which means that Sarah believes as if she has already seen incredibly many heads and tails.

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In addition to the prior distribution, the posterior distribution for \(\theta\) is also a beta distribution. Specifically, it is a beta distribution where \(a\) is equal to the \(a\) of the prior distribution, plus the number of observed succeses/heads. The \(b\) is equal to the \(b\) of the prior distribution, plus the number of observed failures/tails. For instance, Alex’ posterior distribution (Figure 3.5) is a beta distribution with \(a = 1 + 8 = 9\), and \(b = 1 + 2 = 3\), while David’s posterior distribution (Figure 3.9) is a (trunctated) beta distribution with \(a = 3 + 8 = 11\), and \(b = 1 + 2 = 3\). Representing the knowledge updating in terms of the beta distribution also illustrates how Sarah’s prior and posterior distribution are the same: her posterior \(a\) is equal to \(\infty + 8 = \infty\), and her posterior \(b\) is equal to \(\infty + 2 = \infty\).

+

In addition to the prior distribution, the posterior distribution for \(\theta\) is also a beta distribution. Specifically, it is a beta distribution where \(a\) is equal to the \(a\) of the prior distribution, plus the number of observed succeses/heads. The \(b\) is equal to the \(b\) of the prior distribution, plus the number of observed failures/tails. For instance, Alex’s posterior distribution (Figure 3.5) is a beta distribution with \(a = 1 + 8 = 9\), and \(b = 1 + 2 = 3\), while David’s posterior distribution (Figure 3.9) is a (trunctated) beta distribution with \(a = 3 + 8 = 11\), and \(b = 1 + 2 = 3\). Representing the knowledge updating in terms of the beta distribution also illustrates how Sarah’s prior and posterior distribution are the same: her posterior \(a\) is equal to \(\infty + 8 = \infty\), and her posterior \(b\) is equal to \(\infty + 2 = \infty\).

3.5.3 Two-sided vs One-sided Estimation

@@ -411,14 +411,14 @@

Section 2.2 and Section 2.4. Typically, the model with extreme conviction about a “null value”7, which in this case would be Sarah’s model. Then, what we call the alternative hypothesis can be any other model that corresponds to the theory one wants to test: all models we have seen so far make some specific statement about plausible values of \(\theta\). The Bayes factor simply tells us which model predicted the data the best: it is a relative metric that compares two models at a time.

3.6.1 The Savage-Dickey Density Ratio

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When employing such a null/alternative hypothesis setup, there exists a convenient computational trick to obtain the Bayes factor of one alternative model over the null model. When we take the updating of the alternative model, for instance Alex’, we can take the prior density at the value of testing (\(0.5\)) and the posterior density at the value of testing. The ratio of these two values is equal to the Bayes factor comparing Alex and Sarah’s models. The prior density at 0.5 equals 1 (i.e. \(P(\theta = 0.5) = 1\)) and the posterior density at 0.5 equals approximately 0.5 (i.e., (\(P(\theta = 0.5 \mid \text{data}) \approx 0.5\))), so their ratio is approximately \(1 / 0.5 = 2\), in favor of the alternative hypothesis (i.e., \(\text{BF}_{10} \approx 2\)). The Savage-Dickey density ratio also implies the following: when the prior density is higher than the posterior density at the value of testing, we will find evidence in favor of the alternative hypothesis (the magnitude of evidence depends on how much lower the prior density is). This makes sense: the value of testing is the only value that is considered by the null hypothesis, so if that value has suffered a decline in plausibility as a result of the data, then that means the null hypothesis as a whole did not do well. Figure 3.10) below shows Alex’ prior and posterior distribution, with the grey dots indicating the density ratio.

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For a different point of testing, for instance \(0.8\), we can take the Savage-Dickey density ratio at that point. With a test value of 0.8, the null model becomes equal to Paul’s model (his prior distribution was concentrated on the value 0.8), so we are comparing Alex’ model to Paul’s model. In Figure 3.10, the density values are indicated by the purple dots. Here, the null model (i.e., Paul’s model) did very well, since \(0.8\) fits the observed data very well, as indicated by the posterior density being much higher than the prior density at that point (\(\text{BF}_{01} \approx 3\)).

+

When employing such a null/alternative hypothesis setup, there exists a convenient computational trick to obtain the Bayes factor of one alternative model over the null model. When we take the updating of the alternative model, for instance Alex’s, we can take the prior density at the value of testing (\(0.5\)) and the posterior density at the value of testing. The ratio of these two values is equal to the Bayes factor comparing Alex and Sarah’s models. The prior density at 0.5 equals 1 (i.e. \(P(\theta = 0.5) = 1\)) and the posterior density at 0.5 equals approximately 0.5 (i.e., (\(P(\theta = 0.5 \mid \text{data}) \approx 0.5\))), so their ratio is approximately \(1 / 0.5 = 2\), in favor of the alternative hypothesis (i.e., \(\text{BF}_{10} \approx 2\)). The Savage-Dickey density ratio also implies the following: when the prior density is higher than the posterior density at the value of testing, we will find evidence in favor of the alternative hypothesis (the magnitude of evidence depends on how much lower the prior density is). This makes sense: the value of testing is the only value that is considered by the null hypothesis, so if that value has suffered a decline in plausibility as a result of the data, then that means the null hypothesis as a whole did not do well. Figure 3.10) below shows Alex’s prior and posterior distribution, with the grey dots indicating the density ratio.

+

For a different point of testing, for instance \(0.8\), we can take the Savage-Dickey density ratio at that point. With a test value of 0.8, the null model becomes equal to Paul’s model (his prior distribution was concentrated on the value 0.8), so we are comparing Alex’s model to Paul’s model. In Figure 3.10, the density values are indicated by the purple dots. Here, the null model (i.e., Paul’s model) did very well, since \(0.8\) fits the observed data very well, as indicated by the posterior density being much higher than the prior density at that point (\(\text{BF}_{01} \approx 3\)).

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Figure 3.10: Prior and posterior distribution for Alex, with the grey/purple dots indicating the prior/posterior density values for two test values: 0.5 and 0.8. The ratio of the grey values is equal to the Bayes factor comparing Alex’ and Sarah’s models, while the ratio of the purple values is equal to the Bayes factor comparing Alex’ and Paul’s models. This ratio is known as the Savage-Dickey density value.
+
Figure 3.10: Prior and posterior distribution for Alex, with the grey/purple dots indicating the prior/posterior density values for two test values: 0.5 and 0.8. The ratio of the grey values is equal to the Bayes factor comparing Alex’s and Sarah’s models, while the ratio of the purple values is equal to the Bayes factor comparing Alex’s and Paul’s models. This ratio is known as the Savage-Dickey density value.
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