Correct a few more typos

And explain the posterior DF better

docs/historical_samples.md

@@ -19,7 +19,7 @@ kernelspec:
 
 # Historical (Ancient) Samples
 
-Sometimes we wich to infer and date a genetic genealogy from
+Sometimes you may wish to infer and date a genetic genealogy from
 data which includes *historical samples*,
 whose time is older that the current generation (i.e. sample nodes with
 times > 0).
@@ -33,7 +33,7 @@ place any historical samples.
 ## The 2 step approach
 
 Currently, the best way to date such tree sequences is 
-to carry out a two step process, in  which inference and dating is first
+to perform a two step process, in  which inference and dating is first
 performed on the modern samples in order to establish a timescale, followed by
 adding the historical samples and re-inferring using the dated timescale as a
 basis for site ages in the `tsinfer` algorithm. The re-inferred tree sequence

docs/introduction.md

@@ -81,10 +81,10 @@ svg2 = tsdate.date(inferred, population_size=1000, mutation_rate=4e-6).draw_svg(
 haps = "<br>".join(ts.haplotypes())
 cen = 'style="text-align: center; padding: 0.5em 0"'
 HTML(f"""<table>
-    <caption>An example of using `tsinfer` followed by `tsdate` on some DNA sequence data.
+    <caption style="padding: 0 4em">An example of using `tsinfer` followed by `tsdate` on some DNA sequence data.
     You can see that tsdate sets a timescale and changes node times so that mutations (red crosses)
     are more evenly distributed over edges of the genealogy. This results in more realistic local trees
-    (with coaleascences clustered, as expected from theory, at recent times)</caption>
+    (with coalescences clustered, as expected from theory, at recent times)</caption>
     <t style=""><td {cen}><div style="font-size: 0.6em">{haps}</div></td><td></td><td {cen} width="300">{svg1}</td><td></td><td {cen} width="300">{svg2}</td></tr>
     <tr style="font-size: 2em"><td {cen}">DNA sequence</td><td {cen}>→</td><td {cen}>tsinfer</td><td {cen}>→</td><td {cen}>tsdate</td></tr>
     </table>"""

docs/usage.md

@@ -173,10 +173,11 @@ posteriors_df = pd.DataFrame(posteriors)
 posteriors_df.head()  # Show the dataframe
 ```
 
-Since we are using a {ref}`sec_methods_discrete_time` method, each row of this
-posterior dataframe corresponds to a set of probabilities for a given timeslice
-(as specified in the `start_time` and `end_time` columns). Each column corresponds to
-a node (the column header is the node ID in the returned tree sequence).
+Since we are using a {ref}`sec_methods_discrete_time` method, each node
+(numbered column of the dataframe) is associated with a vector of probabilities
+that sum to one: each cell gives the probability that the time of the node
+whose ID is given by the column header lies within the
+timeslice specified by the `start_time` and `end_time` columns.
 
 (sec_usage_popsize)=
 

docs/variable_popsize.md

@@ -76,8 +76,8 @@ specified time intervals). This can them be used to create a prior, via the {fun
 function (see {ref}`sec_priors`).
 
 For example, the following code defines a population that is of effective size
-1 million in the last 200 generations, only two hundred for a period of 10 generations 50,000 generations ago, then
-of size 50,000 for all generations before that, exactly matching the simulated bottleneck
+1 million in the last 50,000 generations, only two hundred for a period of 10 generations 50,000 generations ago, then
+of size 10,000 for all generations before that, exactly matching the simulated bottleneck
 
 ```{code-cell} ipython3
 popsize = tsdate.demography.PopulationSizeHistory(population_size=[1e6, 2e2, 1e4], time_breaks=[50_000, 50_010])
@@ -99,5 +99,6 @@ population size over time, by extimating the number of coalescence points in dif
 intervals, and re-estimating the dates. However, this
 approach has not been fully tested or documented.
 
-If you are interested in doing this, see https://github.com/tskit-dev/tsdate/issues/237#issuecomment-1785655708
+If you are interested in doing this, see
+[GitHub issue #237](https://github.com/tskit-dev/tsdate/issues/237#issuecomment-1785655708)
 for an example.